Geometry of Kirkwood-Dirac classical states: A case study based on discrete Fourier transform (2404.09399v2)
Abstract: The characterization of Kirkwood-Dirac (KD) classicality or non-classicality is very important in quantum information processing. In general, the set of KD classical states with respect to two bases is not a convex polytope[J. Math. Phys. \textbf{65} 072201 (2024)], which makes us interested in finding out in which circumnstances they do form a polytope. In this paper, we focus on the characterization of KD classicality of mixed states for the case where the transition matrix between two bases is a discrete Fourier transform (DFT) matrix in Hilbert space with dimensions $p2$ and $pq$, respectively, where $p, q$ are prime. For the two particular cases we investigate, the sets of extremal points are finite, implying that the set of KD classical states we characterize forms a convex polytope. We show that for $p2$ dimensional system, the set $\rm{KD}{\mathcal{A},\mathcal{B}}+$ is a convex hull of the set $\rm {pure}({\rm {KD}{\mathcal{A},\mathcal{B}}+})$ based on DFT, where $\rm{KD}{\mathcal{A},\mathcal{B}}+$ is the set of KD classical states with respect to two bases and $\rm {pure}({\rm {KD}{\mathcal{A},\mathcal{B}}+})$ is the set of all the rank-one projectors of KD classical pure states with respect to two bases. In $pq$ dimensional system, we believe that this result also holds. Unfortunately, we do not completely prove it, but some meaningful conclusions are obtained about the characterization of KD classicality.
- Ollivier H and Zurek W H 2001 Quantum discord: a measure of the quantumness of correlations Phys. Rev. Lett. 88 017901
- Baumgratz T, Cramer M and Plenio M B 2014 Quantifying coherence Phys. Rev. Lett. 113 140401
- Robertson H P 1929 The uncertainty principle Phys. Rev. 34 163
- Kirkwood J G 1933 Quantum statistics of almost classical assemblies Phys. Rev. 44 31
- Dirac P A 1945 On the analogy between classical and quantum mechanics Rev. Mod. Phys. 17 195
- Brookes B C and Kolmogorov A N 1951 Foundations of the theory of probability The Math. Gaz. 35 292
- Arvidsson-Shukur D R, Drori J C and Halpern N Y 2021 Conditions tighter than noncommutation needed for nonclassicality J. Phys. A: Math. Theor. 54 284001
- Xu J W 2022 Kirkwood-Dirac classical pure states (arXiv:2210.02876)
- Langrenez C, Arvidsson-Shukur D R and De Bièvre S 2023 Characterizing the geometry of the Kirkwood-Dirac positive states (arXiv:2306.00086)
- Lundeen J S and Bamber C 2012 Procedure for direct measurement of general quantum states using weak measurement Phys. Rev. Lett. 108 070402
- Bamber C and Lundeen J S 2014 Observing Diracs classical phase space analog to the quantum state Phys. Rev. Lett. 112 070405
- Pusey M F 2014 Anomalous weak values are proofs of contextuality Phys. Rev. Lett. 113 200401
- Dressel J 2015 Weak values as interference phenomena Phys. Rev. A 91 032116
- Jenne J H and Arvidsson-Shukur D R 2022 Unbounded and lossless compression of multiparameter quantum information Phys. Rev. A 106 042404
- Halpern N Y, Swingle B and Dressel J 2018 Quasiprobability behind the out-of-time-ordered correlator Phys. Rev. A 97 042105
- Levy A and Lostaglio M 2020 Quasiprobability distribution for heat uctuations in the quantum regime PRX Quantum 1 010309
- Lostaglio M 2020 Certifying quantum signatures in thermodynamics and metrology via contextuality of quantum linear response Phys. Rev. Lett. 125 230603
- Spekkens R W 2008 Negativity and contextuality are equivalent notions of nonclassicality Phys. Rev. Lett. 101 020401
- Xu J W 2022 Classi cation of incompatibility for two orthonormal bases Phys. Rev. A 106 022217
- Fiorentino V and Weigert S 2022 Uncertainty relations for the support of quantum states J. Phys. A: Math. Theor. 55 495305
- Hiriart-Urruty J B and Lemaréchal C 2004 Fundamentals of convex analysis Spr. Sci. Bus. Med.
- He J Y and Fu S S 2024 Nonclassicality of the Kirkwood-Dirac quasiprobability distribution via quantum modification terms Phys. Rev. A 109 012215
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Collections
Sign up for free to add this paper to one or more collections.
