Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1510.04682 · Journal of Mathematical Physics

Second-order asymptotics for quantum hypothesis testing in settings beyond i.i.d.—quantum lattice systems and more

Nilanjana Datta, Yan Pautrat, Cambyse Rouzé

Published Jun 1, 2016• 24 citations
arXivHTMLPDF

Abstract

Quantum Stein’s lemma is a cornerstone of quantum statistics and concerns the problem of correctly identifying a quantum state, given the knowledge that it is one of two specific states (ρ or σ). It was originally derived in the asymptotic i.i.d. setting, in which arbitrarily many (say, n) identical copies of the state (ρ⊗n or σ⊗n) are considered to be available. In this setting, the lemma states that, for any given upper bound on the probability αn of erroneously inferring the state to be σ, the probability βn of erroneously inferring the state to be ρ decays exponentially in n, with the rate of decay converging to the relative entropy of the two states. The second order asymptotics for quantum hypothesis testing, which establishes the speed of convergence of this rate of decay to its limiting value, was derived in the i.i.d. setting independently by Tomamichel and Hayashi, and Li. We extend this result to settings beyond i.i.d. Examples of these include Gibbs states of quantum spin systems (with finite-range, translation-invariant interactions) at high temperatures, and quasi-free states of fermionic lattice gases.

DOI
10.1063/1.4953582

Related papers

Generalized Stein's lemma and asymptotic equipartition property for subalgebra entropies

Gao, Li, Mizanur Rahaman

The quantum Stein's lemma is a fundamental result of quantum hypothesis testing in the context of distinguishing two quantum states. The ``generalized quantum Stein's lemma'' asserts that this result is true in a general framework where one of the states is replaced by convex sets of quantum states. Formulated in 2008, the generalized Stein's lemma is one of the most influential results in quantum information theory that links resource convertibility to quantum hypothesis testing. However, in 2023 a logical gap was found in the original proof of the generalized Stein's lemma and since then there has been an enormous effort in resolving this issue. In this work, we show that the assertion of the generalized Stein's lemma is true for the setting where the second hypothesis is the state space of any finite-dimensional subalgebra. This is obtained through a strong asymptotic equipartition property for smooth subalgebra entropies that applies to any fixed smoothing parameter. In fact, we obtain a stronger second-order analysis for the hypothesis-testing relative entropy in this setting. As an application in resource theory, we show that the relative entropy of a subalgebra is the asymptotic dilution cost under suitable operations. This provides a possible route to establishing a connection between different quantum resources based on subalgebras. After finishing this article, we learned that there are two independent works (by Hayashi-Yamasaki, and Lami) that finally resolve the generalized Stein's lemma in its full generality. However, we provide an alternative proof in a special case using different operator-algebraic techniques that may be of independent interest.

arXiv:2401.03090•Jan 5, 2024•0 citations

A generalization of quantum Stein's Lemma

Brandão, FGSL, Martin Bodo Plenio

Given many independent and identically-distributed (i.i.d.) copies of a quantum system described either by the state ρ or σ (called null and alternative hypotheses, respectively), what is the optimal measurement to learn the identity of the true state? In asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying ρ instead of σ, while requiring that the probability that σ is identified in the place of ρ is bounded by a small fixed number. Quantum Stein's Lemma identifies the asymptotic exponential rate at which the specified error probability tends to zero as the quantum relative entropy of ρ and σ. We present a generalization of quantum Stein's Lemma to the situation in which the alternative hypothesis is formed by a family of states, which can moreover be non-i.i.d. We consider sets of states which satisfy a few natural properties, the most important being the closedness under permutations of the copies. We then determine the error rate function in a very similar fashion to quantum Stein's Lemma, in terms of the quantum relative entropy. Our result has two applications to entanglement theory. First it gives an operational meaning to an entanglement measure known as regularized relative entropy of entanglement. Second, it shows that this measure is faithful, being strictly positive on every entangled state. This implies, in particular, that whenever a multipartite state can be asymptotically converted into another entangled state by local operations and classical communication, the rate of conversion must be non-zero. Therefore, the operational definition of multipartite entanglement is equivalent to its mathematical definition. © Springer-Verlag 2010.

