Quantum scrambling via accessible tripartite information

Given a tripartite state ρ in a space $\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_{C}$, its tripartite information is defined as

$$\begin{equation} I_3(\rho) \equiv I(A:B)_\rho + I(A:C)_\rho - I(A:BC)_\rho, \end{equation} \tag{ 1 }$$

where $I(A:B)_\rho$ is the QMI between A and B,

$$\begin{equation} I(A:B)_\rho \equiv S(\rho_A)+S(\rho_B)-S(\rho_{AB}), \end{equation} \tag{ 2 }$$

and $S(\cdot)$ is the von Neumann entropy. An equivalent expression for I₃ is

$$\begin{equation} \begin{gathered} I_3(\rho) = S(\rho_A) + S(\rho_B) + S(\rho_C) \\ - S(\rho_{AB}) - S(\rho_{AC}) - S(\rho_{BC}) + S(\rho_{ABC}), \end{gathered} \end{equation} \tag{ 3 }$$

which highlights its symmetry with respect to the three subspaces.

Another feature of I₃ is that for any pure four-partite state $\vert \psi\rangle\in\mathcal{H}_{ABCD}$, all the tripartite informations computed on reduced tripartite states are identical [32], that is,

$$\begin{equation} I_3(\rho_{ABC}) = I_3(\rho_{ABD}) = I_3(\rho_{ACD}) = I_3(\rho_{BCD}). \end{equation} \tag{ 4 }$$

One possible intuition underlying I₃ is that it quantifies the correlations between A and BC which cannot be accessed via local measurements on B and C. In other words, it quantifies the amount of information about A that is 'hidden' in the correlations between B and C. If $I_3\lt0$, then BC might be expected to contain more information about A than B and C do, if taken separately. It was further argued [4] that the tripartite information might serve as a quantifier of genuine four-partite entanglement for pure states.

We can get some intuition on I₃ by considering its classical counterpart. We will denote the tripartite information computed with respect to a classical probability distribution P as $J_3(P)\equiv J_3(A:B:C)_P$. For a balanced probability distribution over the three-bit events 000 and 111 we have:

$$\begin{equation} p(000) = p(111) = \frac12. \end{equation} \tag{ 5 }$$

This gives $J_3 = 1$. If any one of the bits is uncorrelated with the others, for example if the distribution is

$$\begin{equation} p(001) = p(111) = \frac12, \end{equation} \tag{ 6 }$$

then we get $J_3 = 0$. Finally, a balanced distribution of the form

$$\begin{equation} p(000) = p(011) = p(101) = p(110) = \frac14, \end{equation} \tag{ 7 }$$

gives $J_3 = -1$. We can interpret these results saying that J₃ quantifies the redundancy with which one of the bits is encoded in the other two. In the case of equation (5), the first bit is encoded redundantly in the last two. For equation (6), only one output bit is useful, hence the vanishing redundancy. And for equation (7), 'negative redundancy' means that information can only be retrieved by measuring both of the other two bits at the same time, hence there is more information into the pair of bits than there is when they are taken individually. However, when the mutual information is replaced by the QMI and is applied to generic quantum states, as we will see, this straightforward classical interpretation can get significantly more involved, due to the non-accessible components of the associated QMIs, and the fact that the definition of I₃ does not take into account the choice of encoding basis.

One important issue about I₃ is that it does not directly quantify correlations between input and output measurement probabilities, owing to its definition via QMIs given in equation (1). In other words, the value of I₃ might not be a faithful representation of the amount of information about the input that is lost in correlations between different outputs. This is due to the well-known fact that the correlations measured by the QMI are not generally directly accessible via local measurements [35–38]. Different states can result in identical correlations between measurement outcomes, but nonetheless have different QMIs. For example, $\rho_1 = \mathbb{P}\left[\frac1{\sqrt2}(|00\rangle+|11\rangle)\right]$ and $\rho_2 = \frac12(\mathbb{P}_{00}+\mathbb{P}_{11})$, both produce fully correlated outcomes when both parties measure in the computational basis, and yet $I(A:B)_{\rho_1} = 2 I(A:B)_{\rho_2} = 2$. Here and in the following, we use the shorthand notation

$$\begin{equation} \mathbb{P}(\vert \psi\rangle)\equiv \vert{\psi}\rangle\langle{\psi}\vert \end{equation} \tag{ 8 }$$

to denote the projection onto a ket state $\vert \psi\rangle$. When clear from the context, we will also employ the equivalent notation $\mathbb P_\psi\equiv \mathbb P(|\psi\rangle)$.

