We propose an extension of the classical Rényi divergences to quantum states through an optimization over probability distributions induced by restricted sets of measurements. In particular, we define the notion of locally-measured Rényi divergences, where the set of allowed measurements originates from variants of locality constraints between (distant) parties $A$ and $B$ . We then derive variational bounds on the locally-measured Rényi divergences and systematically discuss when these bounds become exact characterizations. As an application, we evaluate the locally-measured Rényi divergences on variants of highly symmetric data-hiding states, showcasing the reduced distinguishing power of locality-constrained measurements. For $n$ -fold tensor powers, we further employ our variational formulae to derive corresponding additivity results, which gives the locally-measured Rényi divergences operational meaning as optimal rate exponents in asymptotic locally-measured hypothesis testing.We propose an extension of the classical Rényi divergences to quantum states through an optimization over probability distributions induced by restricted sets of measurements. In particular, we define the notion of locally-measured Rényi divergences, where the set of allowed measurements originates from variants of locality constraints between (distant) parties $A$ and $B$. We then derive variational bounds on the locally-measured Rényi divergences and systematically discuss when these bounds become exact characterizations. As an application, we evaluate the locally-measured Rényi divergences on variants of highly symmetric data-hiding states, showcasing the reduced distinguishing power of locality-constrained measurements. For $n$-fold tensor powers, we further employ our variational formulae to derive corresponding additivity results, which gives the locally-measured Rényi divergences operational meaning as optimal rate exponents in asymptotic locally-measured hypothesis testing.

arXiv:2405.05037May 8, 20240 citations

Locally-Measured Rényi Divergences

Tobias Rippchen, Sreejith Sreekumar, Mario Berta

IEEE International Symposium on Information Theory (ISIT)Jul 7, 20243 citations

Rethinking quantum smooth entropies: Tight one-shot analysis of quantum privacy amplification

Bartosz Regula, Marco Tomamichel

We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.

quant-pharXiv:2603.04493Mar 6, 20260 citations

Efficient Quantum Measurements: Computational Max- and Measured Rényi Divergences and Applications

Álvaro Yángüez, Thomas A. Hahn, Jan Kochanowski

Quantum information processing is limited, in practice, to efficiently implementable operations. This motivates the study of quantum divergences that preserve their operational meaning while faithfully capturing these computational constraints. Using geometric, computational, and information theoretic tools, we define two new types of computational divergences, which we termcomputational max-divergence and computational measured Rényi divergences. Both are constrained by a family of efficient binary measurements, and thus useful for state discrimination tasks in the computational setting. We prove that, in the infinite-order limit, the computational measured Rényi divergence coincides with the computational max-divergence, mirroring the corresponding relation in the unconstrained information-theoretic setting. For the many-copy regime, we introduce regularized versions and establish a one-sided computational Stein bound on achievable hypothesis-testing exponents under efficient measurements, giving the regularized computational measured relative entropy an operational meaning. We further define resource measures induced by our computational divergences and prove an asymptotic continuity bound for the computational measured relative entropy of resource. Focusing on entanglement, we relate our results to previously proposed computational entanglement measures and provide explicit separations from the information-theoretic setting. Together, these results provide a principled, cohesive approach towards state discrimination tasks and resource quantification under computational constraints.Quantum information processing is limited, in practice, to efficiently implementable operations. This motivates the study of quantum divergences that preserve their operational meaning while faithfully capturing these computational constraints. Using geometric, computational, and information theoretic tools, we define two new types of computational divergences, which we termcomputational max-divergence and computational measured Rényi divergences. Both are constrained by a family of efficient binary measurements, and thus useful for state discrimination tasks in the computational setting. We prove that, in the infinite-order limit, the computational measured Rényi divergence coincides with the computational max-divergence, mirroring the corresponding relation in the unconstrained information-theoretic setting. For the many-copy regime, we introduce regularized versions and establish a one-sided computational Stein bound on achievable hypothesis-testing exponents under efficient measurements, giving the regularized computational measured relative entropy an operational meaning. We further define resource measures induced by our computational divergences and prove an asymptotic continuity bound for the computational measured relative entropy of resource. Focusing on entanglement, we relate our results to previously proposed computational entanglement measures and provide explicit separations from the information-theoretic setting. Together, these results provide a principled, cohesive approach towards state discrimination tasks and resource quantification under computational constraints.

arXiv:2509.21308IEEE Transactions on Information TheoryApr 2, 20260 citations

Optimal Universal Bounds for Quantum Divergences

Gilad Gour

We identify a universal structural principle underlying the smoothing of classical divergences: the optimizer of the smoothing problem is a clipped probability vector, independently of the specific divergence. This yields a divergence-independent characterization of all smoothed classical divergences and reveals a common geometric structure behind seemingly different quantities. Building on this structural insight, we derive optimal universal bounds for smoothed quantum divergences, including quantum Rényi divergences of arbitrary order and the hypothesis-testing divergence. Our inequalities relate divergences of different orders through bounds of the form $D_β^{\varepsilon} \le D_α+ \mathrm{correction}$ and $D_β^{\varepsilon} \ge D_α+ \mathrm{correction}$ , and we prove that the correction terms are optimal among all universal, state-independent inequalities of this type. Consequently, our results strictly improve previously known bounds whenever those were suboptimal, and in cases where earlier bounds coincide with ours, our analysis establishes their optimality. In particular, we obtain optimal universal bounds for the hypothesis-testing divergence.We identify a universal structural principle underlying the smoothing of classical divergences: the optimizer of the smoothing problem is a clipped probability vector, independently of the specific divergence. This yields a divergence-independent characterization of all smoothed classical divergences and reveals a common geometric structure behind seemingly different quantities. Building on this structural insight, we derive optimal universal bounds for smoothed quantum divergences, including quantum Rényi divergences of arbitrary order and the hypothesis-testing divergence. Our inequalities relate divergences of different orders through bounds of the form $D_β^{\varepsilon} \le D_α+ \mathrm{correction}$ and $D_β^{\varepsilon} \ge D_α+ \mathrm{correction}$, and we prove that the correction terms are optimal among all universal, state-independent inequalities of this type. Consequently, our results strictly improve previously known bounds whenever those were suboptimal, and in cases where earlier bounds coincide with ours, our analysis establishes their optimality. In particular, we obtain optimal universal bounds for the hypothesis-testing divergence.

