Multi-agent paradoxes beyond quantum theory

This condition is theory-independent. It tells us that rational agents can reason about each other's knowledge in the usual way. This is formalized by a weaker version of epistemic modal logic, which we explain in the following (for the full derivation of the form used here see [2]).

Let us start with a simple example. The goal of modal logic is to allow us to operate with chained statements like 'Alice knows that Bob knows that Eve doesn't know the secret key k, and Alice further knows that $k=1$,' which can be expressed as

$$\begin{eqnarray*}&&{K}_{A}\ [({K}_{B}\ \neg {K}_{E}\ k)\ \wedge \ k=1],\end{eqnarray*}$$

where the operators K_i stand for 'agent i knows.' If in addition Alice trusts Bob to be a rational, reliable agent, she can deduce from the statement 'I know that Bob knows that Eve doesn't know the key' that 'I know that Eve doesn't know the key', and forget about the source of information (Bob). This is expressed as

$$\begin{eqnarray*}&&{K}_{A}({K}_{B}\ \neg {K}_{E}\ k)\Longrightarrow {K}_{A}\ \neg {K}_{E}\ \ k.\end{eqnarray*}$$

We should also allow Alice to make deductions of the type 'since Eve does not know the secret key, and one would need to know the key in order to recover the encrypted message m, I conclude that Eve cannot know the secret message,' which can be encoded as

$$\begin{eqnarray*}&&{K}_{A}[(\neg {K}_{E}\ k)\wedge ({K}_{i}\ m\Longrightarrow {K}_{i}\ k,\forall \ i)]\Longrightarrow {K}_{A}\neg {K}_{E}\ m.\end{eqnarray*}$$

Generalizing from this example, this gives us the following structure.

Definition 1 An experimental setup with multiple agents A₁, ...A_N can be described by knowledge operators K₁, ...K_N and statements ϕ ∈ Φ, such that K_i ϕ denotes 'agent A_i knows ϕ.' It should allow agents to make deductions (figure 2(a)), that is

$$\begin{eqnarray*}&&{K}_{i}[\phi \wedge (\phi \Longrightarrow \psi )]\Longrightarrow {K}_{i}\ \psi .\end{eqnarray*}$$

Furthermore, each experimental setup defines a trust relation between agents (figure 2(b)): we say that an agent A_i trusts another agent A_j (and denote it by ${A}_{j}{\rightsquigarrow}{A}_{j}$) iff for all statements ϕ, we have

$$\begin{eqnarray*}&&{K}_{i}({K}_{j}\ \phi )\Longrightarrow {K}_{i}\ \phi .\end{eqnarray*}$$

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For the purposes of following the example of section 4, this informal definition suffices. The full formal version of the axioms of modal logic used here can be found in appendix A ⁶ .

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A note on the complexity cost of reasoning. Note that in general, even the most rational physical agents may be limited by bounded processing power and memory capacity, and will not be able to chain an indefinite number of deductions within sensible time scales. That is, these axioms for reasoning are an idealization of absolutely rational agents with unbounded processing power (see [10] for an overview of this and related issues). If we would like modal logic to apply to realistic, physical agents, we might account for a cost (in time, or in memory) of each logical deduction, and require it to stay below a given threshold, much like a resource theory for complexity. However, in the examples of this paper, agents only need to make a handful of logical deductions, and these complexity concerns do not play a significant role.

This condition is to be instantiated by each physical theory, and is the way that we incorporate the physical theory into the reasoning framework used by agents in a given setting. If all agents use the same theory to model the operational experiment (like quantum mechanics, special relativity, classical statistical physics, or box world), this is included in the common knowledge shared by the agents. For example, in the case of quantum theory, we have that 'everyone knows that the probability of obtaining outcome $| x\rangle$ when measuring a state $| \psi \rangle$ is given by $| \langle x| \psi \rangle {| }^{2}$, and everyone knows that everyone knows this, and so on.'

Definition 2 We model a physical theory shared by all agents ${\{{A}_{i}\}}_{i}$ in a given setting as a set ${\mathbb{T}}$ of statements that are common knowledge shared by all agents (figure 3), i.e.

$$\begin{eqnarray*}&&\phi \in {\mathbb{T}}\ \Longleftrightarrow \ {\left({\{{K}_{i}\}}_{i}\right)}^{n}\ \phi ,\ \ \forall n\in {\mathbb{N}},\end{eqnarray*}$$

where ${\left({\{{K}_{i}\}}_{i}\right)}^{n}$ is the set of all possible sequences of n operators picked from ${\{{K}_{i}\}}_{i}$. For example, $({K}_{1}\ {K}_{5}\ {K}_{1}\ {K}_{2})\in {\left({\{{K}_{i}\}}_{i}\right)}^{4}$ and stands for 'agent A₁ knows that agent A₅ knows that agent A₁ knows that agent A₂ knows.'

Note that the set ${\mathbb{T}}$ of common knowledge may include statements about the settings of the experiment, as well as complex derivations⁷ . To find our paradoxical contradiction, we may only need a very weak version of a full physical theory: for example Frauchiger and Renner only require a possibilistic version of the Born rule, which tells us whether an outcome will be observed with certainty [1]. This will also be the case in box world.

In operational experiments, a reasoning agent can make statements about systems that she studies; consequently, the theory used by the agent must be able to produce a description or a model of such a system, namely, in terms of a set of states. For example, in quantum theory a two-state quantum system with a ground state $| 0\rangle$ and an excited state $| 1\rangle$ (qubit) can be fully described by a set of states $\{| \psi \rangle \}$ in the Hilbert space ${{\mathbb{C}}}^{2}$, where $| \psi \rangle =\alpha | 0\rangle +\beta | 1\rangle$ with $\alpha ,\beta \in {\mathbb{C}}$ and $| \alpha {| }^{2}+| \beta {| }^{2}=1$. Other examples of theories and respective descriptions of states of systems include: GPTs, where e.g. a generalised bit (gbit) is a system completely characterized by two binary measurements which can be performed on it [4] (a review of GPTs can be found in section 3); algebraic quantum mechanics, with states defined as linear functionals $\rho :A\to {\mathbb{C}}$, where A is a $C*$-algebra [5]; or resource theories with some state space Ω, and epistemically defined subsystems [11, 12].

Definition 3 A 'physical system' (or simply 'system') is anything that can be an object of a physical study⁸ . A system can be characterized, according to the theory ${\mathbb{T}}$, by a set of possible states ${{ \mathcal P }}_{S}$. In addition, a system is associated with a set of allowed operations, ${{ \mathcal O }}_{S}:{{ \mathcal P }}_{S}\mapsto {{ \mathcal P }}_{S}$ on these states.

Definition 4 For any two systems S₁ and S₂, the union of the two defines a new system ${S}_{1}\cup {S}_{2}$ or simply ${S}_{1}{S}_{2}$. The operator $\parallel$ denotes parallel composition of states and operations such that ${p}_{{S}_{1}}\parallel {p}_{{S}_{2}}\in {{ \mathcal P }}_{{S}_{1}{S}_{2}}$ whenever ${p}_{{S}_{1}}\in {{ \mathcal P }}_{{S}_{1}}$ and ${p}_{{S}_{2}}\in {{ \mathcal P }}_{{S}_{2}}$ and similarly, ${o}_{{S}_{1}}\parallel {o}_{{S}_{2}}\in {{ \mathcal O }}_{{S}_{1}{S}_{2}}$ whenever ${o}_{{S}_{1}}\in {{ \mathcal O }}_{{S}_{1}}$ and ${o}_{{S}_{2}}\in {{ \mathcal O }}_{{S}_{2}}$. In other words, the state ${p}_{{S}_{1}}\parallel {p}_{{S}_{2}}$ of ${S}_{1}{S}_{2}$ can be prepared by simply preparing the states ${p}_{{S}_{1}}$ and ${p}_{{S}_{2}}$ of the individual systems S₁ and S₂ and the operation ${o}_{{S}_{1}}\parallel {o}_{{S}_{2}}$ can be implemented by locally performing the operations ${o}_{{S}_{1}}$ and ${o}_{{S}_{2}}$ on the individual systems.

We assume no further structure to this operator. Note also that we do not assume that a given composite system can be split into/described in terms of its parts even though combining individual systems in this manner allows us to define certain states of composite systems⁹ . Now we introduce agents into the picture.

Definition 5 A physical setting may be associated with a set ${ \mathcal A }$ of agents. An agent ${A}_{i}\in { \mathcal A }$ is described by a knowledge operator ${K}_{i}\in {{ \mathcal K }}_{{ \mathcal A }}$ and a physical system ${M}_{i}\in {{ \mathcal M }}_{{ \mathcal A }}$, which we call a 'memory.' Each agent may study other systems according to the theory ${\mathbb{T}}$. An agent's memory M_i records the results and the consequences of the studies conducted by A_i. The memory may be itself an object of a study by other agents.

Here we consider measurements both from the perspective of an agent who performs them, and that of another agent who is modeling the first agent's memory.

In an experiment involving measurements, each agent has the subjective experience of only observing one outcome (independently of how others may model her memory), and we can see this as the definition of a measurement: if there is no subjective experience of observing a single outcome, we do not call it a measurement. We can express this experience as statements such as ϕ₀ = 'The outcome was 0, and the system is now in state $| 0\rangle$.' Let us explain further after the formal definition.

Definition 6 A measurement is a type of study that can be conducted by an agent A_i on a system S, the essential result of which is the obtained 'outcome' $x\in {{ \mathcal X }}_{S}$. If witnessed by another agent A_j (who knows that A_i performed the measurement but does not know the outcome), the measurement is characterized by a set of propositions {ϕ_x} ∈ Φ, where ϕ_x corresponds to the outcome x, satisfying:

• ${K}_{j}({K}_{i}(\exists \ x\in {{ \mathcal X }}_{S}:{K}_{i}\ {\phi }_{x}))$,
• ${K}_{j}\ {K}_{i}\ {\phi }_{x}\Longrightarrow {K}_{j}\ {K}_{i}\ \neg ({\phi }_{y}),\ \ \forall \ y\ne x$.

The first condition tells us that A_j knows that from A_i's perspective, she must have observed one outcome x ∈ X, and A_i would have used this knowledge to derive all the relevant conclusions, as expressed by the proposition ϕ_x. For example, if the measurement represents a perfect Z measurement of a qubit, ϕ₀ may include statements like 'the qubit is now in state $| 0\rangle ;$ before the measurement it was not in state $| 1\rangle ;$ if I measure it again in the same way, I will obtain outcome 0;' and so on. Note that this condition does not imply that the measurement outcome stored in A_i's memory is classical for A_j. In fact, in the quantum case A_j may see A_i's memory as a quantum system entangled with the system that A_i measured. Despite this, A_j knows that from A_i's perspective, this outcome appears to be classical, which is what the first condition captures. The second condition implements A_i's experience of observing a single outcome, and the fact that the outside agent A_j knows that this is the case from A_i's perspective. If A_i observes x, they conclude that the conclusions ϕ_y that they would have derived had they observed a different outcome y are not valid and A_j knows that A_i would do so. In the previous example, they would know that it does not hold ϕ₁ = 'the qubit is now in state $| 1\rangle ;$ before the measurement it was not in state $| 0\rangle ;$ if I measure it again I will see outcome 1.' This condition also ensures that the conclusions ${\{{\phi }_{x}\}}_{x}$ are mutually incompatible, i.e. that the measurement is tightly characterized.