The QMI not directly quantifying accessible correlations [37, 38] directly propagates into similar issues for I₃. For example, one might observe that a fully correlated three-qubit separable state of the form $\rho = \frac12(\mathbb{P}_{000}+\mathbb{P}_{111})$ gives $I_3(\rho) = 1$, whereas a GHZ state $\vert \mathrm{GHZ}\rangle\equiv\frac1{\sqrt2}(\vert 000\rangle+\vert 111\rangle)$ gives $I_3(\mathbb{P}_{\mathrm{GHZ}}) = 0$, despite both states translating into an identical degree of retrievability of the information about A from B and C.

Another direct way to see that I₃ might not be the best quantity to consider is to observe that $I_3(\mathbb{P}_\psi) = 0$ for all $\vert \psi\rangle$, that is, all pure states correspond to the same value of the tripartite information, even though pure states might correspond to entirely different ways to distribute information among the three parties. This suggests that the effects of the quantum discord components in I₃ are sufficient to conceal important properties of the correlations between different subsystems.

One of the most popular approaches [4] to quantify the QIS of a unitary dynamics [39] $U:\mathcal{H}_A\otimes\mathcal{H}_B\to\mathcal{H}_C\otimes\mathcal{H}_D$, where $\mathcal{H}_A\otimes\mathcal{H}_B = \mathcal{H}_C\otimes\mathcal{H}_D$, involves evaluating the I₃ on a tripartite state ρ_U defined as

$$\begin{equation} \rho_U\equiv \mathrm{Tr}_M \mathbb{P} \left[ (U_{AB\to CD}\otimes I_{RM}) (\vert \Psi^+\rangle_{AR}\!\otimes\!\vert \Psi^+\rangle_{BM}) \right], \end{equation} \tag{ 9 }$$

where $\vert \Psi^+\rangle_{XY}$ is a maximally entangled state in the partitions $\mathcal{H}_X\otimes \mathcal{H}_Y$. The state in equation (9) has a simple diagrammatic representation, given in figure 1. The intuition behind this method is to map the channel into a state and to study the QIS properties of the channel through the same entropy-based toolkit we use for states. It was then argued [4] that $I_3(\rho_U)\lt0$ witnesses the presence of QIS.

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One way to understand the partial trace used to define ρ_U is to observe that it amounts to probing the properties of the state obtained on the output modes $R,C,D$ (c.f. figure 1) when we do not know anything about the input used in B. Indeed, the same state ρ_U is obtained by letting the input state B be the maximally mixed state. Moreover, by measuring M, the corresponding state conditioned to any measurement outcome is pure, and thus the corresponding I₃ zero. This implies that $I_3(\rho_U)$ can be nonzero only if we introduce some effective partial ignorance about the input state.

Another issue with the use of maximally entangled states in the definition of ρ_U is that the QMI of a bipartite state of the form $(\Phi\otimes I)\mathbb{P}[\vert \Psi^+\rangle]$ quantifies the entanglement-assisted classical transmission rate through the channel Φ [40]. That is to say, $I(A:B)_{\rho_\Phi}$ with $\rho_\Phi\equiv(\Phi\otimes I)\mathbb{P}[\vert \Psi^+\rangle]$ quantifies how much classical information can be sent through the channel, under the assumption that sender and receiver share the entangled state $\vert \Psi^+\rangle$ [41]. It follows that the QMIs in $I_3(\rho_U)$ quantify correlations between input and outputs that are only accessibles assuming these share entangled states, and thus so does $I_3(\rho_U)$. However, in most contexts where QIS is studied, this is not the case. Rather, one is interested in how much information is recoverable from the outputs, without assuming additional knowledge of correlation between them.

