Super-exponential distinguishability of correlated quantum states

In the problem of asymptotic binary i.i.d. state discrimination, the optimal asymptotics of the type I and the type II error probabilities is in general an exponential decrease to zero as a function of the number of samples; the set of achievable exponent pairs is characterized by the quantum Hoeffding bound theorem. A super-exponential decrease for both types of error probabilities is only possible in the trivial case when the two states are orthogonal, and hence can be perfectly distinguished using only a single copy of the system. In this paper we show that a qualitatively different behaviour can occur when there is correlation between the samples. Namely, we use gauge-invariant and translation-invariant quasi-free states on the algebra of the canonical anti-commutation relations to exhibit pairs of states on an infinite spin chain with the properties that a) all finite-size restrictions of the states have invertible density operators, and b) the type I and the type II error probabilities both decrease to zero at least with the speed $e^{-nc\log n}$ with some positive constant $c$, i.e., with a super-exponential speed in the sample size $n$. Particular examples of such states include the ground states of the $XX$ model corresponding to different transverse magnetic fields. In fact, we prove our result in the setting of binary composite hypothesis testing, and hence it can be applied to prove super-exponential distinguishability of the hypotheses that the transverse magnetic field is above a certain threshold vs. that it is below a strictly lower value.


I. INTRODUCTION
In the problem of simple binary state discrimination, an experimenter is presented with a quantum system that is either in some state ω (0) or in another state ω (1) . The experimenter's task is to guess which one the true state of the system is, based on measurements on the system. It is easy to see that even the most elaborate measurement and classical post-processing scheme cannot outperform single 2outcome (binary) measurements when the goal is to minimize the probability of an erroneous decision. More precisely, there are two types of error probabilities to consider: erroneously identifying the state as ω (1) (type I error), or erroneously identifying the state as ω (0) (type II error), and the goal is to minimize some combination of the two. It is easy to see that (in the finite-dimensional case, at least), perfect discrimination (i.e., when both error probabilities are zero) is possible if and only if the density operators of the two states have orthogonal supports.
The error probabilities can be reduced if the experimenter has access to multiple identical copies of the system, and in the asymptotic analysis of the problem one is interested in the achievable asymptotic behaviours of the two error probabilities along all possible sequences of binary measurements (tests) as the number of copies tends to infinity. In general, the best achievable asymptotics is an exponential decrease to zero for both error probabilities; the set of the achievable exponent pairs is described by the quantum Hoeffding bound theorem [4,12,26]. Faster (super-exponential) decrease is possible if and only if the supports of the states are different. For instance, if supp ω (1) ⊆ supp ω (0) then there exists a test sequence along which the type I error is constant zero (hence its exponent is +∞), while the type II error decreases exponentially fast (with the exponent being the Rényi zero-divergence of ω (0) and ω (1) ). A faster than exponential decrease for both error probabilities is possible if and only if the supports are orthogonal, in which case both errors can be made zero trivially for any finite number of copies.
The above are well-known in the i.i.d. (independent and identically distributed) case, i.e., when all the samples are prepared in the same state, and there is no correlation between the different samples. Correlated scenarios can be conveniently described using the concept of the C * -algebra of an infinite spin chain, C Z = ⊗ k∈Z B(H), where H is a finite-dimensional Hilbert space describing a single system. In this case the candidate states ω (0) and ω (1) can be described by positive linear functionals on C Z that take 1 on the identity; their restrictions to any sub-algebra ⊗ k∈Λ B(H) corresponding to a finite subset Λ of samples (equivalently, a finite part of the chain) can be described by density operators in the usual way. A state on the infinite chain is translation-invariant if the density operator of any finite subsystem Λ is the same as that of any of its translates; in particular, the single-site density operators are all the same (i.e., the outcomes of the same measurement performed at different sites are identically distributed). In this picture, a measurement on n consecutive samples is described by a measurement on a length n part of the chain, and the asymptotics is studied in the setting where this length is allowed to go to infinity. Obviously, error exponents are more difficult to determine in the correlated scenario, but rather general results are available in the setting of Stein's lemma, where one of the errors is not required to decrease exponentially [8], and in the setting of the Hoeffding bound for thermal states of translation-invariant finite-range Hamiltonians, and more generally, for states that satisfy a certain factorization property [17]. In these cases, however, the entropic quantities (Umegaki-and quantum Rényi relative entropies) characterizing the achievable exponent pairs are given by regularized formulas, and cannot be explicitly computed in general.
