The resource theory of stabilizer quantum computation

The stabilizer formalism is critical for the results of the present paper. Here we will give a very brief overview of the elements of the theory we require. For an overview of the stabilizer formalism in the context of fault tolerance see [16, 18]. For an overview of the phase space techniques for the stabilizer formalism see [22, 24]. Veitch et al [45] gives an overview of the particular mathematical elements that will be important for this paper.

We begin by defining the generalized Pauli operators for prime dimension and we will build up the formalism from these. Let d be a prime number and define the boost and shift operators

$$\begin{equation} X\left| {j} \right\rangle = \left| {j+1 \bmod d} \right\rangle , \end{equation} \tag{ 1 }$$

$$\begin{equation} Z\left| {j} \right\rangle = {{\omega}}^{j}\left| {j} \right\rangle, \ \omega=\exp\left(\frac{2\pi\imath}{d}\right). \end{equation} \tag{ 2 }$$

From these we can define the generalized Pauli (Heisenberg–Weyl) operators in prime dimension

$$\begin{equation} T_{(a_{1},a_{2})}=\left\{\begin{array}{ll} \imath^{a_{1}a_{2}}Z^{a_{1}}X^{a_{2}},\quad & (a_{1},a_{2})\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}, \ d = 2,\\ \omega^{-\frac{a_{1}a_{2}}{2}}Z^{a_{1}}X^{a_{2}},\quad & (a_{1},a_{2})\in\mathbb{Z}_{d}\times\mathbb{Z}_{d},\ d\ne2, \end{array}\right. \end{equation} \tag{ 3 }$$

where $\mathbb {Z}_{d}$ are the integers modulo d. Note that a slightly different definition is required for qubits. For a system with composite Hilbert space H_a⊗H_b⊗···⊗H_u, the Heisenberg–Weyl operators may be written as

$$\begin{eqnarray*} &&T_{(a_{1},a_{2})\oplus(b_{1},b_{2})\cdots\oplus(u_{1},u_{2})} \equiv T_{(a_{1},a_{2})}\otimes T_{(b_{1},b_{2})}\cdots\otimes T_{(u_{1},u_{2})}. \end{eqnarray*}$$

The Clifford operators $\mathcal {C}_{d}$ are the unitaries that, up to a phase, take the Heisenberg–Weyl operators to themselves, i.e.

$$\begin{equation} U\in\mathcal{C}_{d}\iff\forall\boldsymbol{u}\exists\phi,\boldsymbol{u}': UT_{\boldsymbol{u}}U^{\dagger}=\exp\left(\mathrm{i}\phi\right)T_{\boldsymbol{u}'}. \end{equation} \tag{ 4 }$$

The set of such operators form a group—this is the Clifford group for dimension d. The pure stabilizer states for dimension d are defined as

$$\begin{equation} \left\{ S_{i}\right\} =\left\{ U\left| {0} \right\rangle:U\in\mathcal{C}_{d}\right\} , \end{equation} \tag{ 5 }$$

and we take the full set of stabilizer states to be the convex hull of this set

$$\begin{equation} \mathrm{STAB}\left(\mathcal{H}_{d}\right)=\left\{ \sigma\in L\left(\mathcal{H}_{d}\right):\ \sigma=\sum_{i}p_{i}S_{i}\right\} , \end{equation} \tag{ 6 }$$

where p_i is some probability distribution.

We define stabilizer operations to be any combination of computational basis preparation, computational basis measurement, and Clifford rotations. In particular, this includes all stabilizer state preparations and measurements. This set of operations defines the 'stabilizer subtheory', which is a convex subtheory of the full set of allowed quantum operations on a finite-dimensional system. The only stabilizer measurement we consider directly is measurement in the computational basis. The other measurements in the stabilizer subtheory can be generated, in the usual Heisenberg picture, by conjugation under Clifford rotations.

In section 4, we will need the discrete Wigner function [22, 50], which is defined for quantum systems with finite, odd Hilbert space dimension. The discrete Wigner function is a direct analogue of the usual infinite-dimensional Wigner function [49]. The idea of such representation is to attempt to map quantum theory (states, transformations and measurements) onto a classical probability theory over a phase space, which can be any continuous or discrete set. In any such representation some quantum states and measurements must be mapped to distributions with negative entries [12, 13], i.e. negative 'quasi-probabilities' are unavoidable. The discrete Wigner representation for odd dimensions enjoys the special property that all stabilizer operations can be represented non-negatively, so the Wigner representation gives a classical probability model for the full stabilizer subtheory.

The discrete Wigner representation of a state $\rho \in L(\mathbb {C}^{d^{n}})$ is a quasi-probability distribution over $\left (\mathbb {Z}_{d}\times \mathbb {Z}_{d}\right )^{n}$, which can be thought of as a dⁿ by dⁿ grid (see figure 4 in section 4). The mapping assigning quantum states ρ to Wigner functions $\left \{ W_{\rho }\left (\boldsymbol {u}\right )\right \}$ is given by

$$\begin{equation} W_{\rho}(\boldsymbol{u})=\frac{1}{d^{n}}\mathrm{Tr}(A_{\boldsymbol{u}}\rho), \end{equation} \tag{ 7 }$$

where $\left \{ A_{\boldsymbol {u}}\right \}$ are the phase space point operators. These are defined in terms of the Heisenberg–Weyl operators as

$$\begin{eqnarray} &&A_{\boldsymbol{0}} = \frac{1}{d^{n}}\sum_{\boldsymbol{u}}T_{\boldsymbol{u}},\quad A_{\boldsymbol{u}}=T_{\boldsymbol{u}}A_{\boldsymbol{0}}T_{\boldsymbol{u}}^{\dagger}. \end{eqnarray} \tag{ 8 }$$

These operators are Hermitian so the discrete Wigner representation is real-valued. There are $\left (d^{n}\right )^{2}$ such operators for dⁿ-dimensional Hilbert space, corresponding to the $\left (d^{n}\right )^{2}$ points of discrete phase space.

A quantum measurement with positive operator valued measure (POVM) {E_k} is represented by assigning conditional (quasi-) probability functions over the phase space to each measurement outcome

$$\begin{equation} W_{E_{k}}(\boldsymbol{u})=\mathrm{Tr}(A_{\boldsymbol{u}}E_{k}). \end{equation} \tag{ 9 }$$

In the case where W_Ek(u) ⩾ 0 ∀u, this can be interpreted classically as an indicator function or 'fuzzy measurement' associated with the probability of getting outcome k given that the 'physical state' of the system is at phase space point u, W_Ek(u) = Pr(outcome k|location u).

We say a state ρ has positive representation if $W_{\rho }(\boldsymbol {u})\geqslant 0\ \forall \boldsymbol {u}\in \mathbb {Z}_{d}^{n}\times \mathbb {Z}_{d}^{n}$ and negative representation otherwise. We will say a measurement with POVM M = {E_k} has positive representation if $W_{E_{k}}(\boldsymbol {u})\geqslant 0\ \forall \boldsymbol {u}\in \mathbb {Z}_{p}^{n}\times \mathbb {Z}_{p}^{n},\ \forall E_{k}\in M$ and negative representation otherwise. We are now ready to state a few salient facts about the discrete Wigner representation [15, 22]:

• 1.  
  (Discrete Hudson's theorem) A pure state $\left | {S} \right \rangle$ has positive representation if and only if it is a stabilizer state. Since convex combinations of positively represented states also have positive representation this means, in particular, for any stabilizer state S it holds that Tr(A_uS) ⩾ 0 ∀u.
• 2.  
  Clifford unitaries act as permutations of phase space. This means that if U is a Clifford then

  $$\begin{equation} W_{U\rho U^{\dagger}}(\boldsymbol{v})=W_{\rho}(\boldsymbol{v}'), \end{equation} \tag{ 10 }$$

  for each point v. Only a small subset of the possible permutations of phase space correspond to Clifford operations (namely, the symplectic ones [22]).
• 3.  
  The trace inner product is given as $\mathrm {Tr}(\rho \sigma )=d^{n}\sum _{\boldsymbol {u}}W_{\rho }\left (\boldsymbol {u}\right )W_{\sigma }\left (\boldsymbol {u}\right )$.
• 4.  
  The phase space point operators in dimension dⁿ are tensor products of n copies of the d dimension phase space point operators, e.g. $A_{\left (0,0\right )\oplus \left (0,0\right )}=A_{\left (0,0\right )}\otimes A_{\left (0,0\right )}$.
• 5.  
  The phase point operators satisfy $\mathrm {Tr}\left (A_{\boldsymbol {u}}\right )=1$. This implies $\mathrm {Tr}\left (B\right )=\sum _{\boldsymbol {u}}W_{B}\left (\boldsymbol {u}\right )$ for a Hermitian operator B.
• 6.  
  $\rho = \sum _{\boldsymbol {u}} W_\rho (\boldsymbol {u}) A_{\boldsymbol {u}}$.

This is all we need to know about the discrete Wigner function for the present work. For a much more detailed overview see [22, 23] or for a moderately more detailed overview see [45].

In this paper we are concerned with the transformation of non-stabilizer states using stabilizer operations. In the same way that a state is defined to be entangled if it is not separable we define:

Definition 1. A state is magic if it is not a stabilizer state.

The most general kind of stabilizer operation possible is of the following type:

Definition 2. A stabilizer protocol is any map from $\rho \in \mathcal {S}(\mathcal {H}_{d^{n}})$ to $\sigma \in \mathcal {S}(\mathcal {H}_{d^{m}})$ composed from the following operations:

• 1.  
  Clifford unitaries, ρ → UρU^†.
• 2.  
  Composition with stabilizer states, ρ → ρ⊗S where S is a stabilizer state.
• 3.  
  Computational basis measurement on the final qudit. $\rho \rightarrow \left (\mathbb {I}\otimes |{i}\rangle \!\langle {i}|\right )\rho \left (\mathbb {I}\otimes |{i}\rangle \!\langle {i}|\right )/\mathrm {Tr}\left (\rho \mathbb {I}\otimes |{i}\rangle \!\langle {i}|\right )$ with probability $\mathrm {Tr}\left (\rho \mathbb {I}\otimes |{i}\rangle \!\langle {i}|\right )$.
• 4.  
  Partial trace of the final qudit, $\rho \rightarrow \mathrm {Tr}_{n}\left (\rho \right )$.
• 5.  
  The above quantum operations conditioned on the outcomes of measurements or classical randomness.