Finally, for any unitary U, we have $I_3(\rho_U)\unicode{x2A7D} 0$, which implies that the sign of $I_3(\rho_U)$ does not give useful information in such cases. To see this, given any state of the form equation (9), observe that the reduced states on RM and CD are always product states, and the entropies satisfy

$$\begin{equation} \begin{gathered} S(CDR)=S(M), \\ S(CD) = S(RM) = S(R)+S(M). \end{gathered} \end{equation} \tag{ 10 }$$

It follows that I₃ reads

$$\begin{equation} I_3(CDR) = S(C) + S(D) - S(RC) - S(RD), \end{equation} \tag{ 11 }$$

and thus, using the strong subadditivity of the von Neumann entropy, we always have $I_3(CDR)\le0$. A similar argument was presented in [42]. This is similar to known results in the context of holography: assuming the Ryu–Takajanagi formula holds [43], the tripartite information is always negative in gravitational theories with holographic duals [31]. This feature is dubbed 'monogamy of holographic mutual information' and is considered evidence that quantum gravity should possess some scrambling features.

Example with perfect tensors — A particularly clear way to see that highly negative values of I₃ might not correspond to scrambling dynamics, is to consider a unitary $U:\mathbb{C}^d\otimes\mathbb{C}^d\to \mathbb{C}^d\otimes\mathbb{C}^d$, for some odd d, derived from a four-party perfect tensor, of the form [42]:

$$\begin{equation} U \vert i\rangle\vert j\rangle = \vert i+j\rangle \vert i-j\rangle, \end{equation} \tag{ 12 }$$

where all arithmetics is modulo d. The corresponding state ρ_U gives $I_3(\rho_U) = -2\log d$, which might lead to the conclusion that U maximally scrambles input information. However, if classical information is encoded in the choice of $\vert i\rangle$, fixing the value of $\vert j\rangle$, then local measurements can reliably recover the input information. For example, if we fix j = 0, the evolution reads

$$\begin{equation} U \vert i,0\rangle = \vert i\rangle\vert i\rangle. \end{equation} \tag{ 13 }$$

This shows that there is a way to encode classical information in the input states, such that said information is perfectly recoverable from local measurements of the output.

An alternative way to use the tripartite information to quantify QIS of a unitary U, used in [5, 6], is to focus on correlations with respect to a single input system, using a setup like the one in figure 2. In this approach, given a unitary $U_{AB\to CDE}:\mathcal{H}_A\otimes\mathcal{H}_B\to\mathcal{H}_C\otimes\mathcal{H}_D\otimes\mathcal{H}_E$, the goal is to investigate how information in the input state propagates and is recoverable from different outputs defining a tripartition of the output modes, as shown in figure 2. Given the four-partite state

$$\begin{equation} \vert \psi\rangle_{RCDE} \equiv (U_{AB\to CDE}\otimes I_R) (\vert \Psi^+\rangle_{AR}\otimes \vert 0_B\rangle), \end{equation} \tag{ 14 }$$

with $\vert 0_B\rangle$ some fixed input state in the mode B, the QIS is quantified via

$$\begin{equation} I_3(R:C:D) \equiv I_3(\mathrm{Tr}_E(\mathbb{P}_{\psi})). \end{equation} \tag{ 15 }$$

The state $\vert \psi\rangle_{RCDE}$ can also be equivalently defined via an isometry $V_{A\to CDE}$ as

$$\begin{equation} \vert \psi\rangle_{RCDE} \equiv (V_{A\to CDE}\otimes I_R) \vert \Psi^+\rangle_{AR}. \end{equation} \tag{ 16 }$$

The corresponding QIS is then defined analogously to the previous case, as $I_3(R:C:D) = I_3(\rho_V)$ where $\rho_V = \mathrm{tr}_E(\mathbb P_\psi)$.

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However, as we will show through explicit examples, this 'single-arm approach' to quantifying QIS can also result in misleading values of I₃.

Example with $I_3 = 0$ — Consider the isometry $V:\mathbb{C}^2\to (\mathbb{C}^2)^{\otimes 3}$ defined by

$$\begin{equation} \begin{gathered} V \vert 0\rangle = \frac{\vert 00\rangle+\vert 11\rangle}{\sqrt2}\vert 0\rangle, \qquad V \vert 1\rangle = \frac{\vert 01\rangle+\vert 10\rangle}{\sqrt2}\vert 0\rangle. \end{gathered} \end{equation} \tag{ 17 }$$

The corresponding quantum tripartite information is $I_3(\rho_V) = 0$. On the other hand, the classical probability distribution resulting from encoding information in the computational basis and measuring the first two output qubits, results in a joint distribution identical to equation (7), which gives $J_3 = -1$. Measuring the same two output qubits but encoding information in the basis $\{\vert +\rangle,\vert -\rangle\}$, the evolution reads

$$\begin{equation} \vert \pm\rangle \to \vert \pm\pm\rangle\vert 0\rangle, \end{equation} \tag{ 18 }$$

meaning that information is locally retrievable from both of the first two output qubits, which is akin to the classical example in equation (5), where instead $J_3 = 1$. Furthermore, measuring different combinations of output qubits, we get yet different values of J₃: measuring first and third—or second and third—output qubits, we get $J_3 = 0$ for any choice of input encoding.