A measurement of another agent's memory is also an example of a valid measurement. In other words, agent A_j can choose A_i's lab, consisting of A_i's memory and another system S (which A_i studies), as an object of her study.

Thus, any agent's memory can be modelled by the other agents as a physical system undergoing an evolution that correlates it with the measured system. In quantum theory, this corresponds to the unitary evolution

$$\begin{eqnarray}&&\left(\displaystyle \sum _{x=0}^{N-1}{p}_{x}\ | x{\rangle }_{\mathrm{system}}\right)\otimes | 0{\rangle }_{\mathrm{memory}}\to \displaystyle \sum _{x=0}^{N-1}{p}_{x}\mathop{\underbrace{| x{\rangle }_{\mathrm{system}}\otimes | x{\rangle }_{\mathrm{memory}}}}\limits_{=: \,| \tilde{x}{\rangle }_{{SM}}}.\end{eqnarray} \tag{ 1 }$$

The key aspect here is that the set of states of the joint system of observed system and memory, ${{ \mathcal P }}_{{SM}}=\mathrm{span}\{| x{\rangle }_{\mathrm{system}}\otimes | x{\rangle }_{\mathrm{memory}}\}{}_{x=0}^{N-1}$ is post-measurement isomorphic to the set of states ${{ \mathcal P }}_{S}$ system alone. That is, for every transformation _S that you could apply to the system before the measurement, there is a corresponding transformation _SM acting on the ${{ \mathcal P }}_{{SM}}$ that is operationally identical. By this we mean that an outside observer would not be able to tell if they are operating with _S on a single system before the measurement, or with _SM on system and memory after the measurement. In particular, if _S is itself another measurement on S within a probabilistic theory, it should yield the same statistics as post-measurement _SM. For a quantum example that helps clarify these notions, consider S to be a qubit initially in an arbitrary state $\alpha | 0{\rangle }_{S}+\beta | 1{\rangle }_{S}$. An agent Alice measures S in the Z basis and stores the outcome in her memory A. While she has a subjective experience of seeing only one possible outcome, an outside observer Bob could model the joint evolution of S and A as

$$\begin{eqnarray*}&&\left(\alpha | 0{\rangle }_{S}+\beta | 1{\rangle }_{S}\right)\otimes | 0{\rangle }_{A}\ \to \ \alpha | 0{\rangle }_{S}| 0{\rangle }_{A}+\beta | 1{\rangle }_{S}| 1{\rangle }_{A}.\end{eqnarray*}$$

Suppose now that (before Alice's measurement) Bob was interested in performing an X measurement on S. This would have been a measurement with projectors $\{| +\rangle \langle +{| }_{S},| -\rangle \langle -{| }_{S}\}$, where $| \pm {\rangle }_{S}=\tfrac{1}{\sqrt{2}}(| 0{\rangle }_{S}\pm | 1{\rangle }_{S})$. However, he arrived too late: Alice has already performed her Z measurement on S. If now Bob simply measured X on S he would obtain uniform statistics, which would be uncorrelated with the initial state of S. So what can he do? It may not be very friendly, but he can measure S and Alice's memory A jointly, by projecting onto

$$\begin{eqnarray*}\begin{array}{rcl}| +{\rangle }_{{SA}} & = & \displaystyle \frac{1}{\sqrt{2}}(| 0{\rangle }_{S}| 0{\rangle }_{A}+| 1{\rangle }_{S}| 1{\rangle }_{A})\\ | -{\rangle }_{{SA}} & = & \displaystyle \frac{1}{\sqrt{2}}(| 0{\rangle }_{S}| 0{\rangle }_{A}-| 1{\rangle }_{S}| 1{\rangle }_{A}),\end{array}\end{eqnarray*}$$

which yields the same statistics of Bob's originally planned measurement on S, had Alice not measured it first. This equivalence should also hold in the more general case where the observed system may have been previously correlated with some other reference system: such correlations should be preserved in the measurement process, as modelled from the 'outside' observer Bob.

There are many options to formalize this notion that 'every way that an outside observer could have manipulated the system before the measurement, he may now manipulate a subspace of 'system and observer's memory,' with the same results.' A possible simplification to restrict our options is to take subsystems and the tensor product structure as primitives of the theory, which may apply to GPTs [4] but not for general physical theories (like field theories; for a discussion see [12]). In the interest of time, we will for now restrict ourselves to this case, and leave a more general formulation of this condition as future work. For simplicity, we also restrict ourselves to information-preserving measurements (excluding for now those where some information may have leaked to an environment external to Alice's memory), which are sufficient to derive the contradiction.

Definition 7 Let ${{ \mathcal P }}_{S}$ be a set of states of a system S that is being studied by an agent A_i with a memory M_i, and ${{ \mathcal P }}_{{{SM}}_{i}}$ be a set of states of the joint system SM_i. If for a given initial state ${q}_{{M}_{i}}^{{\rm{in}}}\in {{ \mathcal P }}_{{M}_{i}}$ of the memory, there exists a map ${u}^{q}:{{ \mathcal P }}_{{{SM}}_{i}}\to {{ \mathcal P }}_{{{SM}}_{i}}$ ($\in {{ \mathcal O }}_{{{SM}}_{i}}$) that satisfies the following conditions (1) and (2), then u^q is called an information-preserving memory update.

• 1.  
  Local operations on S before the memory update can be simulated by joint operations on S and M_i after the update. That is, for all ${p}_{S}\in {{ \mathcal P }}_{S},{o}_{S}\in {{ \mathcal O }}_{S},{A}_{j}\in { \mathcal A },\phi$, there exists an operation ${o}_{{{SM}}_{i}}\in {{ \mathcal O }}_{{{SM}}_{i}}$ such that

  $$\begin{eqnarray*}&&{K}_{j}\ \phi [{o}_{S}({p}_{s})]\Rightarrow {K}_{j}\ \phi [{o}_{{{SM}}_{i}}\,\circ \,{u}^{q}({p}_{S}\parallel {q}_{{M}_{i}}^{{\rm{in}}})],\end{eqnarray*}$$

  where ϕ[...] are arbitrary statements that depend on the argument.
• 2.  
  The memory update does not factorize into local operations. That is, there exist no operations $o{{\prime} }_{S}\in {{ \mathcal O }}_{S}$ and $o{{\prime} }_{{M}_{i}}\in {{ \mathcal O }}_{{M}_{i}}$ such that

  $$\begin{eqnarray*}&&{u}^{q}=o{{\prime} }_{S}\parallel o{{\prime} }_{{M}_{i}}.\end{eqnarray*}$$

Condition (1) was explained in previous paragraphs. Condition (2) is required because the trivial map which entails doing nothing to the system and memory (i.e. the identity) satisfies Condition (1) even though such an operation should certainly not be regarded as a memory update. Condition (2) requires that u^q does not factorise into local operations over S and M_i is required in order to rule out such trivial operations that cannot be taken to represent a memory update. See figure 4 for an example of u^q in the quantum case where it is a reversible unitary operation and the initial state of the memory, ${q}_{{M}_{i}}^{{\rm{in}}}$ is $| 0{\rangle }_{{M}_{i}}$. In general, the memory update map u^q need not be reversible; for example, in box world it is an irreversible transformation, as we will see later.

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Note that it is enough to consider the memory update map u^q corresponding to a particular choice of initial state ${q}_{{M}_{i}}^{{\rm{in}}}$ since the map ${u}^{q^{\prime} }$ corresponding to any other state $q{{\prime} }_{{M}_{i}}^{{\rm{in}}}\in {{ \mathcal P }}_{{M}_{i}}$ can be obtained by first locally transforming the memory state into ${q}_{{M}_{i}}^{{in}}$ and then applying u^q. Thus without loss of generality, we will consider only specific initial states in the paper and drop the label q on this map, simply calling it u. For example, in the quantum case, it is enough to consider the memory update with the memory initialised to the state, $| 0{\rangle }_{{M}_{i}}$.

The characterization of measurements introduced in this section is rather minimal. In physical theories like classical and quantum mechanics, measurements have other natural properties that we do not require here. Two striking examples are 'after her measurement, Alice's memory becomes correlated with the system measured in such a way that, for any subsequent operation that Bob could perform on the system, there is an equivalent operation he may perform on her memory' and 'the correlations are such that there exists a joint operation on the system and Alice's memory that would allow Bob to conclude which measurement Alice performed.' While these properties hold in the familiar classical and quantum worlds, we do not know of other physical theories where measurements can satisfy them, and they require Bob to be able to act independently on the system and on Alice's memory, which may not always be possible. For example, we will see that in box world, these two subsystems become superglued after Alice's measurement, and that Bob only has access to them as a whole and not as individual components¹⁰ . As such, we will not require these properties out of measurements, for now. We revisit this discussion in section 5.

Individual states. The so-called generalised bit or gbit is a system completely characterized by two binary measurements which can be performed on it [4]. Such sets of measurements that completely characterise the state of a system are known as fiducial measurements. The state of a gbit is thus fully specified by the vector

$$\begin{eqnarray}&&{{\bf{P}}}_{{\rm{gbit}}}=\left(\begin{array}{c}P(a=0| X=0)\\ P(a=1| X=0)\\ P(a=0| X=1)\\ P(a=1| X=1)\end{array}\right),\end{eqnarray} \tag{ 2 }$$

where X = 0 and X = 1 represent the two choices of measurements and a ∈ {0, 1} are the possible outcomes (figure 5(a)). Analogously, a classical bit is a system characterized by a single binary fiducial measurement

$$\begin{eqnarray}&&{{\bf{P}}}_{{\rm{bit}}}=\left(\begin{array}{c}P(a=0| X=0)\\ P(a=1| X=0)\end{array}\right),\end{eqnarray} \tag{ 3 }$$

and, in quantum theory, a qubit is characterized by three fiducial measurements (corresponding, for example, to three directions X, Y and Z in the Bloch sphere),

$$\begin{eqnarray}&&{{\bf{P}}}_{{\rm{qubit}}}=\left(\begin{array}{c}P(a=0| X=0)\\ P(a=1| X=0)\\ P(a=0| X=1)\\ P(a=1| X=1)\\ P(a=0| X=2)\\ P(a=1| X=2)\end{array}\right).\end{eqnarray} \tag{ 4 }$$

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For normalized states, we have $| {\bf{P}}| ={\sum }_{i}P(a=i| X=j)=1,\forall j$. The set of possible states of a gbit is convex, with extremes

$$\begin{eqnarray}&&{{\bf{P}}}_{00}=\left(\begin{array}{c}1\\ 0\\ 1\\ 0\end{array}\right),\ \ {{\bf{P}}}_{01}=\left(\begin{array}{c}1\\ 0\\ 0\\ 1\end{array}\right),\ \ {{\bf{P}}}_{10}=\left(\begin{array}{c}0\\ 1\\ 1\\ 0\end{array}\right),\ \ {{\bf{P}}}_{11}=\left(\begin{array}{c}0\\ 1\\ 0\\ 1\end{array}\right).\end{eqnarray} \tag{ 5 }$$

These correspond to pure states. In the qubit case, the extremes correspond to all the points on the surface of the Bloch sphere, for example

$$\begin{eqnarray}&&{{\bf{P}}}_{| +\rangle }=\left(\begin{array}{c}1\\ 0\\ 1/2\\ 1/2\\ 1/2\\ 1/2\end{array}\right),\ \ {{\bf{P}}}_{| -\rangle }=\left(\begin{array}{c}0\\ 1\\ 1/2\\ 1/2\\ 1/2\\ 1/2\end{array}\right),\ \ {{\bf{P}}}_{| 0\rangle }=\left(\begin{array}{c}1/2\\ 1/2\\ 1/2\\ 1/2\\ 1\\ 0\end{array}\right)\ \ {{\bf{P}}}_{| 1\rangle }=\left(\begin{array}{c}1/2\\ 1/2\\ 1/2\\ 1/2\\ 0\\ 1\end{array}\right).\end{eqnarray} \tag{ 6 }$$

Note that in box world, pure gbits are deterministic for both alternative measurements, whereas in quantum theory at most one fiducial measurement can be deterministic for each pure qubit, as reflected by uncertainty relations. We denote the set of allowed states of a system A by ${{ \mathcal S }}^{A}$.