Stabilizer protocols encompass magic state distillation protocols as an important special case. For a function to be a valid measure of magic, i.e. a monotone, it must be non-increasing under stabilizer operations, a requirement that can be formalized as:

Definition 3. For each d, let $\mathcal {M}_{d}:\ \mathcal {S}\left (\mathcal {H}_{d}\right )\rightarrow \mathbb {R}$ be a mapping from the set of density operators on $\mathcal {H}_{d}\cong \mathbb {C}^{d}$ to the real numbers. Define ${\mathcal {M}\left (\rho \right )}\equiv \mathcal {M}_{d}\left (\rho \right )\ \forall \rho \in \mathcal {S}\left (\mathcal {H}_{d}\right )$ (for the appropriate d) so that ${\mathcal {M}\left (\cdot \right )}$ is defined for all finite-dimensional Hilbert spaces. If ${\mathcal {M}\left (S\right )}=0$ for all stabilizer states S and it also holds that for all quantum states ρ that $\Lambda \left (\rho \right )=\sum p_{i}\sigma _{i}$ implies ${\mathcal {M}\left (\rho \right )}\geqslant \sum _{i}p_{i}{\mathcal {M}\left (\sigma _{i}\right )}$ for any stabilizer protocol Λ then we say ${\mathcal {M}\left (\cdot \right )}$ is a magic monotone.

There are two important points to notice here. The first is that one need only require operations to not increase magic on average; if $\Lambda \left (\rho \right )=\sigma _{i}$ with probability p_i then we only require ${\mathcal {M}\left (\rho \right )}\geqslant \sum _{i}p_{i}{\mathcal {M}\left (\sigma _{i}\right )}$. In particular this means that post selected measurement can increase magic in the sense that we allow ${\mathcal {M}\left (\left (\mathbb {I}\otimes |{i}\rangle \!\langle {i}|\right )\rho \left (\mathbb {I}\otimes |{i}\rangle \!\langle {i}|\right )/\mathrm {Tr}\left (\rho \mathbb {I}\otimes |{i}\rangle \!\langle {i}|\right )\right )}\geqslant {\mathcal {M}\left (\rho \right )}$ as long as measurement outcome i is obtained with sufficiently small probability. This allows us to analyze non-deterministic protocols. The second point is that we do not require convexity, i.e. it can happen that ${\mathcal {M}\left (p\rho +\left (1-p\right )\sigma \right )}\geqslant p{\mathcal {M}\left (\rho \right )}+\left (1-p\right ){\mathcal {M}\left (\sigma \right )}$. Although convexity is a desirable feature it is possible to have interesting and useful monotones that are not convex (e.g. the logarithmic entanglement negativity [39]).

Convexity constrains the behavior of the monotone on all mixtures of density matrices. The definition of a magic monotone only requires that the measure be non-increasing on mixtures which are formed via stabilizer operations, and only non-increasing relative to the starting states. For instance, we might form a mixture ρ = pρ₀ + (1 − p)ρ₁ by beginning with the state ρ₀⊗ρ₁ and discarding the second state with probability p and the first state with probability 1 − p. A magic monotone must have the property that

$$\begin{equation} \mathcal{M} (\rho_0 \otimes \rho_1) \geqslant \mathcal{M} (\rho), \end{equation} \tag{ 11 }$$

whereas convexity requires that

$$\begin{equation} p \mathcal{M}(\rho_0) + (1-p) \mathcal{M} (\rho_1) \geqslant \mathcal{M} (\rho). \end{equation} \tag{ 12 }$$

Even if $\mathcal {M}$ is additive (i.e. $\mathcal {M} (\rho _0 \otimes \rho _1) = \mathcal {M} (\rho _0) + \mathcal {M} (\rho _1)$), the latter equation is a stronger constraint.

Notice also that because Clifford gates and composition with stabilizer states are reversible within the stabilizer formalism (by the inverse gate and the partial trace respectively) any monotone must actually be invariant under these operations, as opposed to merely non-increasing.

One of the major difficulties with the study of resource monotones is that the actual computation of the value of the monotone for a particular state is often an intractable problem. Although we do not know a simple analytic expression for the relative entropy of magic, it can be computed numerically. For systems with low Hilbert space dimension this is reasonably straightforward. The relative entropy is a convex function in its second argument and we want to perform minimization over the convex set of stabilizer states. This means that a simple numerical gradient search will succeed in finding $\min _{\sigma \in {\mathrm {STAB}\left (\mathcal {H}_{d}\right )}}S\left (\rho \|\sigma \right )$. Each qudit stabilizer state can be written as a convex combination of the N pure qudit stabilizer states. A simple strategy for finding the relative entropy of magic is to do a numerical search over the N − 1 values that define the probability distribution over the stabilizer states. Unfortunately, for a system of n qudits the number of pure stabilizer states is [22]

$$\begin{equation} N=d^{n}\prod_{i=1}^{n}\left(d^{i}+1\right), \end{equation} \tag{ 13 }$$

and this grows too quickly for a numerical search to be viable in general. For example, the original H-type magic state distillation protocol [6] consumes 15 resource states ρ_input to produce a more pure magic state ρ_output. In principal we can bound the quantity of the resource required via,

$$\begin{equation} {{r}_{\mathcal{M}}\left(\rho_{\mathrm{input}}^{\otimes15}\right)}\geqslant p_{i}{{r}_{\mathcal{M}}\left(\rho_{\mathrm{output}}\right)}, \end{equation} \tag{ 14 }$$

where p_i is the success probability of the protocol, but this would require a numerical optimization over more than 2¹³⁶ parameters using the approach just outlined, which is not viable.

For arbitrary resource states it is not clear how to avoid the numerical optimization. However, the states typically used in magic state distillation protocols have a great deal of additional structure that can be exploited. In particular, many protocols have a 'twirling' step where a random Clifford unitary is applied to the resource state to ensure it has the form

$$\begin{equation} \rho_{\epsilon}=\left(1-\epsilon\right)|{M}\rangle\!\langle{M}|+\epsilon\frac{\mathbb{I}}{d}, \end{equation} \tag{ 15 }$$

where |M〉〈M| is invariant under the twirling. If the twirling map is $\mathcal {T}:\rho _{\mathrm {resource}}\rightarrow \sum _{i}p_{i}U_{i}\rho _{\mathrm {resource}}U_{i}^{\dagger }$ for some subset $\left \{ U_{i}\right \}$ of the Clifford operators, then

$$\begin{equation}\fl \min_{\sigma\in\mathrm{STAB}}S\left(\left[\left(1-\epsilon\right)|{M}\rangle\!\langle{M}|+\epsilon\frac{\mathbb{I}}{d} \right] \Big\|\sigma\right) \geqslant \min_{\sigma\in\mathrm{STAB}}S\left(\mathcal{T}\left(\left(1-\epsilon\right)|{M}\rangle\!\langle{M}|+\epsilon\frac{\mathbb{I}}{d}\right) \Big\| \mathcal{T}\left(\sigma\right)\right)\end{equation} \tag{ 16 }$$

$$\begin{equation} = \min_{p\leqslant p_{T}}S\left(\left[\left(1-\epsilon\right)|{M}\rangle\!\langle{M}|+\epsilon\frac{\mathbb{I}}{d}\right] \Big\| \left[p|{M}\rangle\!\langle{M}|+\left(1-p\right)\frac{\mathbb{I}}{d}\right] \right)\end{equation} \tag{ 17 }$$

$$\begin{equation} = S\left( \left[\left(1-\epsilon\right)|{M}\rangle\!\langle{M}|+\epsilon\frac{\mathbb{I}}{d}\right] \Big\| \left[p_{T}|{M}\rangle\!\langle{M}|+\left(1-p_{T}\right)\frac{\mathbb{I}}{d}\right] \right), \end{equation} \tag{ 18 }$$

where p_T is the largest value such that $p_{T}|{M}\rangle \!\langle {M}|+\left (1-p_{T}\right )\frac {\mathbb {I}}{d}$ is a stabilizer state. The reverse inequality is obvious since we are minimizing over a subset of all stabilizer states. This means that the relative entropy of magic can be computed exactly for states of this form by finding p_T . Unfortunately the twirling is only applied to individual qudits so this does not by itself resolve the numerical problems.

Nevertheless, it is possible to give weak bounds according the following observation:

$$\begin{equation} {{r}_{\mathcal{M}}\left(\rho_{\mathrm{output}}\right)} \leqslant {{r}_{\mathcal{M}}\left(\rho_{\mathrm{input}}^{\otimes n}\right)}\end{equation} \tag{ 19 }$$

$$\begin{equation} \leqslant n{{r}_{\mathcal{M}}\left(\rho_{\mathrm{input}}\right)}, \end{equation} \tag{ 20 }$$

where we have used the obvious fact that the relative entropy of magic is subadditive. This bound might not seem weak at all. One might suspect that the relative entropy of magic is genuinely additive so ${{r}_{\mathcal {M}}\left (\rho _{\mathrm {input}}^{\otimes {n}}\right )}\stackrel {?}{=}n{{r}_{\mathcal {M}}\left (\rho _{\mathrm {input}}\right )}$. This seems like a very desirable feature for a monotone to have: n copies of a resource state should contain n times as much resource as a single copy. The relative entropy of magic does not have this feature—it can be the case that ${{r}_{\mathcal {M}}\left (\rho ^{\otimes 2}\right )}<{{r}_{\mathcal {M}}\left (\rho \right )}$. To establish this we consider the qutrit Strange state $|\mathbb {S}\rangle \!\langle \mathbb {S}|$ defined as the pure qutrit state invariant under the symplectic component of the Clifford group (see section 4.4). Twirling by the symplectic subgroup $\mathrm {Sp}\left (2,3\right )$ of the Clifford group⁸ has the effect

$$\begin{equation} \sum_{F}\frac{1}{\left|\mathrm{Sp}\left(2,3\right)\right|}U_{F}\rho U_{F}^{\dagger}=\left(1-\epsilon_{\rho}\right)|\mathbb{S}\rangle\!\langle\mathbb{S}|+\epsilon_{\rho}\frac{\mathbb{I}_{3}}{3}, \end{equation} \tag{ 21 }$$

so we can use our twirling argument to find ${{r}_{\mathcal {M}}(|\mathbb {S}\rangle \!\langle \mathbb {S}|)}$ exactly. A numerical search over the two qutrit stabilizer states turns up a state $\sigma \in {\mathrm {STAB}\left (\mathcal {H}_{9}\right )}$ such that $S\left (|\mathbb {S}\rangle \!\langle \mathbb {S}|^{\otimes 2}\|\sigma \right )<2{{r}_{\mathcal {M}}\left (|\mathbb {S}\rangle \!\langle \mathbb {S}|\right )}$.