Example with $I_3 = 1$ — As another toy example, consider the isometry V acting on single-qubit inputs as

$$\begin{equation} V\vert 0\rangle= \frac{\vert 000\rangle+\vert 111\rangle}{\sqrt2}, \quad V\vert 1\rangle= \frac{\vert 000\rangle-\vert 111\rangle}{\sqrt2}. \end{equation} \tag{ 19 }$$

The associated ρ_V gives $I_3(\rho_V) = 1$. This would seem to indicate that V has no scrambling effects, i.e. input information ends up encoded locally into the output qubits. However, V behaves in exactly the opposite way: unless all three output qubits are measured simultaneously, there is no way to discriminate between the input states $\vert 0\rangle$ and $\vert 1\rangle$, meaning that V fully hides such information into the output correlations. At the same time, we observe that

$$\begin{equation} V\vert +\rangle= \vert 000\rangle, \qquad V\vert -\rangle= \vert 111\rangle, \end{equation} \tag{ 20 }$$

which means that the information encoded in the $\{\vert +\rangle,\vert -\rangle\}$ input basis is encoded in a maximally local way into the output degrees of freedom. This example shows very clearly that, even though there is a choice of input encoding which gives $J_3 = 1$, consistently with the result $I_3(\rho_V) = 1$, one cannot simply state that V 'does not scramble input information': whether this happens depends entirely on the way said information has been encoded in the input states.

Example with $I_3 = -1$ — Finally, consider the isometry defined as

$$\begin{equation} V\vert 0\rangle = \frac{\vert 000\rangle+\vert 111\rangle}{\sqrt2}, \quad V\vert 1\rangle = \frac{\vert 001\rangle+\vert 110\rangle}{\sqrt2}. \end{equation} \tag{ 21 }$$

This gives $I_3(\rho_V) = -1$, which should correspond to a high degree of scrambling. This is true when information is encoded in the computational basis. However, observe that in a different basis this isometry gives

$$\begin{equation} V\vert \pm\rangle = \left( \frac{\vert 00\rangle \pm \vert 11\rangle}{\sqrt2} \right)\vert \pm\rangle, \end{equation} \tag{ 22 }$$

which means that measuring either only the third qubit, or the first two qubits together, allows to recover deterministically the information encoded in the $\{\vert +\rangle,\vert -\rangle\}$ basis. This being in contrast with the maximally negative tripartite information" which would instead suggest V to be maximally scrambling.

Conclusions — The examples in equations (17), (19) and (21) show that the values of I₃ need to be interpreted with some degree of caution, as they do not always faithfully represent how much the corresponding dynamics makes information non-retrievable from local measurements of the output. In the later sections we will show how these issues can be overcome by suitably modifying the definition of QIS in terms of accessible tripartite informations.

The examples in equations (17), (19) and (21) show that the amount of information that can be locally retrieved from a subsystem of the output strongly depends on the choice of encoding states. However, the tripartite information evaluated as in equation (9) or equation (14) is not sensible to the choice of input encoding. In fact, in the setup of figures 1 and 2, the target unitary U (or isometry V) acts on the maximally entangled state $\vert \Psi\rangle_{RA}$. This corresponds to assuming complete ignorance about the input state; as a consequence, the tripartite information evaluated with this method does not depend on the basis used to encode information in the input states.

