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arXiv:1808.01054 · Physical Review A

Partitioned density matrices and entanglement correlators

T. Cox, Philip C. E. Stamp

Published Dec 10, 2018• 4 citations
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Abstract

The density matrix of a nonrelativistic quantum system, divided into $N$ subsystems, is rewritten in terms of the set of all partitioned density matrices for the system. For the case where the different subsystems are distinguishable, we derive a hierarchy of equations of motion linking the dynamics of all the partitioned density matrices, analogous to the ``Schwinger-Dyson'' hierarchy in quantum field theory. The special case of a set of $N$ coupled spin-$1/2$ ``qubits'' is worked out in detail. The equations are then rewritten in terms of a set of ``entanglement correlators,'' which comprises all the possible correlation functions for the system---this case is worked out for coupled spin systems. The equations of motion for these correlators can be written in terms of a first-order differential equation for an entanglement correlator supervector.

DOI
10.1103/physreva.98.062110

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The density matrix of a nonrelativistic quantum system, divided into NNN subsystems, is rewritten in terms of the set of all partitioned density matrices for the system. For the case where the different subsystems are distinguishable, we derive a hierarchy of equations of motion linking the dynamics of all the partitioned density matrices, analogous to the Schwinger-Dyson'' hierarchy in quantum field theory. The special case of a set of $N$ coupled spin-$1/2$ qubits'' is worked out in detail. The equations are then rewritten in terms of a set of ``entanglement correlators,'' which comprises all the possible correlation functions for the system---this case is worked out for coupled spin systems. The equations of motion for these correlators can be written in terms of a first-order differential equation for an entanglement correlator supervector.
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We derive a classification and a measure of classical- and quantum-correlation of multipartite qubit, qutrit, and in general, nnn-level systems, in terms of SU(n)(n)(n) representations of density matrices. We compare the measure for the case of bipartite correlation with concurrence and the entropy of entanglement. The characterization of correlation is in terms of the number of nonzero singular values of the correlation matrix, but that of mixed state entanglement requires additional invariant parameters in the density matrix. For the bipartite qubit case, the condition for mixed state entanglement is written explicitly in terms of the invariant paramters in the density matrix. For identical particle systems we analyze the effects of exchange symmetry on classical and quantum correlation.
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Density matrices are powerful mathematical tools for the description of closed and open quantum systems. Recently, methods for the direct computation of density matrix elements in scalar quantum field theory were developed based on thermo field dynamics (TFD) and the Schwinger-Keldysh formalism. In this article, we provide a more detailed discussion of these methods and derive expressions for density matrix elements of closed and open systems. At first, we look at closed systems by discussing general solutions to the Schrödinger-like form of the quantum Liouville equations in TFD, showing that the dynamical map is indeed divisible, deriving a path integral-based expression for the density matrix elements in Fock space, and explaining why perturbation theory enables us to use the last even in situations where all initial states in Fock space are occupied. Subsequently, we discuss open systems in the same manner after tracing out environmental degrees of freedom from the solutions for closed systems. We find that, even in a general basis, the dynamical map is not divisible, which renders the dynamics of open systems non-Markovian. Finally, we show how the resulting expressions for open systems can be used to obtain quantum master equations, and comment on the artificiality of time integrals over density matrices that usually appear in many other master equations in the literature but are absent in ours.
We provide a description of interacting quantum fields in terms of density matrices for any occupation numbers in Fock space in a momentum basis. As a simple example, we focus on a real scalar field interacting with another real scalar field, and present a practicable formalism for directly computing the density matrix elements of the combined scalar–scalar system. For deriving the main formula, we use techniques from non-equilibrium quantum field theory like thermo-field dynamics and the Schwinger–Keldysh formalism. Our results allow for studies of particle creation/annihilation processes at finite times and other non-equilibrium processes, including those found in the theory of open quantum systems.