scholarly journals Tensorization of the strong data processing inequality for quantum chi-square divergences

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 199
Author(s):  
Yu Cao ◽  
Jianfeng Lu
Keyword(s):  
Data Processing ◽  
Relative Entropy ◽  
Quantum Channel ◽  
Quantum States ◽  
Quantum Channels ◽  
Chi Square ◽  
Quantum Regime ◽  
Quantum Relative Entropy ◽  
Data Processing Inequality ◽  
Arbitrary Quantum

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.

scholarly journals A strengthened data processing inequality for the Belavkin–Staszewski relative entropy

2019 ◽  
Vol 32 (02) ◽  
pp. 2050005 ◽  
Author(s):  
Andreas Bluhm ◽  
Ángela Capel
Keyword(s):  
Data Processing ◽  
Relative Entropy ◽  
Quantum Channels ◽  
Equality Case ◽  
Standard Formula ◽  
Conditional Expectations ◽  
Data Processing Inequality ◽  
Equivalent Conditions ◽  
Arbitrary Quantum

In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). This extends previous results by Carlen and Vershynina for the relative entropy and other standard [Formula: see text]-divergences. To this end, we provide two new equivalent conditions for the equality case of the data processing inequality for the BS-entropy. Subsequently, we extend our result to a larger class of maximal [Formula: see text]-divergences. Here, we first focus on quantum channels which are conditional expectations onto subalgebras and use the Stinespring dilation to lift our results to arbitrary quantum channels.

AN OPERATIONAL ALGEBRAIC APPROACH TO QUANTUM CHANNEL CAPACITY

2008 ◽  
Vol 06 (05) ◽  
pp. 981-996 ◽  
Author(s):  
V. P. BELAVKIN ◽  
X. DAI
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Quantum Channel ◽  
Algebraic Approach ◽  
Quantum Information Theory ◽  
Quantum Channels ◽  
Operational Approach ◽  
Quantum Relative Entropy ◽  
Quantum Mutual Information ◽  
Arbitrary Quantum

An elementary introduction into algebraic approach to unified quantum information theory and operational approach to quantum entanglement as generalized encoding is given. After introducing compound quantum state and two types of informational divergences, namely, Araki–Umegaki (a-type) and of Belavkin–Staszewski (b-type) quantum relative entropic information, this paper treats two types of quantum mutual information via entanglement and defines two types of corresponding quantum channel capacities as the supremum via the generalized encodings. It proves the additivity property of quantum channel capacities via entanglement, which extends the earlier results of Belavkin to products of arbitrary quantum channels for quantum relative entropy of any type.

scholarly journals Relating Entropies of Quantum Channels

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1028
Author(s):  
Dariusz Kurzyk ◽  
Łukasz Pawela ◽  
Zbigniew Puchała
Keyword(s):  
Relative Entropy ◽  
Upper Bound ◽  
Quantum Channel ◽  
Von Neumann Entropy ◽  
Quantum Channels ◽  
Von Neumann ◽  
Channel One ◽  
Depolarizing Channel

In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi–Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.

scholarly journals Quantum Relative Entropy of Tagging and Thermodynamics

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 138
Author(s):  
Jose Diazdelacruz
Keyword(s):  
Angular Momentum ◽  
Relative Entropy ◽  
Physical System ◽  
Total Angular Momentum ◽  
Quantum States ◽  
Restricted Class ◽  
Physical Systems ◽  
Quantum Relative Entropy ◽  
Heat Engines ◽  
Labeling System

Thermodynamics establishes a relation between the work that can be obtained in a transformation of a physical system and its relative entropy with respect to the equilibrium state. It also describes how the bits of an informational reservoir can be traded for work using Heat Engines. Therefore, an indirect relation between the relative entropy and the informational bits is implied. From a different perspective, we define procedures to store information about the state of a physical system into a sequence of tagging qubits. Our labeling operations provide reversible ways of trading the relative entropy gained from the observation of a physical system for adequately initialized qubits, which are used to hold that information. After taking into account all the qubits involved, we reproduce the relations mentioned above between relative entropies of physical systems and the bits of information reservoirs. Some of them hold only under a restricted class of coding bases. The reason for it is that quantum states do not necessarily commute. However, we prove that it is always possible to find a basis (equivalent to the total angular momentum one) for which Thermodynamics and our labeling system yield the same relation.

scholarly journals Optimal bounds on functions of quantum states under quantum channels

2016 ◽  
Vol 16 (9&10) ◽  
pp. 845-861
Author(s):  
Chi-Kwong Li ◽  
Diane Christine Pelejo ◽  
Kuo-Zhong Wang
Keyword(s):  
Relative Entropy ◽  
Scalar Function ◽  
Quantum States ◽  
Quantum Channels ◽  
Trace Distance ◽  
Optimal Bounds ◽  
Class Of Functions

Let ρ1, ρ2 be quantum states and (ρ1, ρ2) 7→ D(ρ1, ρ2) be a scalar function such as the trace distance, the fidelity, and the relative entropy, etc. We determine optimal bounds for D(ρ1, Φ(ρ2)) for Φ blongs to S for different class of functions D(·, ·), where S is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels.

