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 Partially Trace Distance Preserving Maps and Reversible Quantum Channels

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Long Jian ◽  
Kan He ◽  
Qing Yuan ◽  
Fei Wang
Keyword(s):  
Quantum States ◽  
Quantum Channels ◽  
Trace Distance ◽  
Linear Maps ◽  
Unitary Transformations ◽  
Pure States ◽  
Positive Linear Maps

We give a characterization of trace-preserving and positive linear maps preserving trace distance partially, that is, preservers of trace distance of quantum states or pure states rather than all matrices. Applying such results, we give a characterization of quantum channels leaving Helstrom's measure of distinguishability of quantum states or pure states invariant and show that such quantum channels are fully reversible, which are unitary transformations.

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 Maximum relative entropy of coherence for quantum channels

2021 ◽  
Vol 64 (8) ◽  
Author(s):  
Zhi-Xiang Jin ◽  
Long-Mei Yang ◽  
Shao-Ming Fei ◽  
Xianqing Li-Jost ◽  
Zhi-Xi Wang ◽  
...  
Keyword(s):  
Relative Entropy ◽  
Quantum Channels ◽  
Relative Entropy Of Coherence

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.

scholarly journals Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 522
Author(s):  
Bartosz Regula ◽  
Ryuji Takagi ◽  
Mile Gu
Keyword(s):  
Quantum Dynamics ◽  
Distance Measure ◽  
Quantum States ◽  
Quantum Channels ◽  
Linear Combinations ◽  
Operational Approach ◽  
Completely Positive ◽  
Error Mitigation ◽  
Linear Maps ◽  
The Cost

Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels — that is, not completely positive — can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation.

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.

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