In information theory, the data processing inequality is a fundamental property that describes the loss of information through noisy channels. A more refined description is given by the strong data processing inequality (SDPI) for a fixed channel and a full-rank reference state. In classical information theory, the strong data processing inequality tensorizes for a large family of $f$-divergences. However, its quantum counterpart remains largely unexplored. Tensorization of SDPI has previously been shown only for some special cases, and whether it holds more generally for quantum divergences still remains open. In this work, we report two negative results: the tensorization property fails for certain quantum chi-square divergences, and it also does not hold for the quantum relative entropy.
In information theory, the data processing inequality is a fundamental property that describes the loss of information through noisy channels. A more refined description is given by the strong data processing inequality (SDPI) for a fixed channel and a full-rank reference state. In classical information theory, the strong data processing inequality tensorizes for a large family of f-divergences. However, its quantum counterpart remains largely unexplored. Tensorization of SDPI has previously been shown only for some special cases, and whether it holds more generally for quantum divergences still remains open. In this work, we report two negative results: the tensorization property fails for certain quantum chi-square divergences, and it also does not hold for the quantum relative entropy.
Data-processing is a desired property of classical and quantum divergences and information measures. In information theory, the contraction coefficient measures how much the distinguishability of quantum states decreases when they are transmitted through a quantum channel, establishing linear strong data-processing inequalities (SDPI). However, these linear SDPI are not always tight and can be improved in most of the cases. In this work, we establish non-linear SDPI for quantum hockey-stick divergence for noisy channels that satisfy a certain noise criterion. We also note that our results improve upon existing linear SDPI for quantum hockey-stick divergences and also non-linear SDPI for classical hockey-stick divergence. We define Fγ curves generalizing Dobrushin curves for the quantum setting while characterizing SDPI for the sequential composition of heterogeneous channels. In addition, we derive reverse-Pinsker type inequalities for f-divergences with additional constraints on hockey-stick divergences. We show that these non-linear SDPI can establish tighter finite mixing times that cannot be achieved through linear SDPI. Furthermore, we find applications of these in establishing stronger privacy guarantees for the composition of sequential private quantum channels when privacy is quantified by quantum local differential privacy.
It is well-known that any quantum channel E satisfies the data processing inequality (DPI), with respect to various divergences, e.g., quantum χκ2 divergences and quantum relative entropy. More specifically, the data processing inequality states that the divergence between two arbitrary quantum states ρ and σ does not increase under the action of any quantum channel E . For a fixed channel E and a state σ , the divergence between output states E(ρ) and E(σ) might be strictly smaller than the divergence between input states ρ and σ , which is characterized by the strong data processing inequality (SDPI). Among various input states ρ , the largest value of the rate of contraction is known as the SDPI constant. An important and widely studied property for classical channels is that SDPI constants tensorize. In this paper, we extend the tensorization property to the quantum regime: we establish the tensorization of SDPIs for the quantum χκ1/22 divergence for arbitrary quantum channels and also for a family of χκ2 divergences (with κ≥κ1/2 ) for arbitrary quantum-classical channels.
Tight Contraction Rates for Primitive Channels under Quantum f-Divergences
Data-processing inequalities capture the phenomenon that two probability distributions can only become less distinguishable under any common post-processing. For more fine-grained inequalities, one turns to strong data-processing inequality (SDPI) constants, which give the strongest inequalities for a given channel and reference state for a fixed measure of distinguishability. These quantities have been used to quantify the rate at which time-homogeneous Markov chains contract towards a fixed point both in the classical and quantum setting. In this work, we establish that quantum f-divergences satisfy a local reverse Pinsker inequality, which implies the asymptotic contraction rate of a primitive channel to its stationary state is upper bounded by the SDPI constant of any non-commutative χ2-divergence. Using quantum-detailed balance, we establish a sufficient condition for these bounds to be tight. Finally, we apply these results to Petz, Matsumoto, and Hirche-Tomamichel f-divergences, establishing new and strengthening previously known results.
Abstract The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f -divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz–Rényi relative entropies. In this paper, I introduce the optimized quantum f -divergence as a related generalization of quantum relative entropy. I prove that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz’s original approach. Interestingly, the sandwiched Rényi relative entropies are particular examples of the optimized f -divergence. Thus, one benefit of this paper is that there is now a single, unified approach for establishing the data processing inequality for both the Petz–Rényi and sandwiched Rényi relative entropies, for the full range of parameters for which it is known to hold. This paper discusses other aspects of the optimized f -divergence, such as the classical case, the classical-quantum case, and how to construct optimized f -information measures.
The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f-divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz-Renyi relative entropies. In this contribution, I introduce the optimized quantum f-divergence as a related generalization of quantum relative entropy. I prove that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz's original approach. Interestingly, the sandwiched Renyi relative entropies are particular examples of the optimized f-divergence. Thus, one benefit of this approach is that there is now a single, unified approach for establishing the data processing inequality for both the Petz-Renyi and sandwiched Renyi relative entropies, for the full range of parameters for which it is known to hold. Full version of this paper is accessible at arXiv:1710.10252
The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f-divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz-Renyi relative entropies. In this contribution, I introduce the optimized quantum f-divergence as a related generalization of quantum relative entropy. I prove that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz's original approach. Interestingly, the sandwiched Renyi relative entropies are particular examples of the optimized f-divergence. Thus, one benefit of this approach is that there is now a single, unified approach for establishing the data processing inequality for both the Petz-Renyi and sandwiched Renyi relative entropies, for the full range of parameters for which it is known to hold. Full version of this paper is accessible at arXiv:1710.10252