Composite states. The state of a bipartite system AB, denoted by ${{\bf{P}}}^{{AB}}\in {{ \mathcal S }}^{{AB}}$ can be written in the form ${{\bf{P}}}^{{AB}}={\sum }_{i}{r}_{i}\ {{\bf{P}}}_{i}^{A}\otimes {{\bf{P}}}_{i}^{B}$ where r_i are real coefficients¹¹ and ${{\bf{P}}}_{i}^{A}\in {{ \mathcal S }}^{A}$, ${{\bf{P}}}_{i}^{B}\in {{ \mathcal S }}^{B}$ can be taken to be pure and normalised states of the individual systems A and B [4]. Thus, a general 2-gbit state ${{\bf{P}}}_{2}^{{AB}}$ can be written as in figure 5(b) (left), where X, Y ∈  {0, 1} are the two fiducial measurements on the first and second gbit and a, b ∈ {0, 1} are the corresponding measurement outcomes. The PR box ${{\bf{P}}}_{{PR}}$, on the right, is an example of such a 2 gbit state that is valid in box world, which satisfies the condition $a\oplus b={xy}$ [15].

State transformations. Valid operations are represented as matrices that transform valid state vectors to valid state vectors (appendix B). In addition, we only have access to the (single-shot) input/output behaviour of systems, so in practice all valid operations in box world take the form of classical wirings between boxes, which correspond to pre- and post-processing of input and output values, and convex combinations thereof [4]. For example, bipartite joint measurements on a 2-gbit system can be decomposed into convex combinations of classical 'wirings', as shown in figure 6. In contrast, quantum theory allows for a richer structure of bipartite measurements by allowing for entangling measurements (e.g. in the Bell basis), which cannot be decomposed into classical wirings. Bipartite transformations on multi-gbit systems turn out to be classical wirings as well [4]. Reversible operations in particular consist only of trivial wirings: local operations and permutations of systems [16]. One cannot perform entangling operations such as a coherent copy (the quantum CNOT gate) [4, 17], which is required in the original version of the Frauchiger–Renner experiment.

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We will now instantiate our general conditions for agents, memories and measurements (definitions 6 to 7) in box world. As there is no physical theory for the dynamics behind box world, there is plenty of freedom in the choice of implementation. In principle each such choice could represent a different physical theory leading to the same black-box behaviour in the limit of a single agent with an implicit memory. This is analogous to the way in which different versions of quantum theory (Bohmian mechanics, collapse theories, unitary quantum mechanics with von Neumann measurements) result in the same effective theory in that limit.

Definition 8 Let ${\mathbb{T}}$ be the theory that describes box world, according to [4]. As per definition 5, an agent ${A}_{i}\in { \mathcal A }$ is described by a knowledge operator ${K}_{i}\in {{ \mathcal K }}_{{ \mathcal A }}$ and a physical memory ${M}_{i}\in {{ \mathcal M }}_{{ \mathcal A }}$.

We will focus on the case where the memory consists of bit or gbits. Each agent may study other systems according to the theory ${\mathbb{T}}$. An agent's memory records the results and the consequences of the studies conducted by them, and may be an object of a study by other agents.

It is worth mentioning that boxes do not correspond to physical systems, but to input/output functions that can only be evaluated once. As such, the post-measurement state of a physical system is described by a whole new box. The notion of an individual system itself, as we will see, may be unstable under measurements—some measurements glue the system to the observer's memory, in a way that makes individual access to the original system impossible.

Measurement: observer's perspective. From the point of view of the observer who is measuring (say Alice), making a measurement on a system corresponds simply to running the box whose state vector encodes the measurement statistics. Alice may then commit the result of her measurement to a physical memory, like a notebook where she writes 'I measured observable X and obtained outcome a.' To be useful, this should be a memory that may be consulted later, i.e. it could receive an input Y = 'start: open and read the memory', and output the pair (X, a). In the language of GPTs, this means that Alice, from her own perspective, prepares a new box with a trivial input Y = 'start' and two outputs $(X^{\prime} ,a^{\prime} )$, with the behaviour ${\bf{P}}((X^{\prime} ,a^{\prime} )| Y)={\delta }_{X,X^{\prime} }{\delta }_{a,a^{\prime} }$, which depends on her observations (figure 7). She may later run this box (look at her notebook) and recover the measurement data. The exact dimension of the box will depend on how Alice perceives and models her own memory; for example it could consist of two bits, or two gbits, or, if we think that before the measurement she stored the information about the choice of observable elsewhere, it could be a single bit or gbit encoding only the outcome. We leave this open for now, as we do not want to constrain the theory too much at this stage.

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Measurements: inferences. To see the kind of inferences and conclusions that an agent can take from a measurement in box world, it is convenient to look at the example where Alice and Bob share a PR box. Suppose that Alice measured her half of the box with input X = 1 and obtained outcome a = 0. From the PR correlations, ${XY}=a\oplus b$, she can conclude that if Bob measures Y = 0, he must obtain b = 0, and if he measures Y = 1, he must obtain b = 1. This is independent of whether Bob's measurement happens before or after Alice (or even space-like separated). She could reach similar deterministic conclusions for her other choice of measurement and possible outcomes. In the language of definition 6, we have

$$\begin{eqnarray*}\begin{array}{rcl}{\phi }_{X=0,a=0} & = & {\rm{`}}[Y=0\Longrightarrow b=0]\wedge [Y=1\Longrightarrow b=0]{\rm{\mbox{'}}},\\ {\phi }_{X=0,a=1} & = & {\rm{`}}[Y=0\Longrightarrow b=1]\wedge [Y=1\Longrightarrow b=1]{\rm{\mbox{'}}},\\ {\phi }_{X=1,a=0} & = & {\rm{`}}[Y=0\Longrightarrow b=0]\wedge [Y=1\Longrightarrow b=1]{\rm{\mbox{'}}},\\ {\phi }_{X=1,a=1} & = & {\rm{`}}[Y=0\Longrightarrow b=1]\wedge [Y=1\Longrightarrow b=0]{\rm{\mbox{'}}}.\end{array}\end{eqnarray*}$$

Measurement: memory update from an outsider's perspective. Next we need to model how an outside agent, Ursula, models Alice's measurement, in the case where Alice does not communicate her outcome to Ursula¹² . Suppose that all agents share a time reference frame, and Alice makes her measurement at time t = 1. From Ursula's perspective, in the most general case, this will correspond to Alice preparing a new box, with some number of inputs and outputs, which Ursula can later run (figure 8(a)). The exact form of this box will depend on the underlying physical theory for measurements: in the quantum case it corresponds to a box with the measurement statistics of a state that is entangled between the system measured and Alice's memory, as we saw. In classical mechanics, it will correspond to perfect classical correlations between those two subsystems. In the other extreme, we could imagine a theory of very destructive measurements, where after Alice's measurement, the physical system she had measured would vanish. From Ursula's perspective, this could be modelled by a box with a void associated distribution. Now suppose that we would like to have a physical theory where the dimension of systems is preserved by measurements: for example, if the system that Alice measures is instantiated by a box with binary input and output (e.g. a gbit, or half of a PR-box), and Alice's memory, where she stores the outcome of the measurement (as in figure 7) is also represented as a gbit, then we would want the post-measurement box accessible to Ursula to have in total two binary inputs and two binary outputs (or more generally, four possible inputs and four possible outputs). Note that this is not a required condition for a theory to be physical per se—it is just a familiar rule of thumb that gives some persistent meaning to the notion of subsystems and dimensions. In such a theory that supports box world correlations, we find that the allowed statistics of Ursula's box must satisfy the conditions of figure 8(b) (proof in appendix C). These conditions still leave us some wiggle room for possible different implementations.

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Measurements: information-preserving memory update. In order to find a multi-agent paradox, we will need a model of memory update that is information-preserving, in the sense of definition 7. This does not imply that Alice's transformation (as seen by Ursula) be reversible: in fact, we find that in general, it can glue two subsystems such that Ursula will only be able to address them as a whole (since separating them could lead to a violation of no-signalling), but the relevant fact is that Ursula can apply some post-processing in order to obtain a new box with the same behaviour as the pre-measurement system that Alice observed. In figure 9 we give an example of a model that satisfies these conditions, in addition to the conditions of figure 8(b). It is a minimal implementation among many possible, which already allows us to derive such a paradox beyond quantum theory. We further discuss some of the limitations and alternatives to this model in section 5.2. What is important here (and proven in appendix C) is that this model generalizes to the case where Alice measures half of a bipartite state, like a PR box. That is, suppose that Alice and Bob share a PR box. Imagine that at time t = 1 Alice makes her measurement X, obtaining (from her perspective) an outcome a, and that Bob makes his measurement Y at time t = 2, obtaining outcome b. As usual, if Alice and Bob were to communicate at this point, they would find that ${XY}=a\oplus b$, and indeed the propositions ${\phi }_{X,a}$ and ${\phi }_{Y,b}$ that represent their subjective measurement experience would hold. But now suppose that Alice and Bob do not get the chance to communicate and compare their input and outputs; instead, at time t = 3, an observer Ursula, who models Alice's measurement as in figure 9(a), runs the box corresponding to Alice's half of the PR box and Alice's memory, and applies the post-processing of figure 9(b). Ursula's input is $\tilde{X}$ and her output is $\tilde{a}$. Then the claim is that $\tilde{X}Y=\tilde{a}\oplus b$: that is, Ursula and Bob effectively share a PR box. This is proven in appendix C. We now have all the ingredients needed to find a multi-agent epistemic paradox in box world.

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We showed that box world agents reasoning about each others' knowledge can come to a deterministic contradiction, which is stronger than the original paradox, as it can be reached without post-selection, from the point of view of every agent and for any measurement outcome obtained by them.

Post-selection. In contrast to the original Frauchiger–Renner experiment of [1], no post-selection was required to arrive at this contradictory chain of statements as, in fact, all the implications above are symmetric, for example

$$\begin{eqnarray*}&&\tilde{a}=0\ \Longleftrightarrow \ b=0\ \Longleftrightarrow \ a=0\ \Longleftrightarrow \ \tilde{b}=0\ \Longleftrightarrow \ \tilde{a}=1.\end{eqnarray*}$$

As a result, one can arrive at a similar (symmetric) paradoxical chain of statements irrespective of the choice of agent and outcome for the first statement. In other words, irrespective of the outcomes observed by every agent, each agent will arrive at a contradiction when they try to reason about the outcomes of other agents. This is because, as shown in [7], the PR box exhibits strong contextuality and no global assignments of outcome values for all four measurements exists for any choice of local assignments. In contrast, the original paradox of [1] admits the same distribution as that of Hardy's paradox [6]. It is shown in [7] that this distribution is an example of logical contextuality where for a particular choice of local assignments (the ones that are post-selected on in the original Frauchiger–Renner experiment), a global assignment of values compatible with the support of the distribution fails to exist, but this is not true for all local assignments. This makes the paradox even stronger in box world, since it can be found without post-selection and by any of the agents, for any outcome that they observe. In particular, the paradox would already arise in a single run of the experiment. For a simple method to enumerate all possible contradictory statements that the agents may make, see the analysis of the PR box presented in [7].

