Optimized quantum $f$-divergences

The quantum relative entropy [1] is a foundational distinguishability measure in quantum information theory (QIT). It is a function of two quantum states and measures how well one can tell the two states apart by a quantum-mechanical experiment. One important reason for why it has found such widespread application is that it satisfies a data-processing inequality [2, 3]: it does not increase under the action of a quantum channel on the two states. This can be interpreted as saying that two quantum states do not become more distinguishable if the same quantum channel is applied to them, and a precise interpretation of this statement in terms of quantum hypothesis testing is available in [4, 5, 6]. Quantum relative entropy generalizes its classical counterpart [7].

The wide interest in relative entropy sparked various researchers to generalize and study it further, in an attempt to elucidate the fundamental properties that govern its behavior. One notable generalization is Rényi’s relative entropy [8], but this was subsequently generalized even further in the form of the $f$-divergence [9, 10, 11]. For probability distributions $\{p(x)\}_{x}$ and $\{q(x)\}_{x}$ and a convex function $f$, the $f$-divergence is defined as $\sum_{x}q(x)f(p(x)/q(x)),$ in the case that $p(x)=0$ for all $x$ such that $q(x)=0$. The resulting quantity is then non-increasing under the action of a classical channel $r(y|x)$ that produces the output distributions $\sum_{x}r(y|x)p(x)$ and $\sum_{x}r(y|x)q(x)$. Some years after these developments, a quantum generalization of $f$-divergence appeared in [12, 13] In [12, 13] and a later development [14], the quantum data-processing inequality was proved in full generality for arbitrary quantum channels, whenever the underlying function $f$ is operator convex.

Interestingly, when generalizing a notion from classical to QIT, there is often more than one way to do so, and sometimes there could even be an infinite number of ways to do so. This has to do with the non-commutativity of quantum states. For example, there are several different ways that one could generalize the relative entropy to the quantum case, and two prominent formulas were put forward in [1] and [15]. This added complexity for the quantum case could potentially be problematic, but the typical way of determining on which generalizations we should focus is to show that a given formula is the answer to a meaningful operational task. The papers [4, 5] accomplished this for the formula from [1], and since then, researchers have realized more and more just how foundational the formula of [1] is. As a consequence, the formula of [1] is now known as quantum relative entropy.

The situation becomes more intricate when it comes to quantum generalizations of Rényi relative entropy. For many years, the Petz–Rényi relative entropy of [12, 13] has been widely studied and given an operational interpretation [16, 17], again in the context of quantum hypothesis testing. However, in recent years, the sandwiched Rényi relative entropy of [18, 19] has gained prominence, due to its role in establishing strong converses for communication tasks (see, e.g.,[19, 20]). The result of [21] solidified its fundamental meaning in QIT, proving that it has an operational interpretation in the strong converse exponent of quantum hypothesis testing. As such, the situation is that there are two generalizations of Rényi relative entropy that should be considered in QIT, due to their operational role mentioned above.

The same work that introduced the Petz–Rényi relative entropy also introduced a quantum generalization of the notion of $f$-divergence [12, 13] (see also [22]), with the Petz–Rényi relative entropy being a particular example. Since then, other quantum $f$-divergences have appeared [23, 24], now known as minimal and maximal $f$-divergences [25, 24]. However, it has not been known how the sandwiched Rényi relative entropy fits into the paradigm of quantum $f$-divergences.

In this paper, I modify Petz’s definition of quantum $f$-divergence [12, 13, 22], by allowing for a particular optimization (see Definition 1 for details of the modification). As such, I call the resulting quantity the optimized quantum $f$-divergence. I prove that it obeys a quantum data processing inequality, and as such, my perspective is that it deserves to be considered as another variant of the quantum $f$-divergence, in addition to the original, the minimal, and the maximal. Interestingly, the sandwiched Rényi relative entropy is directly related to the optimized quantum $f$-divergence, thus bringing the sandwiched quantity into the $f$-divergence formalism.

One benefit of the results of this paper is that there is now a single, unified approach for establishing the data-processing inequality for both the Petz–Rényi relative entropy and the sandwiched Rényi relative entropy, for the full Rényi parameter ranges for which it is known to hold. This unified approach is based on Petz’s original approach that employed the operator Jensen inequality [26], and it is useful for presenting a succint proof of the data processing inequality for both quantum Rényi relative entropy families.

In the rest of the paper, I begin by defining the optimized quantum $f$-divergence in the next section. In Section III, I prove that the optimized $f$-divergence satisfies the quantum data processing inequality under partial trace whenever the underlying function $f$ is operator anti-monotone with domain $(0,\infty)$ and range $\mathbb{R}$. The core tool underlying this proof is the operator Jensen inequality [26]. In Section IV, I show how the quantum relative entropy and the sandwiched Rényi relative entropies are directly related to the optimized quantum $f$-divergence. Section V then discusses the relation between Petz’s $f$-divergence and the optimized one. I finally conclude in Section VI with a summary.

Let us begin by defining the optimized quantum $f$-divergence. Here I focus exclusively on the case of positive definite operators, and the full version provides details for the more general case of positive semi-definite operators.