A particular class of correlated states where explicit formulas are available can be obtained from translation-invariant and gauge-invariant quasi-free states on the algebra of canonical anti-commutation relations (CAR algebra). Such a state is specified by a measurable function on [0, 2π) with values in [0, 1], called the symbol of the state; see Section II below for details. The achievable exponent pairs were determined for a pair of such states in [25], with explicit expressions for the relevant entropic expressions, in the case where the symbols of the two states, denoted byq andr, are bounded away from 0 and 1 in the sense that η ≤q(x),r(x) ≤ 1 − η for all x ∈ [0, 2π) for some η > 0. In this case the regularized quantum Rényi α-divergences of the two states are finite for every α > 0, and the best achievable asymptotics is an exponential decay for both error probabilities.
Our main contribution in this paper is showing that for certain pairs of quasi-free states, superexponential discrimination is possible. More precisely, we show that if the symbolsq andr are such that there exists a non-degenerate interval on whichq is constant 0 andr is constant 1 then there exists a sequence of tests along which both error probabilities decrease at least with the speed e −nc log n , where n is the sample size. In the same time, unlessq is constant zero andr is constant 1 (up to sets of measure zero), then all the local densities of both states are invertible, and hence it is not only impossible to make both error probabilities vanish for a finite sample size, but if one of the error probabilities is made zero then the other is necessarily equal to 1. This is very different from what can be seen in the i.i.d. case, and to the best of our knowledge, this is the first time that such a behaviour is presented in the literature.
The structure of the paper is as follows. In Section II A we review the necessary basics about quasi-free states on the CAR algebra. In Section II B we explain the notions of error exponents and super-exponential distinguishability for translation-invariant states on the spin chain and on the CAR algebra. In Section III we prove our main result described above. In fact, we state and prove a more general result in the framework of composite state discrimination, showing super-exponential distinguishability of two sets of quasi-free states with invertible local density operators. In Section IV we give various characterizations of super-exponential distinguishability of states in terms of regularized divergences.

II. PRELIMINARIES
A. Quasi-free states on the CAR algebra Here we summarize the necessary basics about quasi-free states on the CAR algebra. For more details and proofs we refer to [1,9,10,29,32].
For a complex Hilbert space H, we will denote the set of bounded operators on H by B(H), and use the notation T (H) := {T ∈ B(H) : 0 ≤ T ≤ I} for the set of tests on H.
For vectors ϕ 1 , . . . , ϕ k in a complex Hilbert space H, let denote their anti-symmetrized tensor product, where S k stands for the set of permutations of k elements and ε(σ) for the sign of the permutation σ. For any k ∈ N \ {0}, the k-th anti-symmetric tensor power of H is where the overline denotes the closure in operator norm, and we define H ∧0 := C. The Hilbert space of a fermionic system with single-particle Hilbert space H is the anti-symmetric Fock space For each ϕ ∈ H, the corresponding creation operator c(ϕ) is the unique bounded linear extension of the map and the corresponding annihilation operator is its adjoint, a(ϕ) := c(ϕ) * . These operators satisfy the canonical anti-commutation relations (CARs), The C*-algebra generated by {a(ϕ) : ϕ ∈ H} is called the algebra of the canonical anti-commutation relations (or CAR-algebra) corresponding to the single-particle Hilbert space H, and is denoted by CAR (H).