Reversibility of the memory update map. As mentioned previously, the memory update map u in the quantum case is quantum CNOT gate which is a unitary and hence reversible. In box world however, this map cannot be reversible since it is known that all reversible maps in box world map product states to product states [16] and hence no reversible u in box world could satisfy Definition 7 of an information preserving memory update. The map we propose here for box world is clearly irreversible as it leads to correlations between the initially uncorrelated system and memory.

Since we lack a physical theory to explain how measurements and transformations are instantiated for generalised non-signalling boxes, and only have access to their input/output behaviour, all allowed transformations consist of pre- and post-processing. In the quantum case, we have in addition to a description of possible input–output correlations, a mathematical framework for the underlying states producing those correlations, the theory of von Neumann measurements and transformations as CPTP maps. In appendix D we briefly show how we one could in principle model the quantum memory updates in the framework of GPTs. In box world, introduction of dynamical features (for example, a memory update algorithm) is less intuitive and requires additional constructions. In the following, we outline the main limitations we found.

Systems versus boxes. In quantum theory, a system corresponds to a physical substrate that can be acted on more than once. For example, Alice could measure a spin first in the Z basis and then in X basis (obviously with different results than if she had measured first X and then Z). The predictions for each subsequent measurement are represented by a different box in the GPT formalism, such that each box encodes the current state of the system in terms of the measurement statistics of a tomographically complete set of measurements. After each measurement, the corresponding box disappears, but quantum mechanics gives us a rule to compute the post-measurement state of the underlying system, which in turn specifies the box for future measurements. On the other hand, the default theory for box world lacks the notion of underlying physical systems and a definite rule to compute the post-measurement vector state of something that has been measured once. Indeed, equations (9a)–(9c) (appendix B) tell us that post-measurement states is only partially specified: for instance, if the measurement performed was fiducial, we know that the block corresponding to that measurement in the post-measurement state would have a '1' corresponding to the outcome obtained and '0' for all other outcomes in the block. However, we still have freedom in defining the entries in the remaining blocks. Our model proposes a possible physical mechanism for updating boxes (which could be read as updating the state of the underlying system), but so far only for the case where we compare the perspectives of different agents, and we leave it open whether Alice has a subjective update rule that would allow her to make subsequent measurements on the same physical system.

Verifying a measurement. In our simple model, the external observer Ursula has no way to know which measurement Alice performed, or whether she measured anything at all—the connection between Alice's and Ursula's views is postulated rather than derived from a physical theory. Indeed, Alice could have simply wired the boxes as in figure 9(a) without actually performing the measurement, and Ursula will not know the difference: she obtains the same joint state of Alice's memory and the system she measured. In contrast, consider the case of quantum mechanics with standard von Neumann measurements. There, Alice's memory gets entangled with the system, and the post-measurement state depends on the basis in which Alice measured her system. For example, if Alice's qubit S starts off in the normalised pure state $| \psi \rangle =\alpha | 0{\rangle }_{S}+\beta | 1{\rangle }_{S}$ and her memory M initialised to $| 0{\rangle }_{M}$, the initial state of her system and memory from Ursula's perspective is $| {\rm{\Psi }}{\rangle }_{{SM}}^{{\rm{in}}}=[\alpha | 0{\rangle }_{S}+\beta | 1{\rangle }_{S}]\otimes | 0{\rangle }_{M}=[\left(\tfrac{\alpha +\beta }{\sqrt{2}}\right)| +{\rangle }_{S}+\left(\tfrac{\alpha -\beta }{\sqrt{2}}\right)| -{\rangle }_{S})]\otimes | 0{\rangle }_{M}$. If Alice measures the system in the Z basis, the post-measurement state from Ursula's perspective is $| {\rm{\Psi }}{\rangle }_{{SM}}^{{\rm{out}},Z}=\alpha | 0{\rangle }_{S}| 0{\rangle }_{M}+\beta | 1{\rangle }_{S}| 1{\rangle }_{M}$, which is an entangled state. If instead, Alice measured in the Hadamard (X) basis, the post-measurement state would be $| {\rm{\Psi }}{\rangle }_{{SM}}^{{\rm{out}},X}=\left(\tfrac{\alpha +\beta }{\sqrt{2}}\right)| +{\rangle }_{S}| 0{\rangle }_{M}+\left(\tfrac{\alpha -\beta }{\sqrt{2}}\right)| -{\rangle }_{S}| 1{\rangle }_{M}$. Clearly the measurement statistics of $| {\rm{\Psi }}{\rangle }_{{SM}}^{{\rm{in}}}$, $| {\rm{\Psi }}{\rangle }_{{SM}}^{{\rm{out}},Z}$ and $| {\rm{\Psi }}{\rangle }_{{SM}}^{{\rm{out}},X}$ are different and Ursula can thus (in principle, with some probability) tell whether or not Alice performed a measurement and which measurement was performed by her. In the absence of a physical theory backing box world, we can still lift this degenerancy between the three situations (Alice did not measure, she measured X = 0, or she measured X = 1) by adding another classical system to the circuitry of 9(a): for example, a trit that stores what Alice did, and which Ursula could consult independently of the glued box of system and Alice's memory. However, we had still have a postulated connection between what's stored in this trit and what Alice actually did, and not one that is physically motivated.

${Supergluing}$ of non-signalling boxes. For the memory update circuit (from Ursula's perspective) of figure 9(a), and the initial state of equation (10), the final state would be ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}=(p\ \ 0\ \ 0\ \ 1-p| p\ \ 0\ \ 0\ \ 1-p| q\ \ 0\ \ 0\ \ 1-q| q\ \ 0\ \ 0\ \ 1-q{)}_{{SM}}^{T}$. Note that while the reduced final state of S does not depend on the input $X^{\prime}$ to M, the reduced final state on Alice's memory M, ${{\bf{P}}}_{{\rm{fin}}}^{M}$ clearly depends on the input X of the system S if $p\ne q$. If X = 0, ${{\bf{P}}}_{{\rm{fin}}}^{M}={(p1-p| p1-p)}^{T}$ and if X = 1, ${{\bf{P}}}_{{\rm{fin}}}^{M}={(q1-q| q1-q)}^{T}$, i.e. the systems S and M are signalling. This is expected since there is clearly a transfer of information from S to M during the measurement as seen in figure 6. However, this means that the state ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$ is not a valid box world state of 2 systems S and M but a valid state of a single system SM i.e. after Alice performs her wiring/measurement, it is not possible to physically separate Alice's system Sfrom her memory M from Ursula's perspective. For if this were possible, there would be a violation of the no-signalling principle and the notion of relativistic causality. In quantum theory, on the other hand it is always possible to perform separate measurements on Alice's system and her memory even after she measures. We call this feature supergluing of post-measurement boxes, where it is no longer possible for Ursula to separately measure S or M, but she can only jointly measure SM as though it were a single system. Note that this is only the case for $p\ne q$ and in our example with the PR-box (section 4), p = q = 1/2 and ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$ remains a valid bipartite non-signalling state in this particular, fine-tuned case of the PR box and there is no supergluing in the particular example described in section 4.

A glass half full. The above-mentioned features of the memory update in box world are certainly not desirable, and not what one would expect to find in a physical theory with meaningful notions of subsystems. An optimistic way to look at these limitations is to see them as providing us with further intuition for why PR boxes have not yet been found in nature. One of the main contributions of this paper is the finding that despite these peculiar features of box world and the fact that it has no entangling bipartite joint measurements (a crucial step in the original quantum paradox), a consistent outside perspective of the memory update exists such that with our generalised assumptions, a multi-agent paradox can be recovered. This indicates that the reversibility of dynamics akin to quantum unitarity is not crucial to derive this kind of paradox.

Other models for physical measurements. Ours is not the first attempt at coming up with a (partial) physical theory that reproduces the statistics of box world. Here we review the approach of Skrzypczyk et al in [18]. There the authors consider a variation of box world that has a reduced set of physical states (which the authors call genuine), which consists of the PR box and all the deterministic local boxes. The wealth of box world state vectors (i.e. the non-signalling polytope, or what we could call epistemic states) is recovered by allowing classical processing of inputs and outputs via classical wirings, as well as convex combinations thereof. In contrast, box world takes all convex combinations of maximally non-signalling boxes (of which the PR box is an example) to be genuine physical states; this becomes relevant as we require the allowed physical operations to map such states to each other. For the restricted state space of [18], the set of allowed operations is larger than in box world, particularly for multipartite settings. For example, there we are allowed maps that implement the equivalent of entanglement swapping: if Bob shares a PR box with Alice, and another with Charlie, there is an allowed map that he can apply on his two halves which leaves Alice and Charlie sharing a PR box, with some probability. It would be interesting to try to model memory update in this modified theory, to see if (1) there is a more natural implementation of measurements within the extended set of operations, and (2) whether this theory allows for multi-agent paradoxes.

While we have shown that a consistency paradox, similar to the one arising in the Frauchiger-Renner setup, can also be adapted for the box world in terms of GPTs, it still remains unclear how to characterize all possible theories where it is possible to find a setup leading to a contradiction. Essentially, one has to restrict the class of such theories and identify the properties of these theories that make such paradoxes possible. It seems that contextuality is a key property of such theories, this is discussed in more detail in section 5.4. Another central ingredient seems to be information-preserving models for physical measurements such as our memory update of definition 7, which allow us to replace counter-factuals with actual measurements, performed in sequence by different agents.

Beyond standard composition of systems. Additionally, it is still an open problem to find an operational way to state the outside view of measurements (and a memory update operation), for theories without a prior notion of subsystems and a tensor rule for composing them. This will allow us to search for multi-agent logical paradoxes in field theories, for example. One possible direction is to use notions of effective and subjective locality, as outlined for example in [12].

Multi-agent logical paradoxes involve chains (or possibly more general structures) of statements that cannot be simultaneously true in a consistent manner. Contextuality, on the other hand, can often be expressed in terms of the inability to consistently assign definite outcome values to a set of measurements [19, 20].

Given the examples of Frauchiger–Renner in quantum theory and the the present one in box world—two contextual theories—our hypothesis is that contextual physical theories, when applied to systems that are themselves reasoning agents, may generally lead to logical multi-agent paradoxes. The fact that such theories may allow a very different description of a measurement process from the points of views of an agent performing the measurement vs an outside agent (who analyses this agent and her system together) also has an important role to play in these paradoxes. In the quantum case this is closely linked to the measurement problem, the problem of reconciling unitary dynamics (outside view) and non-unitary 'collapse' (inside view). The existence of a connection between multi-agent paradoxes and contextuality is hard to miss, but it is the nature of this connection that is unknown i.e. are all proofs of multi-agent logical paradoxes proofs of contextuality, or vice-versa? These questions will be formally addressed in future work. Nevertheless, in the following, we provide an overview of further connections and some more specific open questions in this direction.