The case of greatest interest for us here is when the underlying function $f$ is operator anti-monotone; i.e., for Hermitian operators $A$ and $B$, the function $f$ is such that $A\leq B\Rightarrow f(B)\leq f(A)$ (see, e.g., [27]). This property is rather strong, but there are several functions of interest in quantum physical applications that obey it (see Section IV). One critical property of an operator anti-monotone function with domain $(0,\infty)$ and range $\mathbb{R}$ is that it is also operator convex and continuous (see, e.g., [28]). In this case, we have the following simple proposition, proved in the full version:

Our first main objective is to prove that $\widetilde{Q}_{f}(X\|Y)$ deserves the name “$f$-divergence” or “$f$-relative entropy,” i.e., that it is monotone non-increasing under the action of a completely positive trace-preserving map $\mathcal{N}$:

Such a map $\mathcal{N}$ is also called a quantum channel, due to its purpose in quantum physics as modeling the physical evolution of the state of a quantum system. In QIT contexts, the inequality in (5) is known as the quantum data processing inequality. According to the Stinespring dilation theorem [29], to every quantum channel $\mathcal{N}_{S\rightarrow B}$, there exists an isometry $U_{S\rightarrow BE}^{\mathcal{N}}$ such that

As such, we can prove the inequality in (5) in two steps. Isometric invariance: First show that

for any isometry $U$ and any positive semi-definite $X$ and $Y$. This is done in the full version of this work, using the general definition given there. Monotonicity under partial trace: Then show that

for positive semi-definite operators $X_{AB}$ and $Y_{AB}$ acting on the tensor-product Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, with $X_{A}=\operatorname{Tr}_{B}\{X_{AB}\}$ and $Y_{A}=\operatorname{Tr}_{B}\{Y_{AB}\}$.

We now discuss the second step toward quantum data processing, mentioned above, and here we focus exclusively on positive definite operators:

I now show how several known quantum divergences are particular examples of an optimized quantum $f$-divergence, including the quantum relative entropy [1] and the sandwiched Rényi relative quasi-entropies [18, 19]. The result will be that Theorem 3 recovers quantum data processing for the sandwiched Rényi relative entropies for the full range of parameters for which it is known to hold. Thus, one benefit of Theorem 3 and earlier work of [12, 13, 14] is a single, unified approach, based on the operator Jensen inequality [26], for establishing quantum data processing for all of the Petz– and sandwiched Rényi relative entropies for the full parameter ranges for which data processing is known to hold.

I now discuss in more detail the relation between the optimized quantum $f$-divergence and Petz’s $f$-divergence from [12, 13]. In brief, we find that the Petz $f$-divergence can be recovered by replacing $\tau$ in Definition 1 with $X$.

One main concern is about quantum data processing with the Petz $f$-divergence. To show this, we take $f$ to be an operator anti-monotone function with domain $(0,\infty)$ and range $\mathbb{R}$. As discussed in Section III, one can establish data processing by showing isometric invariance and monotonicity under partial trace. Isometric invariance of $Q_{f}(X\|Y)$ follows from the same proof as given in the full version and was also shown in [14]. Monotonicity of $Q_{f}(X_{AB}\|Y_{AB})$ under partial trace for positive definite $X_{AB}$ and $Y_{AB}$ follows from the operator Jensen inequality [12, 13].

Special and interesting cases of the Petz $f$-divergence are found by taking $f(x)=-\log x,$ $f(x)=-x^{\beta}$ for $\beta\in(0,1]$, and $f(x)=x^{\beta}$ for $\beta\in[-1,0)$. Each of these functions are operator anti-monotone with domain $(0,\infty)$ and range $\mathbb{R}$. As shown in [12, 13], all of the following quantities obey the data processing inequality:

By a reparametrization $\alpha=1-\beta$, the latter two quantities are directly related to the Petz Rényi relative entropy $D_{\alpha}(X\|Y)\equiv\frac{1}{\alpha-1}\log\operatorname{Tr}\{X^{\alpha}Y^{1-\alpha}\}$. Thus, the data processing inequality holds for $D_{\alpha}(X\|Y)$ for $\alpha\in[0,1)\cup(1,2]$ [13, 14].

The main contribution of the present work is the definition of the optimized quantum $f$-divergence and the proof that the data processing inequality holds for it whenever the function $f$ is operator anti-monotone with domain $(0,\infty)$ and range $\mathbb{R}$. The proof of the data processing inequality relies on the operator Jensen inequality [26], and it bears some similarities to the original approach from [12, 13, 14]. Furthermore, I showed how the sandwiched Rényi relative entropies are particular examples of the optimized quantum $f$-divergence. As such, one benefit of this paper is that there is now a single, unified approach, based on the operator Jensen inequality [26], for establishing the data processing inequality for the Petz–Rényi and sandwiched Rényi relative entropies, for the full range of parameters for which it is known to hold.

Acknowledgements. I thank Anna Vershynina for discussions related to the topic of this paper, and I acknowledge support from the NSF under grant no. 1714215.