Note that the relative entropy of entanglement is also strictly subadditive for some states. However, there is a very important difference between the entanglement and magic relative entropy measures: for pure states the relative entropy of entanglement is an additive measure. This fact is at the heart of the importance of the relative entropy distance for the theory of entanglement. As we have just seen this is not true for the relative entropy of magic.

The relative entropy $S\left (\rho \|\sigma \right )$ is 0 if and only if ρ = σ. It is easy to see that this implies that ${{r}_{\mathcal {M}}\left (\rho \right )}$ is faithful in the sense that ${{r}_{\mathcal {M}}\left (\rho \right )}>0$ if and only if ρ is magic. Since ${{r}_{\mathcal {M}}\left (\cdot \right )}$ is a magic monotone, if it is possible to create a magic state σ from a (pure) resource state |ψ〉〈ψ| using a stabilizer protocol it must be the case that ${{r}_{\mathcal {M}}\left (|\psi \rangle \!\langle \psi |\right )}\geqslant {{r}_{\mathcal {M}}\left (\sigma \right )}$. Previous work established the existence of a large class of 'bound magic states'⁹ that cannot be distilled to pure magic states [45]. Together these facts imply that to create any magic state, including bound magic states, a non-zero amount of magic is required. This means, in general, that the amount of magic that can be distilled from a resource state is not equal to the amount of magic required to create it; this is analogous to the well-known result in entanglement theory that the entanglement of formation is not equal to the entanglement of distillation.

Because the relative entropy of magic is subadditive it could be that ${{r}_{\mathcal {M}}^{\infty }\left (\rho \right )}=\lim _{n\rightarrow \infty }\frac {1}{n}{{r}_{\mathcal {M}} (\rho ^{\otimes n})}=0$ for some magic state ρ, i.e. it is not automatic that the regularized relative entropy of magic is faithful. For example, in the resource theory of asymmetry [20], the regularized relative entropy measure is 0 for all states. Happily, for magic theory the relative entropy is well behaved in the asymptotic regime.

Lemma 1. The regularized relative entropy of magic is faithful in the sense that ${{r}_{\mathcal {M}}^{\infty }\left (\rho \right )}=0$ if and only if ρ is a stabilizer state.

Proof. The proof of this fact is a straightforward application of a theorem of Piani [38] showing that the regularized relative entropy measure is faithful for all resource theories where the set of restricted operations includes tomographically complete measurements and the partial trace. The idea is to define a variant of the relative entropy distance that quantifies the distinguishability of states using only stabilizer measurements. This quantity lower bounds the usual relative entropy of magic. Thus by showing that its regularization is faithful we get the claimed result. See appendix A.2 for details.    □

We will need this result for the proof of corollary 1 showing that the regularized relative entropy gives the optimal rate of asymptotic interconversion. It also represents a strengthening of our earlier claim that a non-zero amount of pure state magic is required to create any magic state. For finite size protocols this followed from the faithfulness of the relative entropy of magic, as just explained. The faithfulness of the regularized relative entropy implies that the creation of magic states by an asymptotic stabilizer protocol requires resource magic states to be consumed at a non-zero rate. The analogous problem in entanglement theory was the main motivation for proving that the regularized relative entropy of entanglement is faithful [4, 38].

In the scenario of asymptotic state conversion, it is useful to consider an additional property that a magic measure may possess beyond those required to make it a magic monotone. To understand the additional property, it is simplest to first consider the case of finite state interconversion. Suppose there is some stabilizer protocol Λ that maps n copies of resource state ρ to m copies of a target magic state σ. Then it must be the case that ${\mathcal {M} (\rho ^{\otimes n} )}\geqslant {\mathcal {M} (\sigma ^{\otimes m} )}$ for any magic monotone ${\mathcal {M} (\cdot )}$. If there is also some other stabilizer protocol that maps σ^⊗m to ρ^⊗n then it must be the case that ${\mathcal {M}(\rho ^{\otimes n})}={\mathcal {M}(\sigma ^{\otimes m})}$, which conceptually just means that if ρ and σ are equivalent resources then they have the same magic according to any magic measure. It is rarely possible to exactly interconvert between resource states with only a finite number of copies available. However, it is often the case that we can get close to a reversible interconversion; that is, that conversion from copies of σ to approximate copies of ρ and then back to approximate copies of σ loses only an asymptotically negligible number of copies of the state. Thus, we wish to study measures that satisfy the requirement that asymptotically reversibly interconvertible states have the same resource value.

Typically if we try to convert ρ^⊗n into m copies of σ, the stabilizer protocol ($\Lambda _{n}:\mathcal {H}_{d^{n}}\rightarrow \mathcal {H}_{d^{m}}$) we use will depend on the number n of input states. When converting ρ to σ it is thus necessary to consider a family of stabilizer protocols Λ_n taking ρ^⊗n as input and producing m(n) approximate copies of σ with an error $\|\Lambda _{n}(\rho ^{\otimes n} )-\sigma ^{\otimes m (n )}\|_{1}=\epsilon _{n}$. In the case that the approximation becomes arbitrarily good in the asymptotic limit (i.e. $\lim _{n\rightarrow \infty }\|\Lambda _{n} (\rho ^{\otimes n} )-\sigma ^{\otimes m\left (n\right )}\|_{1}\rightarrow 0$) we say ρ is asymptotically convertible to σ at a rate $R\left (\rho \rightarrow \sigma \right )=\lim _{n\rightarrow \infty }\frac {m(n)}{n}$. We wish to consider magic monotones that are compatible with asymptotic convertibility. In particular, the additional constraint that we will impose is that if ρ is asymptotically convertible to σ then

$$\begin{equation} \lim_{n\rightarrow\infty}\frac{1}{n}\left[{\mathcal{M}\left(\rho^{\otimes n}\right)}-{\mathcal{M}\left(\sigma^{\otimes m\left(n\right)}\right)}\right]\geqslant 0. \end{equation} \tag{ 22 }$$

That is, if asymptotic conversion is possible then on average we must put in at least as much magic as we get out, up to some o(n) discrepancy.

If it is possible to interconvert between σ and ρ at rates $R\left (\sigma \rightarrow \rho \right )=R\left (\rho \rightarrow \sigma \right )^{-1}$ then we say the two resources are asymptotically reversibly interconvertible. Any magic monotone satisfying the additional condition (22) gives the rate of asymptotic interconversion according to the following theorem.

Theorem 2. Let ${\mathcal {M}\left (\cdot \right )}$ be a magic monotone satisfying the condition given by equation (22) and define its asymptotic variant ${\mathrm { \mathcal {M}}^{\infty }\left (\rho \right )}=\lim _{n\rightarrow \infty }{\mathcal {M} (\rho ^{\otimes n})}/n$. Then if it is possible to asymptotically reversibly interconvert between magic states ρ and σ and ${\mathrm { \mathcal {M}}^{\infty }\left (\sigma \right )}$ is non-zero the rate of conversion is given by $R\left (\rho \rightarrow \sigma \right )={\mathrm { \mathcal {M}}^{\infty }\left (\rho \right )}/{\mathrm { \mathcal {M}}^{\infty }\left (\sigma \right )}$.

Proof. This is a special case of a broader theorem that says this result holds in any resource theory. The result was first proved in [28]. That paper dealt primarily with entanglement and missed the requirement that the regularization of the monotone needs to be non-zero. This was pointed out in [20], and the theorem we state here is essentially the application of their theorem 4 to magic theory. The only subtlety is that instead of the condition in equation (22) they require the monotone to be asymptotically continuous, which means $\lim _{n\rightarrow \infty }\|\Lambda _{n}(\rho ^{\otimes n} )-\sigma ^{\otimes m\left (n\right )}\|_{1}\rightarrow 0$ implies

$$\begin{equation} \lim_{n\rightarrow\infty}\frac{{\mathcal{M}\left(\Lambda_{n}\left(\rho^{\otimes n}\right)\right)}-{\mathcal{M}\left(\sigma^{\otimes m\left(n\right)}\right)}}{1+n}\rightarrow0. \end{equation} \tag{ 23 }$$

The first step of their proof is to show that this condition implies equation (22) so we prefer to start with the weaker requirement directly.    □

Corollary 1. If it is possible to asymptotically reversibly interconvert between magic states ρ and σ, the rate at which this can be done is $R\left (\rho \rightarrow \sigma \right )={{r}_{\mathcal {M}}^{\infty }\left (\sigma \right )}/{{r}_{\mathcal {M}}^{\infty }\left (\rho \right )}$, where ${{r}_{\mathcal {M}}^{\infty }\left (\sigma \right )}$ is the regularized relative entropy.