Liar cycles. In [7] relations between logical paradoxes and quantum contextuality are explored; in particular, the authors point out a direct connection between contextuality and a type of classic semantic paradoxes called Liar cycles [21]. A Liar cycle of length N is a chain of statements of the form:

$$\begin{eqnarray}&&{\phi }_{1}={\rm{``}}{\phi }_{2}\ \mathrm{is}\ \mathrm{true}^{\prime\prime} ,{\phi }_{2}={\rm{``}}{\phi }_{3}\ \mathrm{is}\ \mathrm{true}^{\prime\prime} ,...,{\phi }_{N-1}={\rm{``}}{\phi }_{N}\ \mathrm{is}\ \mathrm{true}^{\prime\prime} ,{\phi }_{N}={\rm{``}}{\phi }_{1}\ \mathrm{is}\ \mathrm{false}^{\prime\prime} .\end{eqnarray} \tag{ 7 }$$

It can be shown that the patterns of reasoning which are used in finding a contradiction in the chain of statements above are similar to the reasoning we make use of in FR-type arguments, and can also be connected to the cases of PR box (which corresponds to a Liar cycle of length 4) and Hardy's paradox. This further suggests that multi-agent paradoxes are closely linked to the notion of contextuality.

Relation to logical pre-post selection paradoxes. In [22], it has been shown that every proof of a logical pre-post selection paradox is a proof of contextuality. The exact connection between FR-type paradoxes and logical pre-post selection paradoxes is not known and this would be an interesting avenue to explore which would also provide insights into the relationship between FR paradoxes and contextuality.

In modal logic, a set Σ of possible states (or alternatives, or worlds) is introduced [23]: for example, in a world s₁ the key value is k = 1 and Eve does not know it, and in a state s₂ Eve could know that k = 0. The truth value of a proposition ϕ is then assigned depending on the possible world in Σ, and can differ from one possible world to another. In order to formalize the simple rules agents use for reasoning, we will first provide a structure which serves as a complete picture of the setup the agents are in, and then discuss the elements of the structure.

Definition 9 A Kripke structure M for n agents over a set of statements Φ is a tuple $\langle {\rm{\Sigma }},\pi ,{{ \mathcal K }}_{1},\ldots ,{{ \mathcal K }}_{n}\rangle$ where Σ is a non-empty set of states, or possible worlds, π is an interpretation, and ${{ \mathcal K }}_{i}$ is a binary relation on Σ.

The interpretation π is a map π: Σ × Φ → {true, false}, which defines a truth value of a statement ϕ ∈ Φ in a possible world s ∈ Σ.

${{ \mathcal K }}_{i}$ is a binary equivalence relation on a set of states Σ, where $(s,t)\in {{ \mathcal K }}_{i}$ if agent i considers world t possible given his information in the world s.

The truth assignment tells us if the proposition ϕ ∈ Φ is true or false in a possible world s ∈ Σ; for example, if ϕ= 'Alice has a secret key,' and s is a world where there is an individual named Alice who indeed possesses a secret key, then π(s, ϕ) = true. The truth value of a statement in a given structure M might vary from one possible world to another; we will denote that ϕ is true in world s of a structure M by $(M,s)\,\models \phi \,$, and $\models \phi$ will mean that ϕ is true in any world s of a structure M.

In order to operate the statements agents produce, we have to establish certain rules which are used to compress or judge the statements. These are the axioms of knowledge [24]. They might seem trivial in the light of our everyday reasoning, yet given our awareness of the quantum case, we will treat them carefully. Here we present the reader with a weaker version of the axioms (which includes Trust axiom) that we have developed in previous work [2].

Distribution axiom allows agents combine statement which contain inferences:

Axiom 1 If an agent is aware of a fact ϕ and that a fact ψ follows from ϕ, then the agent can conclude that ψ holds:

$$\begin{eqnarray*}&&(M,s)\,\models \,({K}_{i}\phi \wedge {K}_{i}(\phi \Rightarrow \psi ))\Rightarrow (M,s)\,\models \,{K}_{i}\psi .\end{eqnarray*}$$

Knowledge generalization rule permits agents use commonly shared knowledge:

Axiom 2 All agents know all the propositions that are valid in a structure:

$$\begin{eqnarray*}&&{if}\ (M,s)\,\models \,\phi \ \forall s\ {\rm{then}}\,\models \,{K}_{i}\phi \ \forall i.\end{eqnarray*}$$

Positive and negative introspection axioms highlight the ability of an agent to reflect upon her knowledge:

Axiom 3 Agents can perform introspection regarding their knowledge:

$$\begin{eqnarray*}&&(M,s)\models {K}_{i}\phi \Rightarrow (M,s)\models {K}_{i}{K}_{i}\phi ({\rm{PositiveIntrospection}}),\end{eqnarray*}$$

$$\begin{eqnarray*}&&(M,s)\models \neg {K}_{i}\phi \Rightarrow (M,s)\models {K}_{i}\neg {K}_{i}\phi ({\rm{NegativeIntrospection}}).\end{eqnarray*}$$

We also equip the logical skeleton of the setting with so-called trust structure, which governs the way the information is passed on between agents:

Definition 10 We say that an agent i trusts an agent j (and denote it by $j{\rightsquigarrow}i$ ) if and only if

$$\begin{eqnarray*}&&{K}_{i}\ {K}_{j}\ \phi \Longrightarrow {K}_{i}\ \phi ,\end{eqnarray*}$$

for all ϕ.

In the Frauchiger–Renner setup, as well as in the thought experiment presented in this paper, we consider the following trust structure between agents:

$$\begin{eqnarray}&&A{\rightsquigarrow}B{\rightsquigarrow}U{\rightsquigarrow}W{\rightsquigarrow}A.\end{eqnarray} \tag{ 8 }$$

Further discussion on axioms of modal logic and their application in quantum mechanics can be found in our paper [2].

In section 3, we briefly reviewed states and transformations in GPTs, in particular box world; here we go into further detail. Consider a GPT, ${\mathbb{T}}$. Denoting the set of all allowed states of a system in ${\mathbb{T}}$ by ${ \mathcal S }$, any valid transformation on a normalised GPT state ${\bf{P}}\in { \mathcal S }$ maps it to another normalised GPT state in ${ \mathcal S }$. Consequently, is linear and can be represented by a matrix M such that ${\bf{P}}\to M.{\bf{P}}$ under this transformation and $M.{\bf{P}}\in { \mathcal S }$ [4]. Further, operations that result in different possible outcomes can be associated with a set of transformations, one for each outcome. These also give an operational meaning to unnormalised states where $| {\bf{P}}| ={\sum }_{i}P(a=i| X=j)=c\ \ \forall j,c\in [0,1]$ (i.e. the norm is independent of the value of j). Such an operation M on a normalised initial state ${\bf{P}}$ can be associated with a set of matrices {M_i} such that the unnormalised state corresponding to the ith outcome is ${M}_{i}.{\bf{P}}$. Then the probability of obtaining this outcome is simply the norm of this unnormalised state, $| {M}_{i}.{\bf{P}}|$ and the corresponding normalized final state is ${M}_{i}.{\bf{P}}/| {M}_{i}.{\bf{P}}|$. A set {M_i} represents a valid operation if the following hold [4].

$$\begin{eqnarray}&&0\leqslant | {M}_{i}.{\bf{P}}| \leqslant 1\ \ \forall i,{\bf{P}}\in { \mathcal S },\end{eqnarray} \tag{ 9a }$$

$$\begin{eqnarray}&&\displaystyle \sum _{i}| {M}_{i}.{\bf{P}}| =1\ \ \forall {\bf{P}}\in { \mathcal S },\end{eqnarray} \tag{ 9b }$$

$$\begin{eqnarray}&&{M}_{i}.{\bf{P}}\in { \mathcal S }\ \ \forall i,{\bf{P}}\in { \mathcal S }.\end{eqnarray} \tag{ 9c }$$

This is the analogue of quantum Born rule for GPTs. Box world is a GPT where the state space ${ \mathcal S }$ consists of all normalized states ${\bf{P}}$ whose entries are valid probabilities (i.e. ∈[0,1]) and satisfy the no-signalling constraints i.e. for a N-partite state ${\bf{P}}$, the marginal term ${\sum }_{{a}_{i}}P({a}_{1},..,{a}_{i},..,{a}_{N}| {X}_{1},..,{X}_{i},..,{X}_{N})$ is independent of the setting X_i forall i ∈ {1, ..., N}¹⁴

When the GPT ${\mathbb{T}}$ is box world, the conditions of equations (9a)–(9c) result in the characterization of measurements and transformations in the theory in terms of classical circuits or wirings as shown in [4]. It suffices for the purpose of this paper to take that characterisation as the common knowledge of agents in the theory. In the original quantum paradox [1], the Born rule is taken as common knowledge and here, the common knowledge consists of characterisations that follow from the box world analogue of the born rule (equations (9a)–(9c)). We summarise the results of [4] characterising allowed transformations and measurements in box world and will only consider normalization-preserving transformations.

• Transformations:
  • 1.  
    Single system: All transformations on single box world systems are relabellings of fiducial measurements or outcomes or a convex combination thereof.
  • 2.  
    Bipartite system: Let X and Y be fiducial measurements performed on the transformed bipartite system with corresponding outcomes a and b, then all transformations of 2-gbit systems can be decomposed into convex combinations of classical circuits of the following form: A fiducial measurement $X^{\prime} ={f}_{1}(X,Y)$ is performed on the initial state of the first gbit resulting in the outcome $a^{\prime}$ followed by a fiducial measurement $Y^{\prime} ={f}_{2}(X,Y,X^{\prime} )$ on the initial state of the second gbit resulting in the outcome $b^{\prime}$. The final outcomes are given as $(a,b)={f}_{3}(X,Y,a^{\prime} ,b^{\prime} )$, where f₁, f₂ and f₃ are arbitrary functions.
• Meaurements:
  • 1.  
    Single system: All measurements on single box world systems are either fiducial measurements with outcomes relabelled or convex combinations of such.
  • 2.  
    Bipartite system: All bipartite measurements on 2-gbit systems can be decomposed into convex combinations of classical circuits of the following form (figure 6): A fiducial measurement X is performed on the initial state of the first gbit resulting in the outcome $a^{\prime}$ followed by a fiducial measurement $Y=f(a^{\prime} )$ on the second gbit resulting in the outcome $b^{\prime}$. The final outcome is $a=f^{\prime} (a^{\prime} ,b^{\prime} )$, where f and $f^{\prime}$ are arbitrary functions.

Remark. Note that an agent Alice who measures a box world system only sees a classical final state, which corresponds the classical measurement outcome, since the box is a single-shot input/output function. Alice could use equations (9a)–(9c) to calculate the probabilities of obtaining different outcomes given the measurement she performs and prepare a new box (a new input/output function) depending on the measurement and outcome she just obtained (and has stored in her memory), as in figure 7. An outside agent who does not know Alice's measurement outcome would see correlations between Alice's system and memory and would describe the measurement by an irreversible transformation, more specifically a classical wiring between Alice's system and memory as shown in the following section.

In this section, we describe how a box world agent would measure a system and store the result in a memory. From the perspective of an outside observer (who does not know the outcome of the agent's measurement), we describe the initial and final states of the system and memory before and after the measurement as well as the transformation that implements this memory update in box world. In the quantum case, any initial state of the system S is mapped to an isomorphic joint state of the system S and memory M (see equation (1)) and hence the memory update map that maps the former to the latter (an isometry in this case¹⁵ ) satisfies definition 7 of an information-preserving memory update. We will now characterise the analogous memory update map in box world and show that it also satisfies definition 7.

Theorem 11. In box world, there exists a valid transformation u that maps every arbitrary, normalized state ${{\bf{P}}}_{{\rm{in}}}^{S}$ of the system S to an isomorphic final state ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$ of the system S and memory M and hence constitutes an information-preserving memory update (Definition 7).

