A Computable Gaussian Quantum Correlation for Continuous-Variable Systems

• Main Authors: Liang Liu, Jinchuan Hou, Xiaofei Qi
• Format: Article
• Language: English
• Published: MDPI AG 2021-09-01
• Series: Entropy
• Subjects: continuous-variable systems; Gaussian states; Gaussian geometric discord; Gaussian channels
• Online Access: https://www.mdpi.com/1099-4300/23/9/1190

Generally speaking, it is difficult to compute the values of the Gaussian quantum discord and Gaussian geometric discord for Gaussian states, which limits their application. In the present paper, for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo></mrow></semantics></math></inline-formula>-mode continuous-variable system, a computable Gaussian quantum correlation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">M</mi></semantics></math></inline-formula> is proposed. For any state <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub></semantics></math></inline-formula> of the system, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">M</mi><mo>(</mo><msub><mi>ρ</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> depends only on the covariant matrix of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub></semantics></math></inline-formula> without any measurements performed on a subsystem or any optimization procedures, and thus is easily computed. Furthermore, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">M</mi></semantics></math></inline-formula> has the following attractive properties: (1) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">M</mi></semantics></math></inline-formula> is independent of the mean of states, is symmetric about the subsystems and has no ancilla problem; (2) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">M</mi></semantics></math></inline-formula> is locally Gaussian unitary invariant; (3) for a Gaussian state <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">M</mi><mo>(</mo><msub><mi>ρ</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub></semantics></math></inline-formula> is a product state; and (4) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi mathvariant="script">M</mi><mrow><mo>(</mo><mrow><mo>(</mo><msub><mo>Φ</mo><mi>A</mi></msub><mo>⊗</mo><msub><mo>Φ</mo><mi>B</mi></msub><mo>)</mo></mrow><msub><mi>ρ</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>)</mo></mrow><mo>≤</mo><mi mathvariant="script">M</mi><mrow><mo>(</mo><msub><mi>ρ</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> holds for any Gaussian state <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub></semantics></math></inline-formula> and any Gaussian channels <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Φ</mo><mi>A</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Φ</mo><mi>B</mi></msub></semantics></math></inline-formula> performed on the subsystem A and B, respectively. Therefore, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">M</mi></semantics></math></inline-formula> is a nice Gaussian correlation which describes the same Gaussian correlation as Gaussian quantum discord and Gaussian geometric discord when restricted on Gaussian states. As an application of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">M</mi></semantics></math></inline-formula>, a noninvasive quantum method for detecting intracellular temperature is proposed.