An explicit way to show this is to observe that because A contains the states fed into the evolution, applying a local unitary operation on A, by replacing $\vert \Psi_{AR}^+\rangle$ with $(I_R\otimes U)\vert \Psi_{AR}^+\rangle$, corresponds to changing the basis in which information is encoded. However, a general property of maximally entangled states is that for any local unitary U,

$$\begin{equation} (I_R\otimes U)\vert \Psi_{AR}^+\rangle = (U^T \otimes I_A)\vert \Psi_{AR}^+\rangle. \end{equation} \tag{ 23 }$$

In other words, changing the input entangled state by a local unitary operation on A is equivalent to changing it applying a local unitary operation on R. Since R is an auxiliary space and the choice of basis for it does not affect the value of I₃, this implies that the tripartite information is encoding-independent, as previously claimed. Note that, as we will show in the later sections, defining QIS via accessible mutual informations, does instead allow to naturally capture the importance of the input encoding basis.

Many of the issues discussed in section 2 are associated with the non-accessibility of QMIs. These can be avoided by using instead accessible informations [38, 44]. More precisely, since the goal is quantifying the correlations between an input $\vert \psi\rangle$ and the information retrievable from a subset S of the output system $V\vert \psi\rangle$, it follows that the most relevant quantities are the capacities [38, 40, 44] of the channel $\Phi_S$ defined as

$$\begin{equation} \Phi_S(\rho)\equiv\mathrm{Tr}_{\bar S}(V \rho V^\dagger), \end{equation} \tag{ 24 }$$

where $\mathrm{Tr}_{\bar S}$ denotes the partial trace with respect to all degrees of freedom except for the ones in S. Here V is a generic isometry like the ones introduced in section 2.4. Different kinds of channel capacities quantify achievable rates of classical and quantum information transmission. In the context of QIS, it is not generally clear which type of information is the most relevant. Here we focus on quantifying the amount of information transmissible via a given dynamics and recoverable via local measurements.

However, the classical capacity of a channel quantifies the rate of information transmission achievable over many uses of the channel, when sender and receiver are allowed to use entangled input states and global measurements on the corresponding outcomes. In the context of QIS, this might not be the most appropriate quantity to focus on: both if we think of QIS as the amount of information retrievable about input states, and if we are interested in QIS from the perspective of the black-hole information paradox, considering correlations that are only accessible using input states that are entangled across multiple injections in the channel may not be the relevant figure of merit. Rather, we have to consider correlations accessible without the use of entanglement between sender and receiver, and without nonlocal measurements. The associated quantity is sometimes referred to as the 'accessible information of the channel', or $C_{1,1}$ capacity [38, 44].

Accessible information of a channel — Given a bipartite state ρ, its accessible information is the classical mutual information of the joint probability distribution corresponding to a choice of local measurements, maximized over all possible such choices. That is,

$$\begin{equation} I_{\mathrm{acc}}(\rho) \equiv \sup_{\mu^A,\mu^B} I(A:B)_p, \end{equation} \tag{ 25 }$$

where the classical probability distribution p is given by

$$\begin{equation} p(a,b) = \mathrm{Tr}[(\mu^A_a\otimes\mu^B_b)\rho], \end{equation} \tag{ 26 }$$

and we maximize over the possible local POVMs $\{\mu_a^A\}$ and $\{\mu^B_b\}$. On the other hand, the accessible information of a channel Φ is the maximal accessible information obtainable varying over all possible ways to encode information at the input, and all possible ways to measure the output state to retrieve. This can be written formally as the accessible information of states of the form $(I_R\otimes\Phi)\rho_{cq}$, maximized over all possible classical-quantum states ρ_cq . Note that classical-quantum states are those separable states admitting a decomposition of the form

$$\begin{equation} \rho_{cq}=\sum_x p_x \mathbb{P}_x\otimes \rho_x \end{equation} \tag{ 27 }$$

for some input ensemble $(p_x,\rho_x)$, and with $\mathbb{P}_x$ projections over an orthonormal basis for the auxiliary space R. The introduction of this auxiliary space R is only a formal tool to build into the state the knowledge that the different states ρ_x refer to states fed into the channel at different times. In summary, we define the accessible information of a channel Φ as

$$\begin{equation} I_{\mathrm{acc}}(\Phi) \equiv \sup_{\rho_{cq}} I_{\mathrm{acc}}((I_R\otimes\Phi)\rho_{cq}). \end{equation} \tag{ 28 }$$

We can note the tight connection between this accessible information in (28) and the QMI, we can also write it as

$$\begin{equation} I_{\mathrm{acc}}(\Phi) = \sup_{\rho_{cq},\mu} I\left( (I_R\otimes \Delta_\mu) (I_R\otimes \Phi) \rho_{cq} \right), \end{equation} \tag{ 29 }$$

where $\Delta_\mu(\rho)\equiv \sum_y \mathrm{tr}(\mu_y\rho)\,\mathbb{P}_y$ is the channel describing the decoherence induced by measuring with the POVM µ. It is also worth observing that if we remove the measurement decoherence $\Delta_\mu$, and allow for arbitrary input states ρ, we recover a definition analogous to (15).