•Mar 1, 2010•55 citations

Asymptotic Error Rates in Quantum Hypothesis Testing

K. M. R. Audenaert, M. Nussbaum, Arleta Szkoła +1 more

We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance, which is the corresponding quantity in classical symmetric hypothesis testing. The proof relies on two new techniques introduced by the authors, which are also well suited to tackle the corresponding problem in asymmetric hypothesis testing, yielding the quantum generalisation of the classical Hoeffding bound. This has been done by Hayashi and Nagaoka for the special case where the states have full support. The goal of this paper is to present the proofs of these results in a unified way and in full generality, allowing hypothesis states with different supports. From the quantum Hoeffding bound, we then easily derive quantum Stein’s Lemma and quantum Sanov’s theorem. We give an in-depth treatment of the properties of the quantum Chernoff distance, and argue that it is a natural distance measure on the set of density operators, with a clear operational meaning.

arXiv:0708.4282•Communications in Mathematical Physics•Feb 7, 2008•274 citations

Second-order asymptotics for quantum hypothesis testing

Ke Li

In the asymptotic theory of quantum hypothesis testing, the minimal error probability of the first kind jumps sharply from zero to one when the error exponent of the second kind passes by the point of the relative entropy of the two states in an increasing way. This is well known as the direct part and strong converse of quantum Stein’s lemma. Here we look into the behavior of this sudden change and have make it clear how the error of first kind grows smoothly according to a lower order of the error exponent of the second kind, and hence we obtain the second-order asymptotics for quantum hypothesis testing. This actually implies quantum Stein’s lemma as a special case. Meanwhile, our analysis also yields tight bounds for the case of finite sample size. These results have potential applications in quantum information theory. Our method is elementary, based on basic linear algebra and probability theory. It deals with the achievability part and the optimality part in a unified fashion.

arXiv:1208.1400•The Annals of Statistics•Feb 1, 2014•157 citations

On Composite Quantum Hypothesis Testing

Mario Berta, Fernando G. S. L. Brandão, Christoph Hirche

Abstract We extend quantum Stein’s lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states $$\rho ^{\otimes n}$$ ρ ⊗ n against convex combinations of quantum states $$\sigma ^{\otimes n}$$ σ ⊗ n can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein’s lemma for classical probability distributions and lift the result to the non-commutative setting by using elementary properties of quantum entropy. Finally, our findings also imply an improved recoverability lower bound on the conditional quantum mutual information in terms of the regularized quantum relative entropy—featuring an explicit and universal recovery map.

arXiv:1709.07268•Communications in Mathematical Physics•Jun 10, 2021•34 citations

Generalised quantum Stein's lemma more robust than ever

Filippo Girardi, Kuan-Yi Lee, Masahito Hayashi +1 more

The generalised quantum Stein's lemma is a key result in quantum hypothesis testing, and connects this fundamental primitive of quantum information processing with quantum resource manipulation, a task that is central for technological applications. Prior works have proved this statement in the idealised setting of independent and identically distributed (i.i.d.) sequences of quantum states, and recent extensions consider also sources that are 'close' to i.i.d., according to the strict notion put forth by Mazzola, Sutter, and Renner. For several applications, however, one would need to consider yet more general sources. We establish a version of the generalised quantum Stein's lemma that is conceptually much simpler and general, as it applies to any source that is asymptotically close to an i.i.d. state with respect to the normalised quantum Wasserstein distance of order 1. As an immediate consequence, we solve the Stein exponent of a scenario where the null hypothesis is arbitrarily varying, expressing it in terms of i.i.d. Stein exponents corresponding to arbitrary states in the convex hull of the null hypothesis base set.

