Defining quantum divergences via convex optimization

Fawzi, Hamza; Fawzi, Omar

doi:10.22331/q-2021-01-26-387

Cited by 27 publications

(31 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…( 4) could be removed and thus delivers the desirable quantum channel Stein's Lemma. As a byproduct, the exponentially strong converse property also implies the continuity of the regularized (amortized) Sandwiched Rényi channel divergence at α = 1, resolving an open question in [FF21]. These results extend and deepen our understanding of quantum channel discrimination and quantum channel divergences in the asymptotic regime.…”

Section: Main Contributionsmentioning

confidence: 54%

Towards the ultimate limits of quantum channel discrimination

Fang¹,

Gour²,

Wang³

2021

Preprint

View full text Add to dashboard Cite

This paper studies the difficulty of discriminating quantum channels under operational regimes, proves the quantum channel Stein's lemma (strong converse part), and provides a unified framework to show the operational meaning of quantum channel divergences. First, we establish the exponentially strong converse of quantum channel hypothesis testing under coherent strategies, meaning that any strategy to make the Type II error decays with an exponent larger than the regularized channel relative entropy will unavoidably result in the Type I error converging to one exponentially fast in the asymptotic limit. This result notably delivers the desirable quantum channel Stein's Lemma, enclosing a long-term open problem in quantum information theory. As a byproduct, we show the continuity of the regularized (amortized) Sandwiched Rényi channel divergence at α = 1, resolving another open problem in the field. Second, we develop a framework to show the interplay between the strategies of channel discrimination, the operational regimes, and variants of channel divergences. This framework systematically underlies the operational meaning of quantum channel divergences in quantum channel discrimination. Our work establishes the ultimate limit of quantum channel discrimination, deepening our understanding of quantum channel discrimination and quantum channel divergences in the asymptotic regime. As quantum channel discrimination is strongly connected to many other fundamental tasks in quantum information theory, we expect plentiful applications on related topics such as quantum metrology and quantum communication.

show abstract

Section: Main Contributionsmentioning

confidence: 54%

Towards the ultimate limits of quantum channel discrimination

Fang¹,

Gour²,

Wang³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The following proof resembles a proof from [17], where analogous relations are given for I opt α (A : B) (see (33)), which even show that Theorem 5 can be extended to I opt ∞ (A : B) as they hold for all α. We start with the sandwiched Rényi divergence and use the Schmidt decomposition…”

Section: C4 Proof Of Theoremmentioning

confidence: 68%

“…These are measures of distinguishability of quantum states, which play a pivotal role in information-theoretic tasks, such as single-shot communication protocols [18,19], channel coding [20][21][22][23][24][25] or hypothesis testing [26,27]. In principle, each of the many variants of quantum Rényi divergences [20,[28][29][30][31][32][33] allows us to define a mutual information as we explain in Appendix A. Here, we focus on two particular cases and explain how to compute them in practice, at least when the input state is represented via tensor networks.…”

Section: Introductionmentioning

confidence: 99%

Computable Rényi mutual information: Area laws and correlations

Scalet

Alhambra

Styliaris

et al. 2021

Quantum

View full text Add to dashboard Cite

The mutual information is a measure of classical and quantum correlations of great interest in quantum information. It is also relevant in quantum many-body physics, by virtue of satisfying an area law for thermal states and bounding all correlation functions. However, calculating it exactly or approximately is often challenging in practice. Here, we consider alternative definitions based on Rényi divergences. Their main advantage over their von Neumann counterpart is that they can be expressed as a variational problem whose cost function can be efficiently evaluated for families of states like matrix product operators while preserving all desirable properties of a measure of correlations. In particular, we show that they obey a thermal area law in great generality, and that they upper bound all correlation functions. We also investigate their behavior on certain tensor network states and on classical thermal distributions.

show abstract

“…We postpone this question to future work. Recently, Fawzi and Fawzi used the Kubo-Ando geometric to define new quantum Rényi divergences in terms of a convex optimization program and proved that they satisfy the data processing inequality [43]. It would be interesting to use the non-commutative L p ω spaces to rewrite their expressions as (p, ω)-norms and explore their potential multi-state generalizations.…”

Section: Discussionmentioning

confidence: 99%

Monotonic multi-state quantum $f$-divergences

Furuya

Lashkari

Ouseph

2021

Preprint

View full text Add to dashboard Cite

We use the Tomita-Takesaki modular theory and the Kubo-Ando operator mean to write down a large class of multi-state quantum f -divergences and prove that they satisfy the data processing inequality. For two states, this class includes the (α, z)-Rényi divergences, the f -divergences of Petz, and the measures in [1] as special cases. The method used is the interpolation theory of non-commutative L p ω spaces and the result applies to general von Neumann algebras including the local algebra of quantum field theory. We conjecture that these multi-state Rényi divergences have operational interpretations in terms of the optimal error probabilities in asymmetric multi-state quantum state discrimination.

show abstract

Defining quantum divergences via convex optimization

Cited by 27 publications

References 41 publications

Towards the ultimate limits of quantum channel discrimination

Towards the ultimate limits of quantum channel discrimination

Computable Rényi mutual information: Area laws and correlations

Monotonic multi-state quantum $f$-divergences

Contact Info

Product

Resources

About