In the following, the accessible information will be always understood as evaluated on the evolved classical-quantum state $\tilde{\rho} = I_R\otimes\Phi_{A\rightarrow S\overline{S}}(\rho_{cq})$ :

$$\begin{equation} I_{\text{acc}}(R:S)_{\tilde{\rho}}=I_{\text{acc}}(\text{Tr}_{\overline{S}}\,\tilde{\rho})\,. \end{equation} \tag{ 30 }$$

Whenever the subscript $\Phi(\rho_{cq})$ is omitted, $I_\text{acc}(R:S)$ is understood as optimized over the choice of classical-quantum state ρ_cq .

This latter formulation, in particular, shows the similarities, and differences, with the way the QMI was calculated in sections 2.3 and 2.4. We can also consider the mutual information in equation (28) for fixed choices of input ensembles, which tells us the maximum rate of information transmission that can be substained by the channel for a fixed input encoding, whose letters are the states ρ_x occurring with probabilities p_x . Computing classical capacities—and in particular $C_{1,1}$ capacities—is in general a daunting task, due to the maximization required for the calculation. We focus here on examples where this calculation can be performed, to illustrate the differences that can incur from considering these types of quantities, at least in principle.

Following our discussion on the use of accessible mutual informations, a natural proposal for a measurement of QIS more tightly related to what we seek to quantify is to define the accessible tripartite information $I_{3}^{\,{\mathrm{acc}}}(R:C:D)$. More formally, we define this quantity as

$$\begin{equation} I^{\,{\mathrm{acc}}}_3(R\!:\!C\!:\!D)= I_{\mathrm{acc}}(R\!:\!C) + I_{\mathrm{acc}}(R\!:\!D) - I_{\mathrm{acc}}(R\!:\!CD), \end{equation} \tag{ 31 }$$

for a given tripartition CDE of the output and evolution $\Phi_{A\rightarrow CDE}$. The subspace E could be actually empty, i.e. the output space is actually bipartitioned; in this case, it turns out $I_{\mathrm{acc}}(R\!:\!CD) = \log_2|\mathcal{H}_R|$ for isometric evolutions, since the input information can be always retrieved performing an appropriate global measurement. Each summand in (31) is independently optimized over the classical-quantum states ρ_cq , i.e. over the choice of input states (c.f.r equation (30)). In this definition, each of the three accessible information will also require a separate optimization over the output bases.

Our proposal is to compute $I_{3}^{\,{\mathrm{acc}}}(R:C:D)$, where R represents the classical register, and C and D a pair of output subsystems—which might in general encompass more than one qubit each—in order to understand how much information about R is hidden in correlations between C and D. The choice of output subspaces C and D will dramatically affect the corresponding value of $I_{3}^{\,{\mathrm{acc}}}$, which reflects the fact that information might be encoded in correlations among a specific subset of the output qubits. Figure 3 provides a sketch of the proposed setup.

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The accessible tripartite information quantifies how much more correlations are present between the input register R and the output C and D, rather than between R and CD taken together. This quantity can now therefore be interpreted as a direct analogue of the classical tripartite information discussed in section 2.1. In particular, this also means that $I_{3}^{\mathrm{acc}}$ quantifies how much information about A is hidden in the correlations between C and D, or in other words, how much more information about A is accessible when one can measure C and D together rather than separately. In some circumstances, this does not tell us how much information about inputs is recoverable from outputs: for example, $I_{3}^{\mathrm{acc}}(A:C:D) = 0$ could both mean that there is no correlation at all between inputs and outputs, or that there is maximal correlation, but the input information can be retrieved completely via local measurements. Depending on the circumstances, one might thus be more interested in studying accessible informations between input and individual outputs, rather than tripartite informations. We leave a more detailed analysis of this perspective for future work.