Proof. To simplify the argument, we will describe the proof for the case where S and M are gbits. For higher dimensional systems, a similar argument holds, this will be explained at the end of the proof.

We start with the system in an arbitrary, normalized gbit state ${{\bf{P}}}_{{\rm{in}}}^{S}={\left(p1-p| q1-q\right)}^{T}$ (where the subscript T denotes transpose and p, q ∈ [0, 1]) and the memory initialised to one of the 4 pure states¹⁶ , say ${{\bf{P}}}_{{\rm{in}}}^{M}={{\bf{P}}}_{1}={\left(10| 10\right)}^{T}$. Then the joint initial state, ${{\bf{P}}}_{{\rm{in}}}^{{SM}}={\left(p\ \ 1-p| q\ \ 1-q\right)}_{S}^{T}\otimes {\left(1\ \ 0| 1\ \ 0\right)}_{M}^{T}$ of the system and memory can be written as follows, where ${P}_{{\rm{in}}}(a=i,a^{\prime} =j| X=k,X^{\prime} =l)$ denotes the probability of obtaining the outcomes a = i and $a^{\prime} =j$ when performing the fiducial measurements X = k and $X^{\prime} =l$ on the system and memory respectively, in the initial state ${{\bf{P}}}_{{\rm{in}}}^{{SM}}$.

$$\begin{eqnarray}&&{{\bf{P}}}_{{\rm{in}}}^{{SM}}={\left(\begin{array}{c}{P}_{{\rm{in}}}(a=0,a^{\prime} =0| X=0,X^{\prime} =0)\\ {P}_{{\rm{in}}}(a=0,a^{\prime} =1| X=0,X^{\prime} =0)\\ {P}_{{\rm{in}}}(a=1,a^{\prime} =0| X=0,X^{\prime} =0)\\ {P}_{{\rm{in}}}(a=1,a^{\prime} =1| X=0,X^{\prime} =0)\\ {P}_{{\rm{in}}}(a=0,a^{\prime} =0| X=0,X^{\prime} =1)\\ {P}_{{\rm{in}}}(a=0,a^{\prime} =1| X=0,X^{\prime} =1)\\ {P}_{{\rm{in}}}(a=1,a^{\prime} =0| X=0,X^{\prime} =1)\\ {P}_{{\rm{in}}}(a=1,a^{\prime} =1| X=0,X^{\prime} =1)\\ {P}_{{\rm{in}}}(a=0,a^{\prime} =0| X=1,X^{\prime} =0)\\ {P}_{{\rm{in}}}(a=0,a^{\prime} =1| X=1,X^{\prime} =0)\\ {P}_{{\rm{in}}}(a=1,a^{\prime} =0| X=1,X^{\prime} =0)\\ {P}_{{\rm{in}}}(a=1,a^{\prime} =1| X=1,X^{\prime} =0)\\ {P}_{{\rm{in}}}(a=0,a^{\prime} =0| X=1,X^{\prime} =1)\\ {P}_{{\rm{in}}}(a=0,a^{\prime} =1| X=1,X^{\prime} =1)\\ {P}_{{\rm{in}}}(a=1,a^{\prime} =0| X=1,X^{\prime} =1)\\ {P}_{{\rm{in}}}(a=1,a^{\prime} =1| X=1,X^{\prime} =1)\end{array}\right)}_{{SM}}={\left(\begin{array}{c}p\\ 0\\ 1-p\\ 0\\ p\\ 0\\ 1-p\\ 0\\ q\\ 0\\ 1-q\\ 0\\ q\\ 0\\ 1-q\\ 0\end{array}\right)}_{{SM}}.\end{eqnarray} \tag{ 10 }$$

The rest of the proof proceeds as follows: we first describe a final state ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$ of the system and memory and a corresponding memory update map u that satisfy definition 7 of a generalized information-preserving memory update. Then, we show that this map can be seen an allowed box world transformation which completes the proof.

If an agent performs a measurement on the system, the state of the memory must be updated depending on the outcome and the final state of the system and memory after the measurement must hence be a correlated (i.e. a non-product) state. Although the full state space of the 2 gbit system SM is characterised by the 4 fiducial measurements $(X,X^{\prime} )\in \{(0,0),(0,1),(1,0),(1,1)\}$, definition 7 allows us to restrict possible final states to a useful subspace of this state space that contain correlated states of a certain form. The definition requires that for every map ${{ \mathcal E }}_{S}$ on the system before measurement, there exists a corresponding map ${{ \mathcal E }}_{{SM}}$ on the system and memory after the measurement that is operationally identical. Thus it suffices if the joint final state ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$ belongs to a subspace of the 2 gbit state space for which only 2 of the 4 fiducial measurements are relevant for characterising the state, namely any 2 fiducial measurements on ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$ that are isomorphic to the 2 fiducial measurements on ${{\bf{P}}}_{{\rm{in}}}^{S}$. Note that by definition of fiducial measurements, the outcome probabilities of any measurement can be found given the outcome probabilities of all the fiducial measurements and without loss of generality, we will only consider the case where the agents perform fiducial measurements on their systems.

A natural isomorphism between fiducial measurements on ${{\bf{P}}}_{{\rm{in}}}^{S}$ and those on ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$ to consider here (in analogy with the quantum case) is: $X=i\iff (X,X^{\prime} )=(i,i),\forall i\in \{0,1\}$ i.e. only consider the cases where the fiducial measurements performed on S and M are the same. Now, in order for the states to be isomorphic or operationally equivalent, one requires that performing the fiducial measurements $(X,X^{\prime} )=(i,i)$ on ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$ should give the same outcome statistics as measuring X = 0 on ${{\bf{P}}}_{{\rm{in}}}^{S}$. This can be satisfied through an identical isomorphism on the outcomes: $a=i\iff (a,a^{\prime} )=(i,i),\forall i\in \{0,1\}$. Then the final state of the system and memory, ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$ will be of the form

$$\begin{eqnarray}&&{{\bf{P}}}_{{\rm{fin}}}^{{SM}}={\left(\begin{array}{c}{P}_{{\rm{fin}}}(a=0,a^{\prime} =0| X=0,X^{\prime} =0)\\ {P}_{{\rm{fin}}}(a=0,a^{\prime} =1| X=0,X^{\prime} =0)\\ {P}_{{\rm{fin}}}(a=1,a^{\prime} =0| X=0,X^{\prime} =0)\\ {P}_{{\rm{fin}}}(a=1,a^{\prime} =1| X=0,X^{\prime} =0)\\ {P}_{{\rm{fin}}}(a=0,a^{\prime} =0| X=0,X^{\prime} =1)\\ {P}_{{\rm{fin}}}(a=0,a^{\prime} =1| X=0,X^{\prime} =1)\\ {P}_{{\rm{fin}}}(a=1,a^{\prime} =0| X=0,X^{\prime} =1)\\ {P}_{{\rm{fin}}}(a=1,a^{\prime} =1| X=0,X^{\prime} =1)\\ {P}_{{\rm{fin}}}(a=0,a^{\prime} =0| X=1,X^{\prime} =0)\\ {P}_{{\rm{fin}}}(a=0,a^{\prime} =1| X=1,X^{\prime} =0)\\ {P}_{{\rm{fin}}}(a=1,a^{\prime} =0| X=1,X^{\prime} =0)\\ {P}_{{\rm{fin}}}(a=1,a^{\prime} =1| X=1,X^{\prime} =0)\\ {P}_{{\rm{fin}}}(a=0,a^{\prime} =0| X=1,X^{\prime} =1)\\ {P}_{{\rm{fin}}}(a=0,a^{\prime} =1| X=1,X^{\prime} =1)\\ {P}_{{\rm{fin}}}(a=1,a^{\prime} =0| X=1,X^{\prime} =1)\\ {P}_{{\rm{fin}}}(a=1,a^{\prime} =1| X=1,X^{\prime} =1)\end{array}\right)}_{{SM}}={\left(\begin{array}{c}p\\ 0\\ 0\\ 1-p\\ * \\ * \\ * \\ * \\ * \\ * \\ * \\ * \\ q\\ 0\\ 0\\ 1-q\end{array}\right)}_{{SM}},\end{eqnarray} \tag{ 11 }$$

where ∗are arbitrary, normalised entries and where ${P}_{{\rm{fin}}}(a=i,a^{\prime} =j| X=k,X^{\prime} =l)$ denotes the probability of obtaining the outcomes a = i and $a^{\prime} =j$ when performing the fiducial measurements X = k and $X^{\prime} =l$ on the system and memory respectively, in the final state ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$. This final state can be compressed since the only relevant and non-zero probabilities in ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$ occur when $X=X^{\prime}$ and $a=a^{\prime}$. We can then define new variables $\tilde{X}$ and $\tilde{a}$ such that $X=X^{\prime} =i\iff \tilde{X}=i$ and $a=a^{\prime} =j\iff \tilde{a}=j$ for i, j ∈ {0, 1} and ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$ can equivalently be written as in equation (12) which is clearly of the same form as ${P}_{{\rm{in}}}^{S}$

$$\begin{eqnarray}&&{{\bf{P}}}_{{\rm{fin}}}^{{SM}}\equiv {\left(\begin{array}{c}P(\tilde{a}=0| \tilde{X}=0)\\ P(\tilde{a}=1| \tilde{X}=0)\\ P(\tilde{a}=0| \tilde{X}=1)\\ P(\tilde{a}=1| \tilde{X}=1)\end{array}\right)}_{{SM}}={\left(\begin{array}{c}p\\ 1-p\\ q\\ 1-q\end{array}\right)}_{{SM}}.\end{eqnarray} \tag{ 12 }$$

Hence the initial state of the system, ${{\bf{P}}}_{{\rm{in}}}^{S}={\left(p1-p| q1-q\right)}^{T}$ (which is an arbitrary gbit state) is isomorphic to the final state of the system and memory, ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$ (as evident from equation (12)) with the same outcome probabilities for X = 0,1 and $\tilde{X}=0,1$. This implies that for every transformation ${{ \mathcal E }}_{S}$ on the former, there exists a transformation ${{ \mathcal E }}_{{SM}}$ on the latter such that for all outside agents A_j and for all p, q ∈ [0,1] (i.e. all possible input gbit states on the system), $\ \ {K}_{j}\phi [{{ \mathcal E }}_{S}({{\bf{P}}}_{{\rm{in}}}^{S})]\Rightarrow {K}_{j}\phi [{{ \mathcal E }}_{{SM}}\circ {{\bf{P}}}_{{\rm{fin}}}^{{SM}}]$, where ${{\bf{P}}}_{{SM}}^{{\rm{fin}}}=u({{\bf{P}}}_{S}^{{\rm{in}}})$. Thus any map u that maps ${{\bf{P}}}_{{\rm{in}}}^{{SM}}={{\bf{P}}}_{S}^{{\rm{in}}}\otimes {P}_{{\rm{in}}}^{M}$ to ${{\bf{P}}}_{{SM}}^{{\rm{fin}}}$ satisfies definition 7.

We now find a valid box world transformation that maps the initial state ${{\bf{P}}}_{{\rm{in}}}^{{SM}}$ (equation (10)) to any final state of the form ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$(equation (11)) which would correspond to the memory update map u.