Proof. In [43] it is shown that the relative entropy distance to any convex set of quantum states is asymptotically continuous. Since asymptotic continuity implies equation (22) and the stabilizer states are a convex set, the relative entropy of magic is a magic monotone satisfying condition equation (22). Moreover, we showed in lemma 1 that the regularized relative entropy is non-zero for all magic states.    □

Notice that the relative entropy is only one example of a monotone satisfying the conditions of theorem 2. There could be other monotones for which this result holds. In fact it is conceivable that this result holds for every magic monotone. For any magic monotone with this property, if it possible to asymptotically interconvert between ρ and σ, it must be the case that ${\mathrm { \mathcal {M}}^{\infty }\left (\rho \right )}=C{{r}_{\mathcal {M}}^{\infty }\left (\rho \right )}\Longrightarrow {\mathrm { \mathcal {M}}^{\infty }\left (\sigma \right )}=C{{r}_{\mathcal {M}}^{\infty }\left (\sigma \right )}$ so ${{r}_{\mathcal {M}}^{\infty }\left (\sigma \right )}/{{r}_{\mathcal {M}}^{\infty }\left (\rho \right )}={\mathrm { \mathcal {M}}^{\infty }\left (\sigma \right )}/{\mathrm { \mathcal {M}}^{\infty }\left (\rho \right )}$, i.e. the regularization of such magic measures can differ only up to a multiplicative factor that can vary between sets of quantum states where asymptotic interconversion is possible.

If we have a resource measure ${\mathcal {M}\left (\cdot \right )}$ that is additive then it will be equal to its own regularization, ${\mathcal {M}\left (\cdot \right )}={\mathrm { \mathcal {M}}^{\infty }\left (\cdot \right )}$. If this measure also satisfies equation (22) then it will tell us how to compute the asymptotic interconversion rate whenever asymptotic interconversion is possible. In the particular case that we have a resource theory where asymptotic interconversion is possible between any two resource states then it is easy to see that up to a constant factor there really is a single unique measure of the resource. For instance, this is true of bipartite pure entangled states, and the entanglement entropy [2] is an additive measure that satisfies our condition. Thus the entanglement entropy is the genuinely unique measure of pure state bipartite entanglement. One of the special features of the relative entropy of entanglement is that it reduces to the entanglement entropy on pure states. It is this feature which is ultimately responsible for the privileged status of the relative entropy of entanglement. In the case of magic theory the relative entropy of magic does not reduce to an additive measure on pure states so there is no apparent reason to prefer the relative entropy of magic over any other monotone satisfying the conditions of theorem 2. This stands in contrast to the claim that the relative entropy distance to the set of free states is the unique measure of the resource (e.g. [27]).

The privileged status of the relative entropy magic comes from its role in the asymptotic regime. Since the assumption of infinite state preparations is unreasonable for an actual physical system one might expect that the measure would be of limited practical value. This suspicion is lent additional weight by the fact that, like the regularized relative entropy distance in other resource theories, it is not known how to compute ${{r}_{\mathcal {M}}^{\infty }\left (\rho \right )}$ in general. The regularized relative entropy distance is essentially useless for analyzing the performance of particular distillation protocols. Nevertheless, the monotone is a useful tool for the study of the resource theory of magic. This is the role of the regularized relative entropy distance in the theory of entanglement, where it is a well studied object. We had a taste of this already in our demonstration that the amount of pure state magic required to create a magic state does not equal the amount of pure state magic that can be distilled from that state. It is an exciting direction for future work to see what other insights can be gleaned from the relative entropy of magic and its asymptotic variant.

It is also important to understand exactly what corollary 1 says. The statement is that if asymptotic interconversion is possible then the rate is given by ${{r}_{\mathcal {M}}^{\infty }\left (\rho \right )}/{{r}_{\mathcal {M}}^{\infty }\left (\sigma \right )}$. This 'if clause' is a deceptively strong requirement: it is not guaranteed that asymptotic interconversion will always be possible, or even that it will ever be possible. In particular, every currently known magic state distillation protocol has rate 0 and it is an important open problem to determine if a positive rate distillation protocol exists.

Previous work establishing the existence of bound magic states [45] provides a starting place in the search for a computable monotone. The fundamental tool in that construction is the discrete Wigner function. There it was found that a necessary condition for a magic state to be distillable is that it has negative Wigner representation. However, that work is purely binary in the sense that it does not distinguish degrees of negative representation. It is natural to suspect that a state that is 'nearly' positively represented is less magic than a state with a large amount of negativity in its representation. Here we formalize this intuition by showing that the absolute value of the sum of the negative entries of the discrete Wigner representation of a quantum state is a magic monotone.

Definition 6. The sum negativity of a state ρ is the sum of the negative elements of the Wigner function, ${\mathrm {sn}\left (\rho \right )}\equiv \sum _{\boldsymbol {u}:W_{\rho }(\boldsymbol {u})<0} |W_{\rho }\left (\boldsymbol {u}\right ) |\equiv \frac {1}{2} (\sum _{\boldsymbol {u}} |W_{\rho } (\boldsymbol {u} ) |-1)$.

The right hand side of this expression follows because the normalization of quantum states (Tr ρ = 1) implies $\sum _{\boldsymbol {u}}W_{\rho }\left (\boldsymbol {u}\right )=1$. The advantage of writing the expression in this form is that ${\mathrm { \|}}\rho \|_{W}\equiv \sum _{\boldsymbol {u}} |W_{\rho } (\boldsymbol {u}) |$ is a multiplicative norm and is thus very nice to work with. By this we mean that the composition law is given as

$$\begin{eqnarray} {\rm \|}\rho\otimes\sigma\|_{W} & = & \sum_{\boldsymbol{u},\boldsymbol{v}}\left|W_{\rho\otimes\sigma}\left(\boldsymbol{u}\oplus\boldsymbol{v}\right)\right|\nonumber\\ & = & \sum_{\boldsymbol{u},\boldsymbol{v}}\left|W_{\rho}\left(\boldsymbol{u}\right)W_{\sigma}\left(\boldsymbol{v}\right)\right|\nonumber \\ & = & \left(\sum_{\boldsymbol{u}}\left|W_{\rho}\left(\boldsymbol{u}\right)\right|\right)\left(\sum_{\boldsymbol{v}}\left|W_{\rho}\left(\boldsymbol{v}\right)\right|\right). \end{eqnarray} \tag{ 24 }$$

Since the sum negativity is a linear function of this quantity we can establish that the former is a magic monotone by showing this for the latter.

Theorem 3. The sum negativity is a magic monotone.

Proof. Since sum negativity is clearly 0 for all stabilizer states it suffices to show $\sum _{\boldsymbol {u}} |W_{\rho }\left (\boldsymbol {u}\right ) |$ is non-increasing under stabilizer operations by verifying the required properties. The main components are the use of $\rho =\sum _{\boldsymbol {u}}W_{\rho }(\boldsymbol {u})A_{\boldsymbol {u}}$ and the composition identity equation (24), which is the main motivation for working with this quantity rather than with the sum negativity directly. See appendix B.2 for details.    □

The sum negativity is an intuitively appealing way of using the Wigner function to define a magic monotone, but it has some irritating features. The worst of these is the composition law

$$\begin{equation} {\mathrm{sn}\left(\rho^{\otimes n}\right)}=\textstyle\frac{1}{2}\left[\left(2{\mathrm{sn}\left(\rho\right)}+1\right)^{n}-1\right], \end{equation} \tag{ 25 }$$

which has the troubling feature that a linear increase in the number of resource states implies an exponential increase the amount of resource according to the measure. Happily there is a simple resolution to this problem suggested by the composition law, equation (24). We define a new monotone by a particular function of the sum negativity.

Definition 7. The mana of a quantum state ρ is ${\mathscr {M}\left (\rho \right )}\equiv \log (\sum _{\boldsymbol {u}} |W_{\rho }(\boldsymbol {u}) | )=\log \left (2{\mathrm {sn}\left (\rho \right )}+1\right )$.

Theorem 4. The mana is a magic monotone.

Proof. It is clear that the mana is 0 for all stabilizer states. Most of the other monotone requirements follow because log is a monotonic function, but there is a small subtlety here. Consider a stabilizer protocol that sends ρ → σ_i with probability p_i (e.g. post-selected computational basis measurement). Then we require $\log \left ({\mathrm { \|}}\rho \|_{W}\right )\geqslant \sum _{i}p_{i}\log \left ({\mathrm { \|}}\sigma _{i}\|_{W}\right )$. This need not be true for arbitrary monotonic functions of ∥ρ∥_W but it is easy to see that it follows from the concavity of log and ${\mathrm { \|}}\rho \|_{W}\geqslant \sum _{i}p_{i}{\mathrm { \|}}\sigma _{i}\|_{W}$.    □

From equation (24) this monotone is additive in the sense

$$\begin{equation} {\mathscr{M}\left(\rho\otimes\sigma\right)}={\mathscr{M}\left(\rho\right)}+{\mathscr{M}\left(\sigma\right)}. \end{equation} \tag{ 26 }$$

Beyond its intuitive appeal, additivity is a nice feature for a monotone to have because it makes the bound on distillation efficiency take an especially nice form. How many copies n of a resource magic state ρ are required to distill m copies of a resource magic state σ? Suppose we have a stabilizer protocol $\Lambda (\rho ^{\otimes n})\rightarrow \sigma _{i}~\mathrm {with~probability }~p_{i}$, then the monotone condition combined with additivity shows

$$\begin{equation} \sum_{i}p_{i}{\mathscr{M}\left(\sigma_{i}\right)}\leqslant n{\mathscr{M}\left(\rho\right)}. \end{equation} \tag{ 27 }$$

Taking σ₀ = σ and p₀ = p, the above discussion lets us see:

Theorem 5. Suppose Λ is a stabilizer protocol that consumes resource states ρ to produce m copies of target state σ, succeeding probabilistically. Any such protocol requires at least $\mathbb {E}\left [n\right ]\geqslant m\frac {{\mathscr {M}\left (\sigma \right )}}{{\mathscr {M}\left (\rho \right )}}$ copies of ρ on average.

Proof. Suppose $\Lambda (\rho ^{\otimes k})=\sigma ^{\otimes m}$ with probability p. The fact that the mana is an additive magic monotone implies

$$\begin{equation} k{\mathscr{M}\left(\rho\right)}\geqslant p \; m~{\mathscr{M}\left(\sigma\right)}\Longrightarrow\frac{k}{p}\geqslant m\frac{{\mathscr{M}\left(\sigma\right)}}{{\mathscr{M}\left(\rho\right)}}. \end{equation} \tag{ 28 }$$

Letting l be the number of times we must run the protocol to get a success we have n = kl and

$$\begin{equation} \mathbb{E}\left[l\right]=\frac{1}{p}, \end{equation} \tag{ 29 }$$

from which it follows that $\mathbb {E}\left [n\right ]=\frac {k}{p}\geqslant m\frac {{\mathscr {M}\left (\sigma \right )}}{{\mathscr {M}\left (\rho \right )}}$.    □

We can only bound the average number of copies required because the monotone is only non-increasing on average under stabilizer operations—it might increase conditionally on a specific measurement outcome.