quant-pharXiv:2603.09885Mar 11, 20260 citations

Test-Measured Rényi Divergences

Milán Mosonyi, Fumio Hiai

One possibility of defining a quantum Rényi$\alpha $-divergence of two quantum states is to optimize the classical Rényi$ \alpha $-divergence of their post-measurement probability distributions over all possible measurements (measured Rényi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured Rényi$ \alpha $-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured Rényi$ \alpha $-divergence coincides with the sandwiched Rényi$ \alpha $-divergence when$ \alpha >1 $. Moreover, it also follows from the same theorem that to achieve this, it is sufficient to consider 2-outcome measurements (tests) for any number of copies (this is somewhat surprising, as achieving the measured Rényi$ \alpha $-divergence for$ n $copies might require a number of measurement outcomes that diverges in$ n $, in general). In view of this, it seems natural to expect the same when$ \alpha < 1 $; however, we show that this is not the case. In fact, we show that even for commuting states (classical case) the regularized quantity attainable using 2-outcome measurements is in general strictly smaller than the Rényi$ \alpha $-divergence (which is unique in the classical case). In the general quantum case this shows that the above “regularized test-measured” Rényi$ \alpha $-divergence is not even a quantum extension of the classical Rényi divergence when$ \alpha < 1 $, in sharp contrast to the$ \alpha >1$case.One possibility of defining a quantum Rényi$\alpha $-divergence of two quantum states is to optimize the classical Rényi$\alpha $-divergence of their post-measurement probability distributions over all possible measurements (measured Rényi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured Rényi$\alpha $-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured Rényi$\alpha $-divergence coincides with the sandwiched Rényi$\alpha $-divergence when$\alpha >1$. Moreover, it also follows from the same theorem that to achieve this, it is sufficient to consider 2-outcome measurements (tests) for any number of copies (this is somewhat surprising, as achieving the measured Rényi$\alpha $-divergence for$n$copies might require a number of measurement outcomes that diverges in$n$, in general). In view of this, it seems natural to expect the same when$\alpha < 1$; however, we show that this is not the case. In fact, we show that even for commuting states (classical case) the regularized quantity attainable using 2-outcome measurements is in general strictly smaller than the Rényi$\alpha $-divergence (which is unique in the classical case). In the general quantum case this shows that the above “regularized test-measured” Rényi$\alpha $-divergence is not even a quantum extension of the classical Rényi divergence when$\alpha < 1$, in sharp contrast to the$\alpha >1$case.

arXiv:2201.05477IEEE Transactions on Information TheorySep 26, 202214 citations

A

and

B

. We then derive variational bounds on the locally-measured Rényi divergences and systematically discuss when these bounds become exact characterizations. As an application, we evaluate the locally-measured Rényi divergences on variants of highly symmetric data-hiding states, showcasing the reduced distinguishing power of locality-constrained measurements. For

n

-fold tensor powers, we further employ our variational formulae to derive corresponding additivity results, which gives the locally-measured Rényi divergences operational meaning as optimal rate exponents in asymptotic locally-measured hypothesis testing.

A

and

B

. As our main result, we derive variational characterizations of these locally-measured Rényi divergences. We then evaluate them for variants of data-hiding states, showcasing the reduced distinguishing power of locality-constrained measurements, and give corresponding applications in locally-measured hypothesis testing.

D_β^{\varepsilon} \le D_α+ \mathrm{correction}

and

D_β^{\varepsilon} \ge D_α+ \mathrm{correction}

, and we prove that the correction terms are optimal among all universal, state-independent inequalities of this type. Consequently, our results strictly improve previously known bounds whenever those were suboptimal, and in cases where earlier bounds coincide with ours, our analysis establishes their optimality. In particular, we obtain optimal universal bounds for the hypothesis-testing divergence.

One possibility of defining a quantum Rényi$\alpha

-divergence of two quantum states is to optimize the classical Rényi

\alpha

-divergence of their post-measurement probability distributions over all possible measurements (measured Rényi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured Rényi

\alpha

-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured Rényi

\alpha

-divergence coincides with the sandwiched Rényi

\alpha

-divergence when

\alpha >1

. Moreover, it also follows from the same theorem that to achieve this, it is sufficient to consider 2-outcome measurements (tests) for any number of copies (this is somewhat surprising, as achieving the measured Rényi

\alpha

-divergence for

copies might require a number of measurement outcomes that diverges in

, in general). In view of this, it seems natural to expect the same when

\alpha < 1

; however, we show that this is not the case. In fact, we show that even for commuting states (classical case) the regularized quantity attainable using 2-outcome measurements is in general strictly smaller than the Rényi

\alpha

-divergence (which is unique in the classical case). In the general quantum case this shows that the above “regularized test-measured” Rényi

\alpha

-divergence is not even a quantum extension of the classical Rényi divergence when

\alpha < 1

, in sharp contrast to the

\alpha >1$case.