Note that ϕ → c(ϕ) is complex linear and ϕ → a(ϕ) is complex anti-linear. Thus, if H is separable and is an orthonormal basis (ONB) in it then CAR (H) is the closure of the linear span of the identity and all the multinomials of the form a(e i1 ) * . . . a(e in ) * a(e jm ) . . . a(e j1 ), i 1 < . . . < i n , j 1 < . . . < j m . For any isometry/unitary V : is a unitary from Γ(H) to (C 2 ) ⊗d , and is easy to verify. The map U e (·)U * e : B(F (H)) = CAR (H) → B((C 2 ) ⊗d ) = B(C 2 ) ⊗d is called the Jordan-Wigner isomorphism corresponding to the given ONB. The particle number operator is The eigen-values of N H are 0, . . . , d, with spectral projections where [d] := {1, . . . , d}. Note that N H is defined in a basis-independent way, and the equalities above are valid for any ONB. A state on CAR (H) is a positive linear functional that takes the value 1 on I. For any positive semidefinite (PSD) operator Q ∈ B(H) with Q ≤ I there exists a unique state ω Q on CAR (H) (called the gauge-invariant quasi-free state with symbol Q) with the property It is easy to verify that when H is finite-dimensional, the density operator ω Q of ω Q can be explicitly given as where Q = d j=1 q j |e j e j | is any eigen-decomposition of Q, and U e is the unitary corresponding to the ONB (e j ) d j=1 as in (II.2). Note that for all . This implies immediately that if 1 is not an eigen-value of Q then ω Q can be written as Quasi-free states emerge as equilibrium states of non-interacting fermionic systems. For instance, if the single-particle Hamiltonian H of a system of non-interacting fermions is such that e −βH is trace-class then the Gibbs state of the system at inverse temperature β is the quasi-free state with symbol Q = e −βH Consider now a fermionic chain with a single mode at each site. The single-particle Hilbert space of this system is H = ℓ 2 (Z), the standard basis of which we denote by which is easily seen to be equivalent to U trans Q = QU trans , i.e., the translation-invariance of the symbol Q. For instance, in the above example a translation-invariant single-particle Hamiltonian H yields a translation-invariant quasi-free state as the equilibrium state of the system. Translation-invariant operators on ℓ 2 (Z) commute with each other and they are simultaneously diagonalized by the Fourier transformation That is, every translation-invariant operator A arises in the form A = F * Mâ F , where Mâ denotes the multiplication operator by a bounded measurable functionâ on [0, 2π). As a consequence, the matrix entries of translation-invariant operators in the canonical ONB are constants along diagonals; more explicitly, for any translation-invariant operator A ∈ B(ℓ 2 (Z)), A measurement on a subsystem corresponding to modes at the sites n := {0, . . . , n − 1} has measurement operators in the C * -subalgebra A n ⊆ CAR ℓ 2 (Z) generated by {a(ϕ) : ϕ ∈ H n }, This subalgebra is naturally isomorphic to CAR C n . It is easy to see that if the state of the infinite chain is given by a quasi-free state with symbol Q then the statistics of any such local measurement is given by the quasi-free state ω Qn on CAR C n with symbol Q n := V * n QV n , where V n is the natural embedding of C n into ℓ 2 (Z).
Lemma II.1. Letâ : [0, 2π) → [0, +∞) be a non-negative bounded measurable function, and let A = F * Mâ F be the corresponding translation-invariant operator on ℓ 2 (Z). The following are equivalent: (i) 0 is an eigen-value of V * n AV n for some n ∈ N; (ii) 0 is an eigen-value of V * n AV n for every n ∈ N; (iii) A = 0; (iv)â is equal to 0 almost everywhere.
Proof. The equivalence (iv)⇐⇒(iii) is obvious, as are the implications (iii)=⇒(ii)=⇒(i), and hence we only need to prove (i)=⇒(iv). Assume therefore that V * n AV n ψ = 0 for some ψ ∈ C n \ {0}. Then whenceâ 1/2 F V n ψ = 0 almost everywhere. Since F V n ψ is a non-zero trigonometric polynomial that can only have finitely many zeros, this implies thatâ is 0 almost everywhere.
Ifq is neither almost everywhere zero nor almost everywhere 1 then for every n ∈ N, ω Qn is an invertible density operator on Γ(C n ).
Proof. Applying Lemma II.1 toâ :=q yields that 0 is not an eigen-value of Q n for any n ∈ N. Applying Lemma II.1 toâ := 1 −q yields that 1 is not an eigen-value of Q n , either, for any n ∈ N. Thus, the assertion follows from (II.5).
Finally, a symbol Q on C n is translation-invariant (or rotation-invariant), if it commutes with the n-dimensional translation unitary U trans where the addition is modulo n. Such operators are also called circular, and are simultaneously diagonalized by the n-dimensional discrete Fourier transformation That is, U trans

B. Asymptotic binary state discrimination
The infinite spin chain algebra with single-site finite-dimensional Hilbert space H is defined as Equivalently, ω is a positive linear functional on the C * -algebra C Z (H), with ω(I) = 1, such that for the translation automorphism τ we have ω • τ = ω, and the ω Λ are the density operators of its restrictions onto C Λ (H). Given two sets translation-invariant states Ω (0) = {ω (0,i) } i∈I and Ω (1) = {ω (1,j) } j∈J , a state discrimination protocol of sample size n to decide if the true state of the system belongs to Ω (0) (null-hypothesis H 0 ) or to Ω (1) (alternative hypothesis H 1 ), is specified by a test T n ∈ C [1,n] (H) with 0 ≤ T n ≤ I, representing a measurement with outcomes 0 and 1, with corresponding measurement operators T n and I − T n , respectively. If the outcome of the measurement is k, the experimenter accepts hypothesis H k to be true. The (worst-case) type I error probability of incorrectly rejecting H 0 , and the type II error probability of incorrectly accepting H 0 , respectively, are given by A test T n is projective, if T 2 n = T n . Given a sequence of tests T = (T n ) n∈N , with T n ∈ C [1,n] (H), n ∈ N, the corresponding type I and type II error exponents are defined, respectively, as We say that Ω (0) and Ω (1) can be super-exponentially distinguished, if there exists a test sequence T along which α exp ( T ) = +∞ = β exp ( T ).