The most common case for magic state distillation is nested distillation protocols, which a little thought will show are covered by the bound as a special case. Indeed, this bound covers a broader set of protocols than it might first appear. One might have expected to do better by 'recycling' the output states of the failed protocols. For instance, if

$$\begin{equation} \Lambda\left(\rho^{\otimes k}\right)=\left\{\begin{array}{@{}ll} \sigma^{\otimes m} & \mathrm{with~probability~} p,\\ \tau & \mathrm{with~probability~} 1-p, \end{array}\right. \end{equation} \tag{ 30 }$$

then one expects to reduce the overhead of the total number of copies ρ required by introducing a second stabilizer protocol

$$\begin{equation} \mathcal{E}\left(\tau\otimes\rho^{\otimes k'}\right)=\sigma^{\otimes m} \mathrm{~with~probability~}q. \end{equation} \tag{ 31 }$$

However, by just combining the two steps we have a new protocol $\tilde {\Lambda }(\rho ^{\otimes (k+k')})=\sigma ^{\otimes m}~\mathrm { with~probability }~\tilde {p}=p+\left (1-p\right )q$ and our theorem applies.

Computing the mana of a quantum state is straightforward: we find the Wigner function by taking the trace of ρ with the d² phase space point operators and compute the mana directly. This means that the mana provides a simple way to numerically upper bound the efficiency of distillation protocols, fulfilling the major promise of this section.

Quantifying the magic of a state by the negativity in its Wigner representation is an intuitively appealing idea, but it is not clear that the sum of the negative elements is the best way to do this. For example, we might have instead looked at the maximally negative element of the Wigner function, $\mathrm {maxneg}\left (\rho \right )=-\min _{\boldsymbol {u}}W_{\rho }\left (\boldsymbol {u}\right )$. It is not immediately obvious that the sum negativity is a better way to quantify the magic of a quantum state than the maximal negativity just defined. However, it turns out that the maximal negativity is not a magic monotone, so it is not a useful measure of resources for stabilizer computation. In fact, we will now show that any magic monotone that is determined solely by the values of the negative entries of the Wigner function (and in particular not by the positions in phase space of the negative entries) can be written as a function of only the sum negativity.

The reason that the maximally negative entry is not a magic monotone is that it is not invariant under composition with stabilizer states. Suppose we have some resource state ρ and we compose it with the maximally mixed state on a qudit $\mathbb {I}_{d}/d$. Then $\mathrm {maxneg}\left (\rho \otimes \mathbb {I}_{d}/d\right )=-\min _{\boldsymbol {u},\boldsymbol {v}}W_{\rho }\left (\boldsymbol {u}\right )\cdot W_{\mathbb {I}/d}(\boldsymbol {v})=-\min _{\boldsymbol {u},\boldsymbol {v}}W_{\rho }\left (\boldsymbol {u}\right )\cdot \frac {1}{d^{2}}=\mathrm {maxneg}\left (\rho \right )/d^{2}$. Therefore, this function can decrease under composition with stabilizer states, and thus can increase under partial trace: it is a poor measure of the amount of resource in ρ. The natural requirement that magic monotones must be invariant under composition with arbitrary stabilizer states is an extremely strong one; it forms the backbone of our proof of the uniqueness of the sum negativity.

Theorem 6. Assume ${\mathcal {M}\left (\rho \right )}$ is a function on quantum states that satisfies the following conditions: (i) ${\mathcal {M}\left (\rho \right )}$ is a magic monotone, (ii) ${\mathcal {M}\left (\rho \right )}$ is determined only by the negative values of the Wigner function, and (iii) ${\mathcal {M}\left (\rho \right )}$ is invariant under arbitrary permutations of discrete phase space (i.e. even under permutations that do not correspond to quantum transformations). Then ${\mathcal {M}\left (\rho \right )}$ may be written as a function of only ${\mathrm {sn}\left (\rho \right )}$.

Proof. Consider two quantum states ρ and ρ' that have Wigner representations with different negative entries but ${\mathrm {sn}\left (\rho \right )}={\mathrm {sn}\left (\rho '\right )}$. The idea is to construct stabilizer ancilla states A and A' such that ρ⊗A and ρ'⊗A' have the same negative Wigner function entries. In this case conditions (ii) and (iii) imply ${\mathcal {M}\left (\rho \otimes A\right )}={\mathcal {M}\left (\rho '\otimes A'\right )}$ and since magic monotones are invariant under composition with stabilizer states this means ${\mathcal {M}\left (\rho \right )}={\mathcal {M}\left (\rho '\right )}$, i.e. ${\mathcal {M}\left (\rho \right )}$ is entirely determined by the sum negativity. For details see appendix B.2.    □

For our proof of theorem 6 to succeed it is critical that the value of the monotone does not depend on the locations of the negative entries. All magic monotones must be invariant under Clifford unitaries, ${\mathcal {M} (U\rho U^{\dagger })}={\mathcal {M}\left (\rho \right )} \forall U\in C_{n}$, and these operations correspond to permutations of the phase space. Thus the monotone condition already implies invariance under a subset of possible permutations (namely those that preserve the symplectic inner product). However, we require invariance under arbitrary permutations and there is no compelling reason to expect magic monotones to have this feature in general. It is not clear whether this additional assumption was really necessary; it was introduced because actually working with only the symplectic transformations is extremely challenging. It remains an interesting open problem to either prove uniqueness without this assumption or else give a counterexample in the form of a magic monotone that is determined by just the negative entries of the Wigner representation and does depend on their position. Even if the latter resolution is the case, theorem 6 is useful because it at least shows that sum negativity is the unique 'simple' monotone, in the sense that computing its magnitude does not depend on the detailed symplectic structure of phase space. As simplicity of computation is our primary motivation for the study of Wigner function monotones, this is a significant advantage.

In section 3 we showed that (the regularization of) any monotone satisfying a certain natural asymptotic condition uniquely specifies the rate at which asymptotic interconversion of resource states is possible. Since the mana is additive, it is clearly equal to its own regularization. Thus if it satisfied the condition given by equation (22) we would be able to compute the conversion rates explicitly. Typically it is usually a stronger property that is demanded: asymptotic continuity of the monotone. In appendix B.3 we show that the mana is not asymptotically continuous. However, our counterexample leaves open the possibility that the weaker condition actually required by the theorem holds. It would be very exciting to either prove or disprove this.

To illustrate the use of mana in the evaluation of magic state distillation protocols we have computed the input and output mana of single steps of several (qudit) magic distillation protocols from the literature over a large parameter range. Figures 1 and 2 present qutrit codes from [1, 8] respectively. Figure 3 presents a ququint (d = 5) code from [8]. Notice that none of the protocols come close to meeting the mana bound, which is illustrated as a red line in all three figures.

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It's interesting to compute the qutrit states with maximal sum negativity. Since

$$\begin{equation} {\mathrm{sn}\left(\rho\right)} = -\sum_{\boldsymbol{u}:\mathrm{Tr}\left(\rho A_{\boldsymbol{u}}\right)<0}\mathrm{Tr}\left(\rho A_{\boldsymbol{u}}\right) \end{equation} \tag{ 32 }$$

$$\begin{equation} \hspace{32pt}= -\mathrm{Tr}\left(\rho\sum_{\boldsymbol{u}:\mathrm{Tr}\left(\rho A_{\boldsymbol{u}}\right)<0}A_{\boldsymbol{u}}\right), \end{equation} \tag{ 33 }$$

it is easy to see that the states with maximal sum negativity must be eigenstates of operators $\sum _{\boldsymbol {u}\in S}A_{\boldsymbol {u}}$ where S is some subset of the discrete phase space. An exhaustive search over such subsets reveals two classes of maximally sum negative states.

• 1.  
  The strange states defined to be those with 1 negative Wigner function entry equal to −1/3. There are $\left( {{9}\atop{1}} \right )=9$ such states, e.g.

  $$\begin{equation} \left| \mathbb{S} \right\rangle =\frac{1}{\sqrt{2}}\left(\begin{array}{@{}c@{}}0\\ 1\\ -1 \end{array}\right). \end{equation} \tag{ 34 }$$
• 2.  
  The Norrell states defined to be those with two negative Wigner function entries equal to −1/6. There are $\left( {{9}\atop{2}} \right )=36$ such states, e.g.

  $$\begin{equation} \left| \mathbb{N} \right\rangle =\frac{1}{\sqrt{6}}\left(\begin{array}{@{}c@{}}-1\\ 2\\ -1 \end{array}\right). \end{equation} \tag{ 35 }$$

The maximum value is ${\mathrm {sn}\left (|\mathbb {S}\rangle \!\langle \mathbb {S}|\right )}={\mathrm {sn}\left (|\mathbb {N}\rangle \!\langle \mathbb {N}|\right )}=+1/3$. An example of each type of state is given in figure 4.

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Geometrically each Strange state lies outside a single face of the Wigner simplex and each Norrell state lies outside the intersection of two faces, analogous to the qubit T-type (outside a face) and H-type (outside an edge) states. This analogy is further strengthened since the Norrell states are also the generalized H-type states of [1, 30].

Note that the states with maximal resource value do not need to agree between monotones. In particular,

$$\begin{equation} \frac{{{r}_{\mathcal{M}}\left(|\mathbb{S}\rangle\!\langle\mathbb{S}|\right)}}{{{r}_{\mathcal{M}}\left(|\mathbb{N}\rangle\!\langle\mathbb{N}|\right)}}\approx1.71. \end{equation} \tag{ 36 }$$

Of course this still leaves open the possibility that ${{r}_{\mathcal {M}}^{\infty }\left (|\mathbb {S}\rangle \!\langle \mathbb {S}|\right )}={{r}_{\mathcal {M}}^{\infty }\left (|\mathbb {N}\rangle \!\langle \mathbb {N}|\right )}$.

