. 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.

DOI: 10.1109/isit57864.2024.10619141

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On maximization of measured $f$ -divergence between a given pair of quantum statesOn maximization of measured $f$-divergence between a given pair of quantum states

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arXiv:1412.3676Dec 11, 20142 citations

A

and

B

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.

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.

The measured relative entropy and measured Rényi relative entropy are quantifiers of the distinguishability of two quantum states

ρ

and

σ

. They are defined as the maximum classical relative entropy or Rényi relative entropy realizable by performing a measurement on

ρ

and

σ

, and they have interpretations in terms of asymptotic quantum hypothesis testing. Crucially, they can be rewritten in terms of variational formulas involving the optimization of a concave or convex objective function over the set of positive definite operators. In this paper, we establish foundational properties of these objective functions by analyzing their matrix gradients and Hessian superoperators; namely, we prove that these objective functions are

β

-smooth and

γ

-strongly convex / concave, where

β

and

γ

depend on the max-relative entropies of

ρ

and

σ

. A practical consequence of these properties is that we can conduct Nesterov accelerated projected gradient descent / ascent, a well known classical optimization technique, to calculate the measured relative entropy and measured Rényi relative entropy to arbitrary precision. These algorithms are generally more memory efficient than our previous algorithms based on semi-definite optimization [Huang and Wilde, arXiv:2406.19060], and for well conditioned states

ρ

and

σ

, these algorithms are notably faster.

On maximization of measured

f

-divergence between a given pair of quantum states

This paper deals with maximization of classical

f

-divergence between the distributions of a measurement outputs of a given pair of quantum states.

f

-divergence

D_{f}

between the probability density functions

p_{1}

and

p_{2}

over a discrete set is defined as $D_{f}( p_{1}||p_{2}) :=\sum_{x}p_{2}(x) f\left(p_{1}(x)/p_{2}( x) \right) $. For example, Kullback-Leibler divergence and Renyi type relative entropy are well-known examples with good operational meanings. Thus, finding the maximal value

D_{f}^{\min}

of measured measured

f

-divergence is also an interesting question. But so far the question is solved only for very restricted example of

f

. \ The purposes of the present paper is to advance the study further, by investigating its properties, rewriting the maximization problem to more tractable form, and giving closed formulas of the quantity in some special cases.