Cohering power and decohering power of Gaussian unitary operations

2019 ◽  
Vol 19 (7&8) ◽  
pp. 575-586
Author(s):  
Yangyang Wang ◽  
Xiaofei Qi ◽  
Jinchuan Hou ◽  
Rufen Ma
Keyword(s):  
Relative Entropy ◽  
Continuous Variable ◽  
Quantum States ◽  
Entropy Measure ◽  
Quantum Channels ◽  
Gaussian States ◽  
Basic Properties ◽  
Suitable Measure

Having a suitable measure to quantify the coherence of quantum states, a natural task is to evaluate the power of quantum channels for creating or destroying the coherence of input quantum states. In the present paper, by introducing the maximal coherent Gaussian states based on the relative entropy measure of coherence, we propose the (generalized) cohering power and the (generalized) decohering power of Gaussian unitary operations for continuous-variable systems. Some basic properties are obtained and the cohering power and decohering power of two special kinds of Gaussian unitary operations are calculated.

scholarly journals On Composite Quantum Hypothesis Testing

2021 ◽  
Author(s):  
Mario Berta ◽  
Fernando G. S. L. Brandão ◽  
Christoph Hirche
Keyword(s):  
Hypothesis Testing ◽  
Relative Entropy ◽  
Probability Distributions ◽  
Quantum States ◽  
Quantum Hypothesis ◽  
Error Exponent ◽  
Quantum Relative Entropy ◽  
Asymmetric Quantum ◽  
Alternative Hypotheses ◽  
Quantum Hypothesis Testing

AbstractWe extend quantum Stein’s lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states $$\rho ^{\otimes n}$$ ρ ⊗ n against convex combinations of quantum states $$\sigma ^{\otimes n}$$ σ ⊗ n can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein’s lemma for classical probability distributions and lift the result to the non-commutative setting by using elementary properties of quantum entropy. Finally, our findings also imply an improved recoverability lower bound on the conditional quantum mutual information in terms of the regularized quantum relative entropy—featuring an explicit and universal recovery map.

scholarly journals Volume of the space of qubit-qubit channels and state transformations under random quantum channels

2018 ◽  
Vol 30 (10) ◽  
pp. 1850019 ◽  
Author(s):  
Attila Lovas ◽  
Attila Andai
Keyword(s):  
Lebesgue Measure ◽  
Quantum Channel ◽  
Probability Distributions ◽  
Building Blocks ◽  
The State ◽  
Quantum States ◽  
Qubit State ◽  
Quantum Channels ◽  
State Transformation ◽  
Explicit Formulas

The simplest building blocks for quantum computations are the qubit-qubit quantum channels. In this paper, we analyze the structure of these channels via their Choi representation. The restriction of a quantum channel to the space of classical states (i.e. probability distributions) is called the underlying classical channel. The structure of quantum channels over a fixed classical channel is studied, the volume of the general and unital qubit channels with respect to the Lebesgue measure is computed and explicit formulas are presented for the distribution of the volume of quantum channels over given classical channels. We study the state transformation under uniformly random quantum channels. If one applies a uniformly random quantum channel (general or unital) to a given qubit state, the distribution of the resulted quantum states is presented.

scholarly journals INFORMATION TRANSMISSION VIA ENTANGLED QUANTUM STATES IN GAUSSIAN CHANNELS WITH MEMORY

2006 ◽  
Vol 04 (03) ◽  
pp. 439-452 ◽  
Author(s):  
NICOLAS J. CERF ◽  
JULIEN CLAVAREAU ◽  
JÉRÉMIE ROLAND ◽  
CHIARA MACCHIAVELLO
Keyword(s):  
Quantum Channel ◽  
Quantum States ◽  
Gaussian Channel ◽  
Hard Problem ◽  
Quantum Channels ◽  
Classical Capacity ◽  
Gaussian Channels ◽  
Memoryless Channels ◽  
Entangled Quantum States ◽  
With Memory

Gaussian quantum channels have recently attracted a growing interest, since they may lead to a tractable approach to the generally hard problem of evaluating quantum channel capacities. However, the analysis performed so far has always been restricted to memoryless channels. Here, we consider the case of a bosonic Gaussian channel with memory, and show that the classical capacity can be significantly enhanced by employing entangled input symbols instead of product symbols.

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