As it was shown in [2] (see also [23,Section 5.3] for a detailed exposition) every translation-invariant gauge-invariant quasi-free state ω on CAR ℓ 2 (Z) can be mapped into a translation-invariant stateω on the spin chain C Z (C 2 ) with the preservation of the locality structure. In particular, given two sets Ω (0) = {ω (0,i) } i∈I and Ω (1) = {ω (1,j) } j∈J of such states on CAR ℓ 2 (Z) , and numbers α, β ∈ [0, 1], there exists a (projective) test T n ∈ C [1,n] (C 2 ) such that sup i∈I Trω (0,i) [1,n] (I − T n ) = α, sup j∈J Trω (1,j) [1,n] T n = β, if and only if there exists a (projective) test S n ∈ CAR (H n ) such that sup i∈I ω (0,i) (I − S n ) = α, sup j∈J ω (1,j) (S n ) = β. Hence, in order to explore the achievable error exponent pairs for the pairΩ (0) = {ω (0,i) } i∈I ,Ω (1) = {ω (1,j) } j∈J , one can work directly on the CAR algebra with Ω (0) and Ω (1) . Thus, we introduce the following: . We say that Ω Q and Ω R can be super-exponentially distinguished, if there exists a sequence T n ∈ CAR (H n ), n ∈ N, such that By the above, the sets of states Ω Q and Ω R on the CAR algebra are super-exponentially distinguishable if and only if so are the sets of statesΩ Q andΩ R on the spin chain.

III. SUPER-EXPONENTIAL DISTINGUISHABILITY
In this section we prove the main result of the paper: Theorem III.1. Let {q i } i∈I and {r j } j∈J be measurable functions from [0, 2π) to [0, 1], defining the translation-invariant quasi-free states Ω Q := {ω Q (i) } i∈I , Ω R := {ω R (j) } j∈J on CAR ℓ 2 (Z) . If there exists an interval [µ, ν] ⊆ [0, 2π) of positive length such thatq i is constant 0 andr j is constant 1 on it for every i ∈ I and j ∈ J , then Ω Q and Ω R are super-exponentially distinguishable.
If, moreover,q i is not almost everywhere 0 andr j is not almost everywhere 1 on [0, 2π) for every i ∈ I and j ∈ J , then their local densities ω Q (i) n and ω R (j) n , n ∈ N, are all invertible.
In fact, the above theorem follows immediately from a more detailed statement given in Theorem III.8 below, which we prove in several steps.
The main intuition behind the proof is the following. (For simplicity we take |I| = |J | = 1, Q (1) =: Q, R (1) =: R.) Although the symbols Q, R ∈ B(ℓ 2 (Z)) commute with each other, this is not true anymore for their restrictions Q n and R n onto C n , unless Q or R is a constant multiple of the identity. On the other hand, if instead of restrictions of translation-invariant symbols onto finite-dimensional subspaces we considered translation-invariant symbols Q n , R n on the single-particle Hilbert space C n of a length n finite chain (with periodic boundary conditions, or equivalently, rotation-invariant symbols on a finite ring) then any two such symbols would commute with each other, and would be simultaneously diagonalized by the discrete Fourier transformation; see the end of Section II A. Now, the analogous condition to the one in Theorem III.1 in the finite-dimensional case would be that the functionsq n ,r n ∈ C n satisfŷ q n (k) = 0 = 1 −r n (k), k = l + 1, . . . , l + m for some l, m ∈ C n (modulo n). Hence, for the projection E n := F * n l+m k=l+1 1 {k} 1 {k} F n , we would have Tr E n Q n = 0 = Tr E n (I − R n ). A key technical ingredient of our proof, given in Lemma III.2 and Corollary III.3 below, is that for any such projection, one can construct a test, using the spectral decomposition of the particle number operator on the subspace ran E n , such that the type I and type II error probabilities are upper bounded by a simple expression involving only Tr E n , Tr E n Q n and Tr E n (I − R n ); in particular, if the latter two are 0 then so are the error probabilities. When Q n and R n are the non-commuting restrictions of Q, R ∈ B(ℓ 2 (Z)), we can still follow the above strategy, where instead of making the upper bounds exactly zero, we can make them sufficiently small, as shown in Lemmas III.5, III. 6  For any A ∈ B(H) with 0 ≤ A ≤ I, Proof. Let d := dim H, S := S H , and , (III.11) Using (III.10) and the first bound in (III.11), we get Using that for all k ≤ ⌊d/2⌋, we get Corollary III.3. Let H be a Hilbert space. For every non-zero finite-rank projection E on H there exists an even projection T ∈ span{a(ϕ) : ϕ ∈ ran E} such that for every A ∈ B(H), 0 ≤ A ≤ I, (III.12) Proof. Let V E be the identical embedding of ran E into H. For any even projection S ∈ CAR (ran E), Corollary III.4. Let {Q (i) } i∈I , {R (j) } j∈J ⊆ B ℓ 2 (Z) be symbols of quasi-free states on CAR ℓ 2 (Z) . Assume that there exists a sequence of non-zero projections E n on C n , n ∈ N, such that Then Ω Q and Ω R can be super-exponentially distinguished by even projective tests.