See figure 5 for a plot of the mana values of states in a plane of qutrit state space.

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Our main motivation for studying the mana is that it can be computed explicitly to give concrete bounds on the rate at which magic states can be converted. However, one might suspect that this bound, although non-trivial, is rather arbitrary. For example, it is not clear a priori if the bound given by theorem 5 can ever be saturated, or under what circumstances this might occur. The mana arose very naturally from the negativity of the discrete Wigner function, but it is not immediately clear that the Wigner negativity is the relevant tool for the study of magic theory. However, a number of recent results show that the negativity of the discrete Wigner representation is extremely well motivated in this context. For example, could we have started with some other notion of quasi-probability representation [12] and defined a monotone from that? Recent work [31] has shown (at least for small prime dimension) that this is not the case by connecting the onset of negativity in the discrete Wigner function with the onset of a non-contextuality violation. This means that the subtheory of quantum theory consisting of elements with positive discrete Wigner representation is a maximal classical subtheory in the sense of non-classicality given by contextuality. That is, the set of states with positive discrete Wigner representation is the largest possible subtheory of quantum theory that includes the stabilizer measurements and admits a non-contextual hidden variable theory. In particular this means that any other choice of quasi-probability representation (that represents the stabilizer subtheory non-negatively) would have a positively represented region that is strictly contained within the discrete Wigner function we use here.

For the purposes of magic state distillation we are more interested in the notion of non-classicality given by universal quantum computation. The results of [35, 45] show that there is an intimate connection: the hidden variable model afforded by the discrete Wigner function leads naturally to an efficient classical simulation scheme for quantum circuits with positive Wigner representation. It is not known if access to any negatively represented state suffices to promote stabilizer computation to universal quantum computation, but it is at least apparent that the known classical simulation protocols cannot be extended to deal with this case. In the context of magic state computation it is desirable for the magic measures to give an indication of how useful a state is for quantum computation. In this sense, the fact that the mana is not a faithful monotone is a feature rather than a bug—it picks specifically the set of quantum states that do not admit an efficient simulation scheme under stabilizer operations.

Although the mana is essentially the unique symmetric monotone arising from the negativity of the Wigner function, it is not the only choice of monotone arising from the Wigner function. In particular, one very natural choice is the relative entropy distance to the set of states with positive Wigner representation, ${{r}_{\mathcal {W}}\left (\rho \right )=\min _{\sigma :W_{\sigma }\left (\boldsymbol {u}\right )\geqslant 0\,\forall \boldsymbol {u}}S\left (\rho \|\sigma \right )}$. It is easy to check that all of the results of section 3 go through for this new monotone, subject to obvious modifications in the statement of the theorems.

The major inspiration for the monotones of this section was earlier work showing that states with positive Wigner representation cannot be distilled by stabilizer protocols. In the theory of entanglement it is known that states with positive partial transpose (ppt) cannot be distilled by local operations and classical communication (LOCC) protocols [25], and this inspired the introduction of the entanglement negativity $\mathcal {N}\!\left (\rho \right )$, a measure of the violation of the ppt condition, as a measure of entanglement [47]. As with the sum negativity, the major advantage of this measure is that it is computable, allowing for explicit upper bounds on the efficiency of entanglement distillation. The entanglement negativity grows exponentially in the number of resource states, prompting the definition of an additive variant $\mathcal {LN}\left (\rho \right )\equiv \log \left (2\mathcal {N}\left (\rho \right )+1\right )$—exactly as in the present case. Like the mana this measure has the strange features that it is neither convex nor asymptotically continuous¹⁰. The close analogy we have uncovered suggests that it may be possible to adapt much of the work on entanglement negativity to the magic case: this is an interesting direction for future work.

There is at least one way in which the sum negativity is better behaved than the entanglement negativity. All separable states are local, but this does not mean that all entangled states are non-local in the sense that they enable violation of a Bell inequality. In [37] Peres conjectured that any ppt state should admit a local hidden variable model; proving or disproving this conjecture is one of the major outstanding problems in the study of entanglement. In our case the equivalent conjecture would be that any state with positive Wigner representation admits a non-contextual hidden variable model. But in our case the answer is obvious: the Wigner representation itself is this non-contextual hidden variable theory! Moreover, as noted above, recent work [31] has shown (at least for small prime dimension) that magic states admit such a model only if they have positive Wigner representation. The direct resolution of this question (which has proven difficult to solve for other resource theories) is a consequence of our use of the Wigner function (quasi-probability) technology. However, the quasi-probability techniques used in this section have no known analogue in other resource theories. The possibility of exporting this technology to the study of other resource theories, in particular entanglement theory, is a fascinating and promising direction for future work.

A closely related problem is to determine a qubit analogue for the mana. Because it is possible to violate a contextuality inequality (e.g. a GHZ inequality) using qubit stabilizers, there can be no qubit analogue for the discrete Wigner function (see also [48]). This is because the discrete Wigner function is a non-contextual hidden variable theory. Nevertheless, it may be possible to find a computable monotone of a similar flavor.

Theorem 7. The relative entropy of magic is a magic monotone.

Proof. We need to verify that this function is non-increasing under stabilizer operations.

• 1.  
  Invariance under Clifford unitaries. For any unitary, $S(U\rho U^{\dagger }\|U\sigma U^{\dagger })=S(\rho \|\sigma)$. If U is a Clifford and σ is a stabilizer state then UσU^† will also be a stabilizer state, ergo ${{r}_{\mathcal {M}}(U\rho U^{\dagger })}=\min _{\sigma }S(U\rho U^{\dagger }\|\sigma )=\min _{\sigma }S(U\rho U^{\dagger }\|U\sigma U^{\dagger })=\min _{\sigma }S (\rho \|\sigma )={{r}_{\mathcal {M}}\left (\rho \right )}$.
• 2.  
  Non-increasing on average under stabilizer measurement. Without loss of generality, we consider computational basis measurement on the final qudit. Let $\{V_{i}\}=\{\mathbb {I}\otimes |{i}\rangle \!\langle {i}|\}$ be the measurement POVM and label outcome probabilities $p_{i}=\mathrm {Tr}\left (V_{i}\rho \right )\!,\ q_{i}=\mathrm {Tr}\left (V_{i}\sigma \right )$ as well as post-measurement states ρ_i = V_iρV^†_i and σ_i = V_iσV^†_i. In [44] it is shown that

  $$\begin{equation} \sum_{i}p_{i}S\left(\frac{\rho_{i}}{p_{i}} \Big\| \frac{\sigma_{i}}{q_{i}}\right)\leqslant S\left(\rho\|\sigma\right). \end{equation} \tag{ A.1 }$$

  Since σ_i/q_i is a stabilizer state whenever σ is a stabilizer state this implies measurement does not increase the relative entropy of magic on average.
• 3.  
  Non-increasing under partial trace. From the strong subadditivity property of the von Neumann entropy [34] we have $S\left (\mathrm {Tr}_{B}\left (\rho \right )\|\mathrm {Tr}_{B}\left (\sigma \right )\right )\le S\left (\rho \|\sigma \right )$ from which the result follows immediately.
• 4.  
  Invariance under composition with stabilizer states. $S\left (\rho \otimes A\|\sigma \otimes A\right )=S\left (\rho \|\sigma \right )$ for any quantum state A, from which it follows ${{r}_{\mathcal {M}}\left (\rho \otimes A\right )}\le {{r}_{\mathcal {M}}\left (\rho \right )}$. Equality follows because the relative entropy of magic is non-increasing under the partial trace, i.e. ${{r}_{\mathcal {M}}\left (\rho \right )}\le {{r}_{\mathcal {M}}\left (\rho \otimes A\right )}$.

    □

We now turn to the asymptotic variant of the relative entropy of magic, ${{r}_{\mathcal {M}}^{\infty }(\rho )}=\lim _{n\rightarrow \infty }{{r}_{\mathcal {M}} (\rho ^{\otimes n})}/n$. We show that this quantity is non-zero if and only if ρ is a magic state, which in particular implies that magic must be consumed at a non-zero rate to create magic states. We will also need this result for theorem 2.

Theorem 8. The regularized relative entropy of magic is faithful in the sense that ${{r}_{\mathcal {M}}^{\infty }\left (\rho \right )}=0$ if and only if ρ may be written as a convex combination of stabilizer states.

Proof. We recover this result as a special case of the main theorem of [38]. That paper introduces a variant of the relative entropy measure that quantifies the distinguishability of a quantum state from the set of free states using a restricted set of measurements. Let {M_i} be a measurement POVM and define the map

$$\begin{equation} \mathcal{M}\left(\rho\right)=\sum_{i}p_{i}\left(\rho\right)|{i}\rangle\!\langle{i}|,\ p_{i}\left(\rho\right)=\mathrm{Tr}\left(\rho M_{i}\right), \end{equation} \tag{ A.2 }$$

where $\left \{ \left | {i} \right \rangle \right \}$ is any orthonormal set and $\mathcal {M}$ is a map associated to measurement {M_i}. Letting $\mathbb {M}$ be the set of restricted measurements we can define

$$\begin{equation} \mathbb{M}S\left(\rho\|\sigma\right)\equiv\max_{\mathcal{M}\in\mathbb{M}}S\left(\mathcal{M}\left(\rho\right)\|\mathcal{M}\left(\sigma\right)\right). \end{equation} \tag{ A.3 }$$

The significance of this quantity is from [38, theorem 1]:    □

Theorem 9. Consider a restricted set of operations inducing a resource theory. Let $\mathbb {M}$ be the restricted set of measurements (here the stabilizer measurements) and P the set of free states (here the stabilizer states). If the set of free states is closed under restricted measurement and the partial trace then it holds that the regularization of the relative entropy distance to the set of free states $r_{P}^{\infty }\left (\rho \right )$ satisfies

$$\begin{equation} r_{P}^{\infty}\left(\rho\right)\geqslant \min_{\sigma\in P}\mathbb{M}S\left(\rho\|\sigma\right). \end{equation} \tag{ A.4 }$$

The stabilizer formalism satisfies the conditions of the theorem. Moreover, since the stabilizer measurements contain an informationally complete measurement it holds that $\mathbb {M}S\left (\rho \|\sigma \right )>0$ whenever ρ is a magic state. This implies ${{r}_{\mathcal {M}}^{\infty }\left (\rho \right )}>0$ whenever ρ is a magic state. ${{r}_{\mathcal {M}}^{\infty }\left (\rho \right )}=0$ for all stabilizer states ρ, so the claimed result follows.

