Total versus quantum correlations in a two-mode Gaussian state

Abstract In Li and Luo (2007 Phys. Rev. A 76 032327), the inequality <?CDATA $(1/2){ \mathcal T }\geqslant { \mathcal Q }$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mi mathvariant="italic"></mml:mi> <mml:mo>≥</mml:mo> <mml:mi mathvariant="italic"></mml:mi> </mml:math> was identified as a fundamental postulate for a consistent theory of quantum versus classical correlations for arbitrary measures of total <?CDATA ${ \mathcal T }$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> </mml:math> and quantum <?CDATA ${ \mathcal Q }$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> </mml:math> correlations in bipartite quantum states. Besides, Hayden et al (2006 Commun. Math. Phys. 265 95) have conjectured that, in some conditions within systems endowed with infinite-dimensional Hilbert spaces, quantum correlations may dominate not only half of total correlations but total correlations itself. Here, in a two-mode Gaussian state, quantifying <?CDATA ${ \mathcal T }$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> </mml:math> and <?CDATA ${ \mathcal Q }$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> </mml:math> respectively by the quantum mutual information <?CDATA ${{ \mathcal I }}^{G}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msup> </mml:math> and the entanglement of formation (EoF) <?CDATA ${{ \mathcal E }}_{F}^{G}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> , we verify that <?CDATA ${{ \mathcal E }}_{F,R}^{G}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> is always less than <?CDATA $(1/2){{ \mathcal I }}_{R}^{G}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> when <?CDATA ${{ \mathcal I }}^{G}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msup> </mml:math> and <?CDATA ${{ \mathcal E }}_{F}^{G}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> are defined via the Rényi-2 entropy. While via the von Neumann entropy, <?CDATA ${{ \mathcal E }}_{F,V}^{G}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> may even dominate <?CDATA ${{ \mathcal I }}_{V}^{G}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> itself, which partly consolidates the Hayden conjecture, and partly, provides strong evidence hinting that the origin of this counterintuitive behavior should intrinsically be related to the von Neumann entropy by which the EoF <?CDATA ${{ \mathcal E }}_{F,V}^{G}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> is defined, rather than related to the conceptual definition of the EoF <?CDATA ${{ \mathcal E }}_{F}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> </mml:msub> </mml:math> . The obtained results show that—in the special case of mixed two-mode Gaussian states—quantum entanglement can be faithfully quantified by the Gaussian Rényi-2 EoF <?CDATA ${{ \mathcal E }}_{F,R}^{G}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> .