Proof. Let T n be the projection corresponding to V n E n V * n as in Corollary III.3, where V n is the canonical embedding of C n into ℓ 2 (Z). Since Tr E n = Tr V n E n V * n , Tr E n Q (i) . (III.14) The statement follows by taking the infima over the respective index sets, and then the liminf in n in the above inequalities, and noting that 0 ≤ Tr En 2n log 8 ≤ 1 2 log 8.
Hence, in order to complete the proof of Theorem III.1, it is sufficient to show that if Ω Q and Ω R are as in Theorem III.1 then a sequence of projections as in Corollary III.4 exists. For this, we will need some simple facts about Fourier transforms; see, e.g., [31] for details.
In particular, recall that the n-th partial sum of the Fourier series of an integrable function on [0, 2π) is given by is the Dirichlet kernel, and ⋆ stands for the convolution. The n-th Césaro mean of the partial sums is where F n (x) := 1 n n−1 k=0 D n (x) = 1 2πn sin 2 (nx/2) sin 2 (x/2) is the Fejér kernel. The following may be known; however, as we have not found a reference in the literature, we provide a detailed proof. Recall that V n is the canonical embedding of C n into ℓ 2 (Z).
Lemma III.5. Letâ be a bounded measurable complex-valued function on [0, 2π) and A = F * Mâ F . Then the diagonal matrix entries of F n V * n AV n F * n are given by Proof. Let A n := V * n AV n . By (II.6) and (II.7), where in the third equality above we replaced the summation over j, l with a single summation over m = j − l.
Let Ω Q and Ω R be as in Theorem III.1. Then there exists a positive constant c and a sequence of even projections T n ∈ span{a(ϕ) : ϕ ∈ H n } such that In particular, Ω Q and Ω R can be super-exponentially distinguished by even projective tests. If, moreover,q i is not almost everywhere 0 andr j is not almost everywhere 1 on [0, 2π) for every i ∈ I and j ∈ J , then their local densities ω Q (i) n and ω R (j) n , n ∈ N, are all invertible.
Proof. Let T n be as in the proof of Corollary III.4 with E n := E n,δ , n ∈ N, for some δ as in Lemma III.7. The inequalities in (III.13)-(III.14) combined with (III.15) and (III.16) yield (III. 17) The assertion about the invertibility of the density operators follows immediately from Corollary II.2.
Example III.9. Consider the XX model with local Hamiltonian on B(C 2 ) [1,n] given by where σ x,k is the Pauli x operator That is, the experimenter's task is to test whether the transverse magnetic field is below h 0 or above h 1 , by making measurements on a finite part of the chain. It is straightforward to verify thatq h is constant zero on [µ := arccos f (h 1 ), ν := arccos f (h 0 )] for every h ≤ h 0 , whiler h is constant one on [µ, ν] for every h ≥ h 1 , and hence, by Theorem III.1, the two hypotheses can be tested with super-exponentially decreasing error probabilities. By Corollary II.2, the local densities ω Q (h) n are invertible for every h 1 ≤ h < 1, and the local densities ω R (h) n are invertible for every −1 < h ≤ h 0 . A variant of the above problem is when the experimenter's task is to test whether the transverse magnetic field is between h 0 and h ′ 0 or between h 1 and h ′ It is straightforward to verify that this problem satisfies the conditions in Theorem III.1 with µ = arccos h 1 , ν = arccos h ′ 0 , and therefore the two hypotheses can be tested with super-exponentially decreasing error probabilities, and, moreover, all local densities are invertible for every size n.