The main ingredient in establishing both the sum negativity ${\mathrm {sn}\left (\rho \right )}$ and the mana ${\mathscr {M}\left (\rho \right )}$ as magic monotones is to show that ${\mathrm { \|}}\rho \|_{W}=\sum _{\boldsymbol {u}}\left |W_{\rho }\left (\boldsymbol {u}\right )\right |$ is non-increasing under stabilizer operations.

Theorem 10. ${\mathrm { \|}}\rho \|_{W}=\sum _{\boldsymbol {u}}\left |W_{\rho }\left (\boldsymbol {u}\right )\right |$ is a convex and non-increasing under stabilizer operations.

Proof. We need to verify that this function is non-increasing under stabilizer operations:

• 1.  
  Invariance under Clifford unitaries. The action of Clifford unitaries on the phase space of the Wigner function is a permutation, u → Fu. Thus, ${\mathrm { \|}}U\rho U^{\dagger }\|_{W}=\sum _{\boldsymbol {u}}\left |W_{U\rho U^{\dagger }}\left (\boldsymbol {u}\right )\right |=\sum _{\boldsymbol {u}}\left |W_{\rho }\left (F\boldsymbol {u}\right )\right |=\sum _{\boldsymbol {u}}\left |W_{\rho }\left (\boldsymbol {u}\right )\right |={\mathrm { \|}}\rho \|_{W}$.
• 2.  
  Non-increasing on average under stabilizer measurement. We consider computational basis measurement on the final qudit. The expected value of ${\mathrm { \|}}\tilde {\rho }\|_{W}$ for the post measurement state $\tilde {\rho }$ is

  $$\begin{equation}\fl \mathbb{E}\left[{\rm \|}\tilde{\rho}\|_{W}\right] = \sum_{i}\mathrm{Tr}\left(\rho\mathbb{I}\otimes|{i}\rangle\!\langle{i}|\right){\rm \|}\left(\mathbb{I}\otimes|{i}\rangle\!\langle{i}|\right) \rho \left(\mathbb{I}\otimes|{i}\rangle\!\langle{i}|\right)/\mathrm{Tr}\left(\rho\mathbb{I}\otimes|{i}\rangle\!\langle{i}|\right)\|_{W} \end{equation} \tag{ B.1 }$$

  $$\begin{equation} = \sum_{i}{\rm \|}\left(\mathbb{I}\otimes|{i}\rangle\!\langle{i}|\right) \rho\left(\mathbb{I}\otimes|{i}\rangle\!\langle{i}|\right)\|_{W} \end{equation} \tag{ B.2 }$$

  and by writing $\left (\mathbb {I}\otimes |{i}\rangle \!\langle {i}|\right )\rho \left (\mathbb {I}\otimes |{i}\rangle \!\langle {i}|\right )$ as

  $$\begin{equation} \fl \left(\mathbb{I}\otimes|{i}\rangle\!\langle{i}|\right)\rho \left(\mathbb{I}\otimes|{i}\rangle\!\langle{i}|\right) = \sum_{\boldsymbol{u},\boldsymbol{v}}W_{\rho}\left(\boldsymbol{u}\oplus\boldsymbol{v}\right)\left\langle{i} \right| {A_{\boldsymbol{v}}} \left| {i} \right\rangle \cdot A_{\boldsymbol{u}}\otimes|{i}\rangle\!\langle{i}| \end{equation} \tag{ B.3 }$$

  $$\begin{equation} = \sum_{\boldsymbol{u}}\left(\sum_{\boldsymbol{v}}W_{\rho}\left(\boldsymbol{u}\oplus\boldsymbol{v}\right)\left\langle{i} \right| {A_{\boldsymbol{v}}} \left| {i} \right\rangle \right)A_{\boldsymbol{u}}\otimes\sum_{\boldsymbol{w}}\left(\frac{1}{d}\left\langle{i} \right| {A_{\boldsymbol{w}}} \left| {i} \right\rangle \right)A_{\boldsymbol{w}} \end{equation} \tag{ B.4 }$$

  we find

  $$\begin{equation}\fl \mathbb{E}\left[{\rm \|}\tilde{\rho}\|_{W}\right] = \sum_{i}\sum_{\boldsymbol{u},\boldsymbol{w}}\left|\left(\sum_{\boldsymbol{v}}W_{\rho}\left(\boldsymbol{u}\oplus\boldsymbol{v}\right)\left\langle{i} \right| {A_{\boldsymbol{v}}} \left| {i} \right\rangle \right)\left(\frac{1}{d}\left\langle{i} \right| {A_{\boldsymbol{w}}} \left| {i} \right\rangle \right)\right|\end{equation} \tag{ B.5 }$$

  $$\begin{eqnarray}&&\hspace{-22pt} = \sum_{i}\sum_{\boldsymbol{u}}\left(\sum_{\boldsymbol{w}}\frac{1}{d}\left\langle{i} \right| {A_{\boldsymbol{w}}} \left| {i} \right\rangle \right)\left|\left(\sum_{\boldsymbol{v}}W_{\rho}\left(\boldsymbol{u}\oplus\boldsymbol{v}\right)\left\langle{i} \right| {A_{\boldsymbol{v}}} \left| {i} \right\rangle \right)\right|\ \ \mathrm{(}\because\left\langle{i} \right| {A_{\boldsymbol{w}}} \left| {i} \right\rangle \geqslant 0\mathrm{)}\nonumber\\ \end{eqnarray} \tag{ B.6 }$$

  $$\begin{eqnarray}&&\hspace{-22pt} \leqslant \sum_{i}\sum_{\boldsymbol{u}}\sum_{\boldsymbol{v}}\left|W_{\rho}\left(\boldsymbol{u}\oplus\boldsymbol{v}\right)\left\langle{i} \right| {A_{\boldsymbol{v}}} \left| {i} \right\rangle \right| \ \ \nonumber\\ &&\quad \left({\because} \textrm{triangle inequality and}\ \ \sum_{\boldsymbol{w}}\frac{1}{d}\left\langle{i} \right| {A_{\boldsymbol{w}}} \left| {i} \right\rangle =1\right)\end{eqnarray} \tag{ B.7 }$$

  $$\begin{equation}\hspace{-22pt} = \sum_{\boldsymbol{u},\boldsymbol{v}}\left(\sum_{i}\left\langle{i} \right| {A_{\boldsymbol{v}}} \left| {i} \right\rangle \right)\left|W_{\rho}\left(\boldsymbol{u}\oplus\boldsymbol{v}\right)\right|\mathrm{ \ \ (}\because\left\langle{i} \right| {A_{\boldsymbol{w}}} \left| {i} \right\rangle \geqslant 0\mathrm{)}\end{equation} \tag{ B.8 }$$

  $$\begin{equation}\hspace{-22pt} = {\rm \|}\rho\|_{W}\ \ \ \left(\because\sum_{i}\left\langle{i} \right| {A_{\boldsymbol{v}}} \left| {i} \right\rangle =1\right). \end{equation} \tag{ B.9 }$$
• 3.  
  Invariance under composition with stabilizer states. Let σ be any state with positive Wigner representation. Then

  $$\begin{equation} {\rm \|}\rho\otimes\sigma\|_{W} = {\rm \|}\rho\|_{W}{\rm \|}\sigma\|_{W}\end{equation} \tag{ B.10 }$$

  $$\begin{equation} \hspace{52pt}= {\rm \|}\rho\|_{W}, \end{equation} \tag{ B.11 }$$

  since ${\mathrm { \|}}\sigma \|_{W}=\sum _{\boldsymbol {u}}\left |W_{\sigma }\left (\boldsymbol {u}\right )\right |=\sum _{\boldsymbol {u}}W_{\sigma }\left (\boldsymbol {u}\right )=1$ for positively represented states. All stabilizer states are positively represented so they are included as a special case.
• 4  
  Non-increasing under partial trace. We trace out the final qudit B of the system. If $\rho =\sum _{\boldsymbol {u},\boldsymbol {v}}W_{\rho }\left (\boldsymbol {u}\oplus \boldsymbol {v}\right )A_{\boldsymbol {u}}\otimes A_{\boldsymbol {v}}$ then $\mathrm {Tr}_{B}\left (\rho \right )=\sum _{\boldsymbol {u}}\left (\sum _{\boldsymbol {v}}W_{\rho }\left (\boldsymbol {u}\oplus \boldsymbol {v}\right )\right )A_{\boldsymbol {u}}$, so

  $$\begin{equation} {\rm \|}\mathrm{Tr}_{B}\left(\rho\right)\|_{W} = \sum_{\boldsymbol{u}}\left|\sum_{\boldsymbol{v}}W_{\rho}\left(\boldsymbol{u}\oplus\boldsymbol{v}\right)\right|\end{equation} \tag{ B.12 }$$

  $$\begin{equation}\hspace{60pt} \leqslant {\rm \|}\rho\|_{W}, \end{equation} \tag{ B.13 }$$

  by the triangle inequality.
• 5.  
  Convexity.

  $$\begin{equation} {\rm \|}p\rho+\left(1-p\right)\sigma\|_{W} = \sum_{\boldsymbol{u}}\left|pW_{\rho}\left(\boldsymbol{u}\right)+\left(1-p\right)W_{\sigma}\left(\boldsymbol{u}\right)\right|\end{equation} \tag{ B.14 }$$

  $$\begin{equation}\hspace{7.8pc} \leqslant p{\rm \|}\rho\|_{W}+\left(1-p\right){\rm \|}\sigma\|_{W} \end{equation} \tag{ B.15 }$$

  by the triangle inequality.

    □

We next establish that this was essentially the only choice we could have made to (simply) quantify the magic of a quantum state via its Wigner representation.