Example with $I_3 = 0$ — Consider as an explicit example the isometry given in equation (17). While the quantum tripartite information would give $I_3 = 0$ in this case, using accessible tripartite informations we get different values depending on the partition chosen for the output space. We have

$$\begin{equation} \begin{gathered} I_{\mathrm{acc}}(R:1) = I_{\mathrm{acc}}(R:2) =1,\,\quad I_{\mathrm{acc}}(R:3) = 0, \\ I_{\mathrm{acc}}(R:12)= I_{\mathrm{acc}}(R:13) = I_{\mathrm{acc}}(R:23) = 1, \end{gathered} \end{equation} \tag{ 32 }$$

and thus

$$\begin{equation} \begin{gathered} I_3^{\,{\mathrm{acc}}}(R\!:\!1\!:\!2) {=} 1,\;\; I_3^{\,{\mathrm{acc}}}(R\!:\!1\!:\!3) {=} I_3^{\,{\mathrm{acc}}}(R\!:\!2\!:\!3) {=} 0 \\ I_3^{\,{\mathrm{acc}}}(R\!:\!1\!:\!23) {=} I_3^{\,{\mathrm{acc}}}(R\!:\!2\!:\!13) {=} 1,\; I_3^{\,{\mathrm{acc}}}(R\!:\!12\!:\!3) {=} 0 \end{gathered} \end{equation} \tag{ 33 }$$

These reflect how input qubits are encoded in the output space in a way that is partially nonlocal. Note how, using the fully quantum tripartite information I₃ discussed in section 2.3, we only obtain a single quantity, which cannot capture the complexity of the information encoding we get even in this extremely simple toy example.

Example with $I_3 = 1$ — In this case, all accessible informations are equal to 1, corresponding to the fact that the input basis $\vert \pm\rangle$ is transmitted with maximal redundancy on the output states. The accessible tripartite informations are correspondingly also all maximal, showing that this isometry is as non-scrambling as one can be. More explicitly: equation (19)

$$\begin{align} I_{\mathrm{acc}}(R:1) &= I_{\mathrm{acc}}(R:2) = I_{\mathrm{acc}}(R:3) = 1, \nonumber\\ I_{\mathrm{acc}}(R:12) & = I_{\mathrm{acc}}(R:13) = I_{\mathrm{acc}}(R:23) = 1. \end{align} \tag{ 34 }$$

In this case all accessible tripartite informations are $I_3^{\mathrm{acc}}(R:i:j) = 1$.

Example with $I_3 = -1$ — Let us now consider again the toy example in equation (21), which gave $I_3 = -1$. The individual accessible informations are

$$\begin{align} I_{\mathrm{acc}}(R:1) &= I_{\mathrm{acc}}(R:2) = 0, \quad I_{\mathrm{acc}}(R:3) = 1,\nonumber \\ I_{\mathrm{acc}}(R:12) &= I_{\mathrm{acc}}(R:13) = I_{\mathrm{acc}}(R:23) = 1, \end{align} \tag{ 35 }$$

and the accessible tripartite informations read

$$\begin{align} I_3^{\,{\mathrm{acc}}}(R\!:\!1\!:\!2) &= {-}1, \;\; I_3^{\,{\mathrm{acc}}}(R\!:\!2\!:\!3){=}I_3^{\,{\mathrm{acc}}}(R\!:\!1\!:\!3){=}0 \nonumber\\ I_3^{\,{\mathrm{acc}}}(R\!:\!1\!:\!23) &= I_3^{\,{\mathrm{acc}}}(R\!:\!2\!:\!13) {=} 0,\,\, I_3^{\,{\mathrm{acc}}}(R\!:\!12\!:\!3) {=} 1 \end{align} \tag{ 36 }$$

These results are consistent with the information about input states being encoded redundantly in both third and first two qubits. Note that these values of $I_3^{\,{\mathrm{acc}}}$ are not all independent from one another: for example, $I_3^{\,{\mathrm{acc}}}(R:12:3) = 1$ and $I_3^{\,{\mathrm{acc}}}(R:1:3) = I_3^{\,{\mathrm{acc}}}(R:2:3) = 0$ tell us that the input is encoded in both third and first two qubits, but that measuring only second or third qubits individually we do not find the same redundancy; it must then follow that the encoding in the first two qubits is nonlocal, as then verified by $I_3^{\,{\mathrm{acc}}}(R:1:2) = -1$.