IV. COMMENTS ON ORTHOGONALITY
In this section we discuss some relations between three concepts: a) the orthogonality of a pair of states, b) their super-exponential distinguishability, and c) certain distinguishability measures taking infinite value on the given pair. We start with an overview of the well-known relations between these for density operators on a finite-dimensional Hilbert space, and then discuss a possible extension to pairs of translation-invariant states on an infinite spin chain.
In particular, Theorem IV.1 remains valid if we replace the lim inf with lim sup in the definition of the error exponents in (II.8), and define super-exponential distinguishability accordingly, and we also replace the lim inf with lim sup in the definition of the regularized distinguishability measures in (IV.23).
Remark IV.3. Two states (positive linear normalized functionals) ω (0) and ω (1) on a C * -algebra are defined to be orthogonal in [30,Definition 1.14.1] if ω (0) − ω (1) = 2, where the norm is the usual functional norm; this is equivalent to χ(ω (0) ω (1) ) = +∞ in our notation. The above arguments show that this notion of orthogonality may not be the best suited for the study of asymptotic state discrimination on an infinite spin chain; in particular, any two translation-invariant product states are orthogonal according to this definition, irrespective of whether the density operators of their local restrictions are orthogonal or not.
In contrast, if we define ω (0) and ω (1) on an infinite spin chain to be orthogonal if χ(ω (0) ω (1) ) = +∞ then for translation-invariant product states this becomes equivalent to the usual orthogonality of their single-site restrictions ω (0) [1] and ω (1) [1] . Another appealing feature of this notion of orthogonality of states is that it is equivalent to various regularized distinguishability measures being +∞, according to Theorem IV.1, which gives a nice generalization of the analogous single-site characterizations of orthogonality given in (IV.20) and (IV.22). Of course, this notion of orthogonality is limited to pairs of translation-invariant states on an infinite spin chain, and does not make sense in general for pairs of states on an abstract C * -algebra.
Remark IV.4. Clearly, if any (and hence all) of (i)-(vi) in Theorem IV.1 holds then we have D α (ω (0) ω (1) ) = +∞ for any quantum Rényi α-divergence with α ∈ (0, 1) that is monotone non-increasing under 2-outcome measurements, i.e., under the type of CPTP maps given in (IV.21). Here, we say that D α is a quantum Rényi α-divergence if it is defined on all pairs of density operators on any finite-dimensional Hilbert space, and for commuting states it reduces to the classical Rényi α-divergence of the diagonal elements of the two density operators in a common eigen-basis. One such example is Matsumoto's maximal α-divergence [22] for every α ∈ (0, 1); however, at the moment we do not know if the regularized maximal α-divergence being +∞ implies the other properties listed in Theorem IV.1.

V. CONCLUSION
We have shown that translation-invariant quasi-free states with defining functionsq andr are superexponentially distinguishable if there is an interval [µ, ν] of non-zero length such that one of the functions is constant 0 and the other one is constant 1 on this interval. We have shown that in this case both errors decreases at least as fast as e −nc log n in the sample size n; it is however, an open question whether this is in fact the optimal asymptotics, or a faster decrease, e.g., e −cn 1+δ with some δ > 0 can be attained. This can be asked for the class of functions that we considered, but it is also natural to ask if there is any upper bound on the speed of convergence to zero for general pairs of translation-invariant states on a spin chain.
It is known that a translation-invariant quasi-free state ω Q is pure (i.e., an extremal point of the convex set of states) if and only if the corresponding functionq is an indicator function, i.e.,q = 1 BQ for some measurable subset B Q of [0, 2π) (see, e.g., [10]). Two such pure states ω Q and ω R are different if and only if B Q and B R are different in the measure-theoretic sense, i.e., the Lebesgue measure of (B Q \B R )∪(B R \B Q ) is positive. This motivates to ask whether the following extension of our result is true: Ifq andr are measurable functions from [0, 2π) to [0, 1] such that there exists a measurable set B ⊆ [0, 2π) of positive Lebesgue measure on whichq is constant 0 andr is constant 1 then ω Q and ω R can be super-exponentially distinguished. In particular, this would imply the super-exponential distinguishability of any two different pure translation-invariant quasi-free states.