Sum negativity is the unique phase space measure of magic.

Theorem 11. Assume ${\mathcal {M}\left (\rho \right )}$ is a function on quantum states that satisfies the following conditions: (i) ${\mathcal {M}\left (\rho \right )}$ is a magic monotone; (ii) ${\mathcal {M}\left (\rho \right )}$ is determined only by the negative values of the Wigner function; and (iii) ${\mathcal {M}\left (\rho \right )}$ is invariant under arbitrary permutations of discrete phase space (i.e., even under permutations that do not correspond to quantum transformations). Then ${\mathcal {M}\left (\rho \right )}$ may be written as a function of only ${\mathrm {sn}\left (\rho \right )}$.

Proof. Let ρ have negative entries $-N_{1},-N_{2},\ldots ,-N_{k}$ and ρ' have negative entries $-N_{1}',-N_{2}',\ldots ,-N_{k'}'$, with

$$\begin{equation} N\equiv{\mathrm{sn}\left(\rho\right)}=\sum N_{i}=\sum N_{i}'={\mathrm{sn}\left(\rho'\right)}. \end{equation} \tag{ B.16 }$$

A and A' will be ancilla states acting on m qudits, with $m=\max \left \{ \lceil \log _{d}k\rceil ,\lceil \log _{d}k'\rceil \right \}$; d is the size of each qudit.

$$\begin{equation} A = \sum_{i=1}^{k'}(N_{i}'/N)|{i}\rangle\!\langle{i}|,\end{equation} \tag{ B.17 }$$

$$\begin{equation} A' = \sum_{i=1}^{k}(N_{i}/N)|{i}\rangle\!\langle{i}|. \end{equation} \tag{ B.18 }$$

These are valid states since the sum of the N_i and N_i' is the same. The Wigner function of A consists of columns labeled by i with entries N_i'/rN, with r = d^m; each column contains r such elements. It also has d^m − k' columns filled with zeros. Similarly for A', except it has d^m − k zero columns and the non-zero columns have r copies of N_i/rN instead.

The negative Wigner function entries for the state ρ⊗A are of the form −N_iN_j'/rN, for all i and j. Each of these appears r times. The negative Wigner function entries for ρ'⊗A' are of the form −N_j'N_i/rN, for all i and j. Again, each appears r times. These entries could be in different locations, but since the function we are calculating does not depend on location of negative entries, only their values, it follows that

$$\begin{equation} {\mathcal{M}\left(\rho\right)}={\mathcal{M}\left(\rho\otimes A\right)}={\mathcal{M}\left(\rho'\otimes A'\right)}={\mathcal{M}\left(\rho'\right)}. \end{equation} \tag{ B.19 }$$

Therefore, ${\mathcal {M}\left (\rho \right )}$ is a function only of ${\mathrm {sn}\left (\rho \right )}$.    □

In practice a perfect conversion is generally not possible, $\|\Lambda (\rho ^{\otimes m})-\sigma ^{\otimes n}\|_{1}>0$ for even the best choice of stabilizer protocol Λ. A state $\tilde {\sigma }_{n}$ that is close enough to σ^⊗n can be used in place of σ^⊗n in information theoretic tasks so a better notion of conversion would be: how many copies of ρ are required to produce a state $\Lambda (\rho ^{\otimes m})=\tilde {\sigma }_{n}$ that is 'close enough' to σ^⊗n. A natural notion of closeness is $\|\tilde {\sigma }_{n}-\sigma ^{\otimes n}\|_{1}<\epsilon$ for some operationally relevant . It is conceivable that there is some choice of $\tilde {\sigma }_{n}$ in the epsilon ball around σ^⊗n such that ${\mathscr {M}\left (\tilde {\sigma }_{n}\right )}\ll {\mathscr {M}\left (\sigma ^{\otimes n}\right )}$, in which case ${\mathscr {M}\left (\sigma \right )}$ would have little operational significance. Happily, it is not difficult to show that ${\mathscr {M}\left (\rho \right )}$ is continuous with respect to the 1-norm in the sense that for a sequence of states $\rho _{k},\sigma _{k}\in \mathcal {S}\left (\mathcal {H}_{d}\right )$ $\left \{ \|\rho _{k}-\sigma _{k}\|\right \}_{k}\rightarrow 0\Longrightarrow \left \{ \left |{\mathscr {M}\left (\rho _{k}\right )}-{\mathscr {M}\left (\sigma _{k}\right )}\right |\right \} _{k}\rightarrow 0$, so for a target state of fixed dimension there is some well-defined sense in which closeness in the 1-norm implies that the mana of two states is close.

In the case of asymptotic conversion of states this notion needs some massaging. Formally, let $\Lambda _{n}:\mathcal {S}\left (\mathcal {H}_{d^{m(n)}}\right )\rightarrow \mathcal {S}\left (\mathcal {H}_{d^{n}}\right )$ be stabilizer protocols satisfying

$$\begin{eqnarray} &&\lim_{n\rightarrow\infty}\|\Lambda_{n}\left(\rho^{\otimes m(n)}\right)-\sigma^{\otimes n}\| \rightarrow 0. \end{eqnarray} \tag{ B.20 }$$

In particular we would like to avoid a situation where $\lim _{n\rightarrow \infty }{\mathscr {M} (\Lambda _{n} (\rho ^{\otimes m(n)}))}\ll {\mathscr {M}(\sigma ^{\otimes n})}$. One way that this requirement can be formalized is the property of asymptotic continuity. A function is said to be asymptotically continuous if for sequences ρ_n,σ_n on $\mathcal {H}_{n}$, lim_n→∞∥ρ_n − σ_n∥ → 0 implies

$$\begin{equation} \lim_{n\rightarrow\infty}\frac{f(\rho_{n})-f(\sigma_{n})}{1+\log(\dim\mathcal{H}_{n})}\rightarrow0. \end{equation} \tag{ B.21 }$$

This notion is the commonly accepted generalization of continuity to the asymptotic regime and is of particular importance because if the mana could be shown to be asymptotically continuous it would give the asymptotic conversion rate, as in theorem 2. Unhappily, it is very difficult to show this. This is mostly because it is false.

Theorem 12. ${\mathscr {M}\left (\sigma \right )}$is not asymptotically continuous.

Proof. Define $\tilde {\sigma }_{n}=\left (1-\delta _{n}\right )\sigma ^{\otimes n}+\delta _{n}\eta _{n}$, with lim_n→∞δ_n → 0. Asymptotic continuity would imply

$$\begin{equation} \lim_{n\rightarrow\infty}\frac{{\mathscr{M}\left(\tilde{\sigma}_{n}\right)}-{\mathscr{M}\left(\sigma^{\otimes n}\right)}}{n}\rightarrow0, \end{equation} \tag{ B.22 }$$

but we will show this need not be the case. Suppose σ is negative on points $\mathcal {N}=\left \{ \boldsymbol {u}:\ W_{\sigma }(\boldsymbol {u})<0\right \}$. Let η be the state with maximal sum negativity satisfying $W_{\eta }(\boldsymbol {u})<0\iff \boldsymbol {u}\in \mathcal {N}$ (ie. η is negative on the same points as σ). Then,

$$\begin{equation} {\rm \|}\tilde{\sigma}\|_{W} = \sum_{\boldsymbol{u}}|\left(1-\delta_{n}\right)W_{\sigma^{\otimes n}}(\boldsymbol{u})+\delta_{n}W_{\eta^{\otimes n}}(\boldsymbol{u})|\end{equation} \tag{ B.23 }$$

$$\begin{equation} \hspace{32pt}= \sum_{\boldsymbol{u}}\left(\left(1-\delta_{n}\right)|W_{\sigma^{\otimes n}}(\boldsymbol{u})|+\delta_{n}|W_{\eta^{\otimes n}}(\boldsymbol{u})|\right)\end{equation} \tag{ B.24 }$$

$$\begin{equation}\hspace{32pt} = \left(1-\delta_{n}\right){\rm \|}\sigma^{\otimes n}\|_{W}+\delta_{n}{\rm \|}\eta^{\otimes n}\|_{W}\end{equation} \tag{ B.25 }$$

$$\begin{equation}\hspace{32pt} = \left(1-\delta_{n}\right){\rm \|}\sigma\|_{W}^{n}+\delta_{n}{\rm \|}\eta\|_{W}^{n}. \end{equation} \tag{ B.26 }$$

Here we have exploited that the sign of W_η^⊗n(u) and the sign of W_σ^⊗n(u) are always the same. Subbing this in

$$\begin{equation} \frac{{\mathscr{M}\left(\tilde{\sigma}_{n}\right)}-{\mathscr{M}\left(\sigma^{\otimes n}\right)}}{n}=\frac{1}{n}\log\left(\left(1-\delta_{n}\right)+\delta_{n}\left(\frac{{\rm \|}\eta\|_{W}}{{\rm \|}\sigma\|_{W}}\right)^{n}\right), \end{equation} \tag{ B.27 }$$

but by assumption ∥η∥_W > ∥σ∥_W unless ∥σ∥_W is maximal for all states that are negative on $\mathcal {N}$, so the limit need not go to 0. Thus asymptotic continuity cannot hold generally.    □

This result is not actually terribly surprising. Suppose we have a preparation apparatus that always prepares σ^⊗n. Now further suppose that we rebuild our apparatus so that with probability δ_n it will instead produce η^⊗n with a far greater amount of negativity. Then it is intuitively obvious that we should be able to extract more negativity from the new apparatus just by sacrificing a few copies of the output state to determine whether we have produced σ or η. Of course as n goes to infinity this will only work if δ_n goes to zero slowly enough, but this argument does clarify the irrelevance of asymptotic continuity.

Essentially asymptotic continuity fails because it is possible that access to a very large amount of resource, even with small probability, can dramatically improve our preparation procedure. Notice that the opposite is not (obviously) true: if our machine fails with a very small probability this does not make it useless. Indeed, if we had a promise of the form $\tilde {\sigma }_{n}=\left (1-\delta \right )\sigma ^{\otimes n}+\delta \eta ^{\otimes n}$ then we could just sacrifice some small number of registers to check that the output state was in fact σ^⊗n.