quant-pharXiv:2609.17309•Sep 16, 2026•0 citations
Quantum Stein’s lemma is a cornerstone of quantum statistics and concerns the problem of correctly identifying a quantum state, given the knowledge that it is one of two specific states (ρ or σ). It was originally derived in the asymptotic i.i.d. setting, in which arbitrarily many (say, n) identical copies of the state (ρ⊗n or σ⊗n) are considered to be available. In this setting, the lemma states that, for any given upper bound on the probability αn of erroneously inferring the state to be σ, the probability βn of erroneously inferring the state to be ρ decays exponentially in n, with the rate of decay converging to the relative entropy of the two states. The second order asymptotics for quantum hypothesis testing, which establishes the speed of convergence of this rate of decay to its limiting value, was derived in the i.i.d. setting independently by Tomamichel and Hayashi, and Li. We extend this result to settings beyond i.i.d. Examples of these include Gibbs states of quantum spin systems (with finite-range, translation-invariant interactions) at high temperatures, and quasi-free states of fermionic lattice gases.
The quantum Stein's lemma is a fundamental result of quantum hypothesis testing in the context of distinguishing two quantum states. The ``generalized quantum Stein's lemma'' asserts that this result is true in a general framework where one of the states is replaced by convex sets of quantum states. Formulated in 2008, the generalized Stein's lemma is one of the most influential results in quantum information theory that links resource convertibility to quantum hypothesis testing. However, in 2023 a logical gap was found in the original proof of the generalized Stein's lemma and since then there has been an enormous effort in resolving this issue. In this work, we show that the assertion of the generalized Stein's lemma is true for the setting where the second hypothesis is the state space of any finite-dimensional subalgebra. This is obtained through a strong asymptotic equipartition property for smooth subalgebra entropies that applies to any fixed smoothing parameter. In fact, we obtain a stronger second-order analysis for the hypothesis-testing relative entropy in this setting. As an application in resource theory, we show that the relative entropy of a subalgebra is the asymptotic dilution cost under suitable operations. This provides a possible route to establishing a connection between different quantum resources based on subalgebras. After finishing this article, we learned that there are two independent works (by Hayashi-Yamasaki, and Lami) that finally resolve the generalized Stein's lemma in its full generality. However, we provide an alternative proof in a special case using different operator-algebraic techniques that may be of independent interest.
Given many independent and identically-distributed (i.i.d.) copies of a quantum system described either by the state ρ or σ (called null and alternative hypotheses, respectively), what is the optimal measurement to learn the identity of the true state? In asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying ρ instead of σ, while requiring that the probability that σ is identified in the place of ρ is bounded by a small fixed number. Quantum Stein's Lemma identifies the asymptotic exponential rate at which the specified error probability tends to zero as the quantum relative entropy of ρ and σ. We present a generalization of quantum Stein's Lemma to the situation in which the alternative hypothesis is formed by a family of states, which can moreover be non-i.i.d. We consider sets of states which satisfy a few natural properties, the most important being the closedness under permutations of the copies. We then determine the error rate function in a very similar fashion to quantum Stein's Lemma, in terms of the quantum relative entropy. Our result has two applications to entanglement theory. First it gives an operational meaning to an entanglement measure known as regularized relative entropy of entanglement. Second, it shows that this measure is faithful, being strictly positive on every entangled state. This implies, in particular, that whenever a multipartite state can be asymptotically converted into another entangled state by local operations and classical communication, the rate of conversion must be non-zero. Therefore, the operational definition of multipartite entanglement is equivalent to its mathematical definition. © Springer-Verlag 2010.
We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance, which is the corresponding quantity in classical symmetric hypothesis testing. The proof relies on two new techniques introduced by the authors, which are also well suited to tackle the corresponding problem in asymmetric hypothesis testing, yielding the quantum generalisation of the classical Hoeffding bound. This has been done by Hayashi and Nagaoka for the special case where the states have full support. The goal of this paper is to present the proofs of these results in a unified way and in full generality, allowing hypothesis states with different supports. From the quantum Hoeffding bound, we then easily derive quantum Stein’s Lemma and quantum Sanov’s theorem. We give an in-depth treatment of the properties of the quantum Chernoff distance, and argue that it is a natural distance measure on the set of density operators, with a clear operational meaning.
In the asymptotic theory of quantum hypothesis testing, the minimal error probability of the first kind jumps sharply from zero to one when the error exponent of the second kind passes by the point of the relative entropy of the two states in an increasing way. This is well known as the direct part and strong converse of quantum Stein’s lemma. Here we look into the behavior of this sudden change and have make it clear how the error of first kind grows smoothly according to a lower order of the error exponent of the second kind, and hence we obtain the second-order asymptotics for quantum hypothesis testing. This actually implies quantum Stein’s lemma as a special case. Meanwhile, our analysis also yields tight bounds for the case of finite sample size. These results have potential applications in quantum information theory. Our method is elementary, based on basic linear algebra and probability theory. It deals with the achievability part and the optimality part in a unified fashion.
Abstract We extend quantum Stein’s lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states ρ⊗n\rho ^{\otimes n}ρ⊗n ρ ⊗ n against convex combinations of quantum states σ⊗n\sigma ^{\otimes n}σ⊗n σ ⊗ n can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein’s lemma for classical probability distributions and lift the result to the non-commutative setting by using elementary properties of quantum entropy. Finally, our findings also imply an improved recoverability lower bound on the conditional quantum mutual information in terms of the regularized quantum relative entropy—featuring an explicit and universal recovery map.
The generalised quantum Stein's lemma is a key result in quantum hypothesis testing, and connects this fundamental primitive of quantum information processing with quantum resource manipulation, a task that is central for technological applications. Prior works have proved this statement in the idealised setting of independent and identically distributed (i.i.d.) sequences of quantum states, and recent extensions consider also sources that are 'close' to i.i.d., according to the strict notion put forth by Mazzola, Sutter, and Renner. For several applications, however, one would need to consider yet more general sources. We establish a version of the generalised quantum Stein's lemma that is conceptually much simpler and general, as it applies to any source that is asymptotically close to an i.i.d. state with respect to the normalised quantum Wasserstein distance of order 1. As an immediate consequence, we solve the Stein exponent of a scenario where the null hypothesis is arbitrarily varying, expressing it in terms of i.i.d. Stein exponents corresponding to arbitrary states in the convex hull of the null hypothesis base set.