Noting that all bipartite transformations in box world can be decomposed to a classical circuit of a certain form (see appendix B.1 or the origi-nal paper [4] for details), In figure C1, we construct an explicit circuit of this form that converts ${{\bf{P}}}_{{\rm{in}}}^{{SM}}$ to ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$. By construction, we only need to consider the case of $X=X^{\prime}$ since for $X\ne X^{\prime}$, the entries of ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}$ can be arbitrary and are irrelevant to the argument. For $X\ne X^{\prime}$, one can consider any such circuit description and it is easy to see that ${{\bf{P}}}_{{\rm{in}}}^{{SM}}={\left(p\ \ 1-p| q\ \ 1-q\right)}_{S}^{T}\otimes {\left(1\ \ 0| 1\ \ 0\right)}_{M}^{T}$ is indeed transformed into ${{\bf{P}}}_{{\rm{fin}}}^{{SM}}=(p\ \ 0\ \ 0\ \ 1-p| \ast \ \ \ast \ \ \ast \ \ \ast | \ast \ \ \ast \ \ \ast \ \ \ast | q\ \ 0\ \ 0\ \ 1-q{)}_{{SM}}^{T}$ through the map u defined by these sequence of steps. For example, if the circuit description for the $X\ne X^{\prime}$ case is same as that for the $X=X^{\prime}$ case, then the resultant memory update map is equivalent to the circuit of figure 9(a) which corresponds to performing a fixed measurement $X^{\prime} =0$ on the initial state of M and a classical CNOT on the output wire of M controlled by the output wire of S¹⁷ . The final state in that case is $(p\ \ 0\ \ 0\ \ 1-p| p\ \ 0\ \ 0\ \ 1-p| q\ \ 0\ \ 0\ \ 1-q| q\ \ 0\ \ 0\ \ 1-q{)}_{{SM}}^{T}$.

For higher dimensional systems S with n > 2 fiducial measurements, X ∈ {0, ..., n − 1} and k > 2 outcomes taking values a ∈ {0, ..., k − 1}, let b_n and b_k be the number of bits required to represent n and k in binary respectively. Then the memory M would be initialized to b_k copies of the pure state ${{\bf{P}}}_{{\rm{in}},n}^{M}=(1\ \ 0| ...| 1\ \ 0{)}_{M}^{T}$ which contains n identical blocks (one for each of the n fiducial measurements). One can then perform the procedure of figure C1 'bitwise' combining each output bit with one pure state of M and apply the same argument to obtain the result. For the specific case of the memory update transformation of figure 9(a), this would correspond to a bitwise CNOT on the output wires of S and M.□

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So far, we have considered a single agent measuring a system in her lab. We can also consider situations where multiple agents jointly share a state and measure their local parts of the state, updating their corresponding memories. One might wonder whether the initial correlations in the shared state are preserved once the agents measure it to update their memories (clearly the local measurement probabilities remain unaltered as we saw in this section). The answer is affirmative and this is what allows us to formulate the Frauchiger–Renner paradox in box world as done in the section 4, even though a coherent copy analogous to the quantum case does not exist here.

Theorem 12. Suppose that Alice and Bob share an arbitrary bipartite state ${{\bf{P}}}_{{\rm{in}}}^{{PR}}$ (which may be correlated), locally perform a fiducial measurement on their half of the state and store the outcome in their local memories A and B. Then the final joint state ${{\bf{P}}}_{{\rm{fin}}}^{\tilde{A}\tilde{B}}$ of the systems $\tilde{A}:= {PA}$ and $\tilde{B}:= {RB}$ as described by outside agents is isomorphic to ${{\bf{P}}}_{{\rm{in}}}^{{PR}}$ with the systems $\tilde{A}$ and $\tilde{B}$ taking the role of the systems P and R i.e. local memory updates by Alice and Bob preserve any correlations initially shared between them.

Proof. In the following, we describe the proof for the case where the bipartite system shared by Alice and Bob consists of 2 gbits, however, the result easily generalises to arbitrary higher dimensional systems by the argument presented in the last paragraph of the proof of theorem 11.

Let ${{\bf{P}}}_{{\rm{in}}}^{{PR}}$ be an arbitrary 2 gbit state with entries ${P}_{{\rm{in}}}({ab}={ij}| {XY}={kl})$ (i, j, k, l ∈ {0, 1}), which correspond to the joint probabilities of Alice and Bob obtaining the outcomes a = i and b = j when measuring X = k and Y = l on the P and R subsystems when sharing that initial state. Let $X^{\prime} ,a^{\prime} \in \{0,1\}$ and $Y^{\prime} ,b^{\prime} \in \{0,1\}$ be the fiducial measurements and outcomes for the memory systems A and B (also gbits) respectively. We describe the measurement and memory update process for each agent separately and characterise the final state of Alice's and Bob's systems and memories after the process as would appear to outside agents who do not have access to Alice and Bob's measurement outcomes. This analysis does not depend on the order in which Alice and Bob perform the measurement as the correlations are symmetric between them, so without loss of generality, we can consider Bob's measurement first and then Alice's.

Suppose that Bob's memory B is initialised to the state ${{\bf{P}}}_{{\rm{in}}}^{B}={{\bf{P}}}_{1}^{B}={\left(1\ \ 0| 1\ \ 0\right)}_{B}^{T}$. Then the joint initial state of the Alice's and Bob's system and Bob's memory as described by an agent Wigner outside Bob's lab is ${{\bf{P}}}_{{\rm{in}}}^{{PRB}}={{\bf{P}}}_{{\rm{in}}}^{{PR}}\otimes {{\bf{P}}}_{1}^{B}$. This can be expanded as follows where ${P}_{{\rm{in}}}({abb}^{\prime} ={ijk}| {XYY}^{\prime} ={lmn})$ represents the probability of obtaining the binary outcomes a = i, b = j, $b^{\prime} =k$ when performing the binary fiducial measurements X = l,Y = m, $Y^{\prime} =n$ on the initial state ${{\bf{P}}}_{{\rm{in}}}^{{PRB}}$.

$$\begin{eqnarray}&&{{\bf{P}}}_{{\rm{in}}}^{{PRB}}={\left(\begin{array}{c}{P}_{{\rm{in}}}({abb}^{\prime} =000| {XYY}^{\prime} =000)\\ {P}_{{\rm{in}}}({abb}^{\prime} =001| {XYY}^{\prime} =000)\\ {P}_{{\rm{in}}}({abb}^{\prime} =010| {XYY}^{\prime} =000)\\ {P}_{{\rm{in}}}({abb}^{\prime} =011| {XYY}^{\prime} =000)\\ {P}_{{\rm{in}}}({abb}^{\prime} =100| {XYY}^{\prime} =000)\\ {P}_{{\rm{in}}}({abb}^{\prime} =101| {XYY}^{\prime} =000)\\ {P}_{{\rm{in}}}({abb}^{\prime} =110| {XYY}^{\prime} =000)\\ {P}_{{\rm{in}}}({abb}^{\prime} =111| {XYY}^{\prime} =000)\\ \cdot \\ \cdot \\ \cdot \\ {P}_{{\rm{in}}}({abb}^{\prime} =000| {XYY}^{\prime} =111)\\ {P}_{{\rm{in}}}({abb}^{\prime} =001| {XYY}^{\prime} =111)\\ {P}_{{\rm{in}}}({abb}^{\prime} =010| {XYY}^{\prime} =111)\\ {P}_{{\rm{in}}}({abb}^{\prime} =011| {XYY}^{\prime} =111)\\ {P}_{{\rm{in}}}({abb}^{\prime} =100| {XYY}^{\prime} =111)\\ {P}_{{\rm{in}}}({abb}^{\prime} =101| {XYY}^{\prime} =111)\\ {P}_{{\rm{in}}}({abb}^{\prime} =110| {XYY}^{\prime} =111)\\ {P}_{{\rm{in}}}({abb}^{\prime} =111| {XYY}^{\prime} =111)\end{array}\right)}_{{PRB}}={\left(\begin{array}{c}{P}_{{\rm{in}}}({ab}=00| {XY}=00)\\ 0\\ {P}_{{\rm{in}}}({ab}=01| {XY}=00)\\ 0\\ {P}_{{\rm{in}}}({ab}=10| {XY}=00)\\ 0\\ {P}_{{\rm{in}}}({ab}=11| {XY}=00)\\ 0\\ \cdot \\ \cdot \\ \cdot \\ {P}_{{\rm{in}}}({ab}=00| {XY}=11)\\ 0\\ {P}_{{\rm{in}}}({ab}=01| {XY}=11)\\ 0\\ {P}_{{\rm{in}}}({ab}=10| {XY}=11)\\ 0\\ {P}_{{\rm{in}}}({ab}=11| {XY}=11)\end{array}\right)}_{{PRB}}\end{eqnarray} \tag{ 13 }$$

${{\bf{P}}}_{{\rm{in}}}^{{PRB}}$ has 8 blocks ${G}_{{XYY}^{\prime} }$, one for each value of $(X,Y,Y^{\prime} )$ and is a product state with 4 equal pairs of blocks, ${G}_{000}^{{\rm{in}}}={G}_{001}^{{\rm{in}}}$, ${G}_{010}^{{\rm{in}}}={G}_{011}^{{\rm{in}}}$, ${G}_{100}^{{\rm{in}}}={G}_{101}^{{\rm{in}}}$, ${G}_{110}^{{\rm{in}}}={G}_{111}^{{\rm{in}}}$ since both measurements on the initial state of B give the same outcome.

Now, the outside observer Wigner will describe the transformation on RB through the memory update map u of figure C1. Let ${{\bf{P}}}_{{\rm{fin}}}^{{PRB}}$ be the final state that results by applying this map to the systems RB in the initial state ${{\bf{P}}}_{{\rm{in}}}^{{PRB}}$. Any transformation on a system characterised by n fiducial measurements with k outcomes each can be represented by a nk × nk block matrix where each block is a k × k matrix (see [4] for further details), for the system RB, n = k = 4 and the memory update map u_RB would be a 16 × 16 block matrix of the following form where each u_ij is a 4 × 4 matrix.

$$\begin{eqnarray*}{u}_{{RB}}={\left(\begin{array}{ccccc}{u}_{11} & \cdot & \cdot & \cdot & {u}_{14}\\ \cdot & & & & \cdot \\ \cdot & & & & \cdot \\ \cdot & & & & \cdot \\ {u}_{41} & \cdot & \cdot & \cdot & {u}_{44}\end{array}\right)}_{{RB}}.\end{eqnarray*}$$

Here, the first 4 rows decide the entries in the first block of the transformed matrix, the next 4, the second block and so on. Noting that the memory update transformation (figure C1) merely permutes elements within the relevant blocks (and does not mix elements between different blocks), the only non-zero blocks of u_RB are the diagonal ones u_ii. Further, by the same argument as in theorem 11, the only relevant entries in the transformed state are when the same fiducial measurement is performed on Bob's system R and memory B i.e. only cases where $Y=Y^{\prime}$. The remaining measurement choices maybe arbitrary for the final state (just as they are for $X\ne X^{\prime}$ in equation (11)). This means that among the 4 diagonal blocks, only 2 of them are relevant. The 4 fiducial measurements on RB are ${YY}^{\prime} =00,01,10,11$ and in that order, only the first and fourth are relevant since they correspond to $Y=Y^{\prime}$. Within these relevant blocks (in this case u₁₁ and u₄₄), the operation is a CNOT on the output $b^{\prime}$ controlled by the output b and we have the following matrix representation of the memory update map u of figure C1 ¹⁸ .

