Geometric measures of information for quantum state characterization

We analyze the geometry of a joint distribution over a set of discrete random variables. We briefly review Shannon's entropy, conditional entropy, mutual information and conditional mutual information. We review the entropic information distance formula of Rokhlin and Rajski. We then define an analogous information area. We motivate this definition and discuss its properties. We extend this definition to higher-dimensional volumes. We briefly discuss the potential utility for these geometric measures in quantum information processing.

[1]  John Preskill,et al.  Quantum computing and the entanglement frontier , 2012, 1203.5813.

[2]  Edward E. Daub,et al.  Reflections on science by Nakaya Ukichiro : an advanced Japanese reader = 中谷宇吉郎の科学観 , 2003 .

[3]  C. Caves,et al.  Concurrence-based entanglement measures for isotropic states , 2003 .

[4]  Schumacher,et al.  Information and quantum nonseparability. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[5]  Warner A. Miller,et al.  Quantum information geometry in the space of measurements , 2018, Commercial + Scientific Sensing and Imaging.

[6]  J. Siewert,et al.  Entanglement of three-qubit Greenberger-Horne-Zeilinger-symmetric states. , 2012, Physical review letters.

[7]  C. Rajski,et al.  A Metric Space of Discrete Probability Distributions , 1961, Inf. Control..

[8]  Pérès.  Separability Criterion for Density Matrices. , 1996, Physical review letters.

[9]  Terhal,et al.  Entanglement of formation for isotropic states , 2000, Physical review letters.

[10]  Péter Gács,et al.  Information Distance , 1998, IEEE Trans. Inf. Theory.

[11]  Tzu-Chieh Wei.  Relative entropy of entanglement for multipartite mixed states: Permutation-invariant states and Dür states , 2008 .

[12]  M. S. Sarandy,et al.  Global quantum discord in multipartite systems , 2011, 1105.2548.

[13]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[14]  Sergei Bravyi.  Entanglement entropy of multipartite pure states , 2003 .

[15]  J. Wheeler.  Information, physics, quantum: the search for links , 1999 .

[16]  V. Rokhlin.  LECTURES ON THE ENTROPY THEORY OF MEASURE-PRESERVING TRANSFORMATIONS , 1967 .

[17]  Robert L. Wolpert,et al.  Statistical Inference , 2019, Encyclopedia of Social Network Analysis and Mining.

[18]  K. Jacobs,et al.  Statistical Inference, Distinguishability of Quantum States, And Quantum Entanglement , 1997, quant-ph/9703025.

[19]  W. H. Zurek,et al.  Thermodynamic cost of computation, algorithmic complexity and the information metric , 1989, Nature.

[20]  Marcus Huber,et al.  Detection of high-dimensional genuine multipartite entanglement of mixed states. , 2009, Physical review letters.

[21]  W. Wootters.  Entanglement of Formation of an Arbitrary State of Two Qubits , 1997, quant-ph/9709029.

[22]  Shahabeddin M. Aslmarand,et al.  Quantum Reactivity: A Measure of Quantum Correlation , 2019 .

[23]  Bill Broyles.  Notes , 1907, The Classical Review.