Symmetry-Like Relation of Relative Entropy Measure of Quantum Coherence
Chengyang Zhang, Zhihua Guo, Huaixin Cao
Entropy · 2020
doi:10.3390/e22030297 (opens in a new tab)
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Abstract
Quantum coherence is an important physical resource in quantum information science, and also as one of the most fundamental and striking features in quantum physics. To quantify coherence, two proper measures were introduced in the literature, the one is the relative entropy of coherence <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>C</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mi>diag</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>−</mo> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> and the other is the <inline-formula> <math display="inline"> <semantics> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula>-norm of coherence <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>C</mi> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo>∑</mo> <mrow> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> </msub> <mrow> <mo>|</mo> <msub> <mi>ρ</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>|</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. In this paper, we obtain a symmetry-like relation of relative entropy measure <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>C</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>ρ</mi> <mrow> <msub> <mi>A</mi> <mn>1</mn> </msub> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>⋯</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> of coherence for an <i>n</i>-partite quantum states <inline-formula> <math display="inline"> <semantics> <msup> <mi>ρ</mi> <mrow> <msub> <mi>A</mi> <mn>1</mn> </msub> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>⋯</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> </mrow> </msup> </semantics> </math> </inline-formula>, which gives lower and upper bounds for <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>C</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. As application of our inequalities, we conclude that when each reduced states <inline-formula> <math display="inline"> <semantics> <msup> <mi>ρ</mi> <msub> <mi>A</mi> <mi>i</mi> </msub> </msup> </semantics> </math> </inline-formula> is pure, <inline-formula> <math display="inline"> <semantics> <msup> <mi>ρ</mi> <mrow> <msub> <mi>A</mi> <mn>1</mn> </msub> <mo>⋯</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> </mrow> </msup> </semantics> </math> </inline-formula> is incoherent if and only if the reduced states <inline-formula> <math display="inline"> <semantics> <msup> <mi>ρ</mi> <msub> <mi>A</mi> <mi>i</mi> </msub> </msup> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>tr</mi> <msub> <mi>A</mi> <mi>i</mi> </msub> </msub> <msup> <mi>ρ</mi> <mrow> <msub> <mi>A</mi> <mn>1</mn> </msub> <mo>⋯</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> are all incoherent. Meanwhile, we discuss the conjecture that <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>C</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mi>C</mi> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> for any state <inline-formula> <math display="inline"> <semantics> <mi>ρ</mi> </semantics> </math> </inline-formula>, which was proved to be valid for any mixed qubit state and any pure state, and open for a general state. We observe that every mixture <inline-formula> <math display="inline"> <semantics> <mi>η</mi> </semantics> </math> </inline-formula> of a state <inline-formula> <math display="inline"> <semantics> <mi>ρ</mi> </semantics> </math> </inline-formula> satisfying the conjecture with any incoherent state <inline-formula> <math display="inline"> <semantics> <mi>σ</mi> </semantics> </math> </inline-formula> also satisfies the conjecture. We also observe that when the von Neumann entropy is defined by the natural logarithm ln instead of <inline-formula> <math display="inline"> <semantics> <msub> <mo form="prefix">log</mo> <mn>2</mn> </msub> </semantics> </math> </inline-formula>, the reduced relative entropy measure of coherence <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mover accent="true"> <mi>C</mi> <mo stretchy="false">¯</mo> </mover> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo>−</mo> <msub> <mi>ρ</mi> <mi>diag</mi> </msub> <mo form="prefix">ln</mo> <msub> <mi>ρ</mi> <mi>diag</mi> </msub> <mo>+</mo> <mi>ρ</mi> <mo form="prefix">ln</mo> <mi>ρ</mi> </mrow> </semantics> </math> </inline-formula> satisfies the inequality <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mover accent="true"> <mi>C</mi> <mo stretchy="false">¯</mo> </mover> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mi>C</mi> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> for any state <inline-formula> <math display="inline"> <semantics> <mi>ρ</mi> </semantics> </math> </inline-formula>.
Abstract licence: CC BY
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