Monogamy and tradeoff relations for wave-particle duality information

• URL: http://arxiv.org/abs/2401.01235v1
• Date: Tue, 2 Jan 2024 15:03:55 GMT
• Title: Monogamy and tradeoff relations for wave-particle duality information
• Authors: Shailja Kapoor, Sohail, Gautam Sharma and Arun K. Pati
• Abstract summary: We show that the information content of a quantum system is nothing but the Hilbert-Schmidt distance between the state of the quantum system and the maximally mixed state. We then employ the quantum Pinsker's inequality and the reverse Pinsker's inequality to derive a new complementarity and a reverse complementarity relation between the single-party information content and the entanglement present in a bipartite quantum system in a pure state.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The notions of predictability and visibility are essential in the mathematical formulation of wave particle duality. The work of Jakob and Bergou [Phys. Rev. A 76, 052107] generalises these notions for higher-dimensional quantum systems, which were initially defined for qubits, and subsequently proves a complementarity relation between predictability and visibility. By defining the single-party information content of a quantum system as the addition of predictability and visibility, and assuming that entanglement in a bipartite system in the form of concurrence mutually excludes the single-party information, the authors have proposed a complementarity relation between the concurrence and the single-party information content. We show that the information content of a quantum system defined by Jakob and Bergou is nothing but the Hilbert-Schmidt distance between the state of the quantum system of our consideration and the maximally mixed state. Motivated by the fact that the trace distance is a good measure of distance as compared to the Hilbert-Schmidt distance from the information theoretic point of view, we, in this work, define the information content of a quantum system as the trace distance between the quantum state and the maximally mixed state. We then employ the quantum Pinsker's inequality and the reverse Pinsker's inequality to derive a new complementarity and a reverse complementarity relation between the single-party information content and the entanglement present in a bipartite quantum system in a pure state. As a consequence of our findings, we show that for a bipartite system in a pure state, its entanglement and the predictabilities and visibilities associated with the subsystems cannot be arbitrarily small as well as arbitrarily large.

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