$$\begin{eqnarray}{u}_{{RB}}={\left(\begin{array}{cccc}{CN} & 0 & 0 & 0\\ 0 & * & 0 & 0\\ 0 & 0 & * & 0\\ 0 & 0 & 0 & {CN}\end{array}\right)}_{{RB}},\ \ {CN}=\left(\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{array}\right),\end{eqnarray} \tag{ 14 }$$

where 0 represents the 4 × 4 null matrix and blocks labelled ∗ can be arbitrary. The final state ${{\bf{P}}}_{{\rm{fin}}}^{{PRB}}$ as seen by Wigner is then

$$\begin{eqnarray}&&{{\bf{P}}}_{{\rm{fin}}}^{{PRB}}=({{ \mathcal I }}_{P}\otimes {u}_{{RB}}){{\bf{P}}}_{{\rm{in}}}^{{PRB}}=({{ \mathcal I }}_{P}\otimes {u}_{{RB}})\left[{{\bf{P}}}_{{\rm{in}}}^{{PR}}\otimes {\left(\begin{array}{c}1\\ 0\\ 1\\ 0\end{array}\right)}_{B}\right],\end{eqnarray} \tag{ 15 }$$

where ${{ \mathcal I }}_{P}$ is the identity transformation on the P system. Since the CN blocks are the only relevant blocks in u_RB and each block of ${{\bf{P}}}_{{\rm{in}}}^{{PRB}}$ has the same pattern of non-zero and zero entries (equation (13)), it is enough to look at the action of ${{ \mathcal I }}_{P}\otimes {CN}$ on the first block ${G}_{000}^{{\rm{in}}}$ of ${{\bf{P}}}_{{\rm{in}}}^{{PRB}}$. Noting that ${{ \mathcal I }}_{P}$ is a 2 × 2 identity matrix, we have

$$\begin{eqnarray*}\begin{array}{rcl}({{ \mathcal I }}_{P}\otimes {CN}){G}_{000}^{{\rm{in}}} & = & \left(\begin{array}{cccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\end{array}\right)\left(\begin{array}{c}{P}_{{\rm{in}}}({ab}=00| {XY}=00)\\ 0\\ {P}_{{\rm{in}}}({ab}=01| {XY}=00)\\ 0\\ {P}_{{\rm{in}}}({ab}=10| {XY}=00)\\ 0\\ {P}_{{\rm{in}}}({ab}=11| {XY}=00)\\ 0\end{array}\right)\,=\,{G}_{000}^{{\rm{fin}}}\\ & = & \left(\begin{array}{c}{P}_{{\rm{in}}}({ab}=00| {XY}=00)\\ 0\\ 0\\ {P}_{{\rm{in}}}({ab}=01| {XY}=00)\\ {P}_{{\rm{in}}}({ab}=10| {XY}=00)\\ 0\\ 0\\ {P}_{{\rm{in}}}({ab}=11| {XY}=00)\end{array}\right)\,=\,\left(\begin{array}{c}{P}_{{\rm{fin}}}({abb}^{\prime} =000| {XYY}^{\prime} =000)\\ {P}_{{\rm{fin}}}({abb}^{\prime} =001| {XYY}^{\prime} =000)\\ {P}_{{\rm{fin}}}({abb}^{\prime} =010| {XYY}^{\prime} =000)\\ {P}_{{\rm{fin}}}({abb}^{\prime} =011| {XYY}^{\prime} =000)\\ {P}_{{\rm{fin}}}({abb}^{\prime} =100| {XYY}^{\prime} =000)\\ {P}_{{\rm{fin}}}({abb}^{\prime} =101| {XYY}^{\prime} =000)\\ {P}_{{\rm{fin}}}({abb}^{\prime} =110| {XYY}^{\prime} =000)\\ {P}_{{\rm{fin}}}({abb}^{\prime} =111| {XYY}^{\prime} =000)\end{array}\right),\end{array}\end{eqnarray*}$$

where ${P}_{{\rm{fin}}}({abb}^{\prime} ={ijk}| {XYY}^{\prime} ={lmn})$ represents the probability of obtaining the outcomes a = i, b = j, $b^{\prime} =k$ when performing the fiducial measurements X = l,Y = m, $Y^{\prime} =n$ on the final state ${{\bf{P}}}_{{\rm{fin}}}^{{PRB}}$ and ${G}_{000}^{{\rm{fin}}}$ is the first block of this final state. Clearly the only non-zero outcome probabilities are when $b=b^{\prime}$ and this allows us to compress the final state by defining $\tilde{b}=i\iff b=b^{\prime} =i$ for i ∈ {0, 1} and we have the following

$$\begin{eqnarray*}&&({{ \mathcal I }}_{P}\otimes {CN}){G}_{000}^{{\rm{in}}}\equiv \left(\begin{array}{c}{P}_{{\rm{in}}}({ab}=00| {XY}=00)\\ {P}_{{\rm{in}}}({ab}=01| {XY}=00)\\ {P}_{{\rm{in}}}({ab}=10| {XY}=00)\\ {P}_{{\rm{in}}}({ab}=11| {XY}=00)\end{array}\right)=\left(\begin{array}{c}{P}_{{\rm{fin}}}(a\tilde{b}=00| {XYY}^{\prime} =000)\\ {P}_{{\rm{fin}}}(a\tilde{b}=01| {XYY}^{\prime} =000)\\ {P}_{{\rm{fin}}}(a\tilde{b}=10| {XYY}^{\prime} =000)\\ {P}_{{\rm{fin}}}(a\tilde{b}=11| {XYY}^{\prime} =000)\end{array}\right)={G}_{00}^{{\rm{in}}}.\end{eqnarray*}$$

Here ${G}_{00}^{{\rm{in}}}$ is the first block of the initial state ${{\bf{P}}}_{{\rm{in}}}^{{PR}}$ and we have that the first block of the final state of PRB is equivalent (up to zero entries) to the first block of the initial state over PR alone or ${G}_{000}^{{\rm{fin}}}={G}_{00}^{{\rm{in}}}$. Among the 8 blocks of ${{\bf{P}}}_{{\rm{fin}}}^{{PRB}}$, only the 4 blocks ${G}_{000}^{{\rm{fin}}}$, ${G}_{011}^{{\rm{fin}}}$, ${G}_{100}^{{\rm{fin}}}$ and ${G}_{111}^{{\rm{fin}}}$ are the relevant ones (since $Y=Y^{\prime}$ for these) and we can similarly show that ${G}_{011}^{{fin}}\equiv {G}_{01}^{{in}}$, ${G}_{100}^{{\rm{fin}}}\equiv {G}_{10}^{{\rm{in}}}$ and ${G}_{111}^{{\rm{fin}}}\equiv {G}_{11}^{{\rm{in}}}$ for the remaining 3 relevant blocks. Defining $\tilde{Y}=i\iff Y=Y^{\prime} =i$ for i ∈ {0, 1}, we obtain

$$\begin{eqnarray}&&{{\bf{P}}}_{{\rm{fin}}}^{{PRB}}={{\bf{P}}}_{{\rm{fin}}}^{P\tilde{B}}\equiv \left(\begin{array}{c}{P}_{{\rm{fin}}}(a\tilde{b}=00| X\tilde{Y}=00)\\ {P}_{{\rm{fin}}}(a\tilde{b}=01| X\tilde{Y}=00)\\ {P}_{{\rm{fin}}}(a\tilde{b}=10| X\tilde{Y}=00)\\ {P}_{{\rm{fin}}}(a\tilde{b}=11| X\tilde{Y}=00)\\ {P}_{{\rm{fin}}}(a\tilde{b}=00| X\tilde{Y}=01)\\ {P}_{{\rm{fin}}}(a\tilde{b}=01| X\tilde{Y}=01)\\ {P}_{{\rm{fin}}}(a\tilde{b}=10| X\tilde{Y}=01)\\ {P}_{{\rm{fin}}}(a\tilde{b}=11| X\tilde{Y}=01)\\ {P}_{{\rm{fin}}}(a\tilde{b}=00| X\tilde{Y}=10)\\ {P}_{{\rm{fin}}}(a\tilde{b}=01| X\tilde{Y}=10)\\ {P}_{{\rm{fin}}}(a\tilde{b}=10| X\tilde{Y}=10)\\ {P}_{{\rm{fin}}}(a\tilde{b}=11| X\tilde{Y}=10)\\ {P}_{{\rm{fin}}}(a\tilde{b}=00| X\tilde{Y}=11)\\ {P}_{{\rm{fin}}}(a\tilde{b}=01| X\tilde{Y}=11)\\ {P}_{{\rm{fin}}}(a\tilde{b}=10| X\tilde{Y}=11)\\ {P}_{{\rm{fin}}}(a\tilde{b}=11| X\tilde{Y}=11)\end{array}\right)=\left(\begin{array}{c}{P}_{{\rm{in}}}({ab}=00| {XY}=00)\\ {P}_{{\rm{in}}}({ab}=01| {XY}=00)\\ {P}_{{\rm{in}}}({ab}=10| {XY}=00)\\ {P}_{{\rm{in}}}({ab}=11| {XY}=00)\\ {P}_{{\rm{in}}}({ab}=00| {XY}=01)\\ {P}_{{\rm{in}}}({ab}=01| {XY}=01)\\ {P}_{{\rm{in}}}({ab}=10| {XY}=01)\\ {P}_{{\rm{in}}}({ab}=11| {XY}=01)\\ {P}_{{\rm{in}}}({ab}=00| {XY}=10)\\ {P}_{{\rm{in}}}({ab}=01| {XY}=10)\\ {P}_{{\rm{in}}}({ab}=10| {XY}=10)\\ {P}_{{\rm{in}}}({ab}=11| {XY}=10)\\ {P}_{{\rm{in}}}({ab}=00| {XY}=11)\\ {P}_{{\rm{in}}}({ab}=01| {XY}=11)\\ {P}_{{\rm{in}}}({ab}=10| {XY}=11)\\ {P}_{{\rm{in}}}({ab}=11| {XY}=11)\end{array}\right)={{\bf{P}}}_{{\rm{in}}}^{{PR}}\end{eqnarray} \tag{ 16 }$$

Equation (16) shows that final state ${{\bf{P}}}_{{\rm{fin}}}^{P\tilde{B}}$ of Alice's system P, Bob's system R and Bob's memory B after Bob's local memory update is isomorphic to the initial state ${{\bf{P}}}_{{\rm{in}}}^{{PR}}$ shared by Alice and Bob, having the same outcome probabilities as the latter for all the relevant measurements. Thus the initial correlations present in ${{\bf{P}}}_{{\rm{in}}}^{{PR}}$ are preserved after Bob locally updates his memory according to the update procedure of figure C1. One can now repeat the same argument for Alice's local memory update taking ${{\bf{P}}}_{{\rm{fin}}}^{P\tilde{B}}\otimes (1\ \ 0| 1\ \ 0{)}_{A}^{T}$ to be the initial state and by analogously defining $\tilde{s}=i\iff s=s^{\prime} =i$ for s∈{a, X},i ∈  {0, 1}, we have the required result that the final state after both parties perform their local memory updates (as described by outside agents Ursula and Wigner) is isomorphic and operationally equivalent to the initial state shared by the parties before the memory update

$$\begin{eqnarray}&&{{\bf{P}}}_{{\rm{fin}}}^{{PARB}}={{\bf{P}}}_{{\rm{fin}}}^{\tilde{A}\tilde{B}}\equiv {{\bf{P}}}_{{\rm{in}}}^{{PR}}.\end{eqnarray} \tag{ 17 }$$

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