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arXiv:quant-ph/9504019 · Physical Review Letters

Unitary Evolution between Pure and Mixed States

Benni Reznik

Published Feb 19, 1996• 23 citations
arXivHTMLPDF

Abstract

We propose an extended quantum mechanical formalism that is based on a wave operator $\stackrel{^}{\ensuremath{\varrho}}$, which is related to the ordinary density matrix via $\ensuremath{\rho}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\stackrel{^}{\ensuremath{\varrho}}{\stackrel{^}{\ensuremath{\varrho}}}^{\ifmmode\dagger\else\textdagger\fi{}}$. This formalism allows a (generalized) unitary evolution between pure and mixed states. It also preserves much of the connection between symmetries and conservation laws. The new formalism is illustrated for the case of a two-level system.

DOI
10.1103/physrevlett.76.1192

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We propose an extended quantum mechanical formalism that is based on a wave operator \stackrel{^}{\ensuremath{\varrho}}, which is related to the ordinary density matrix via \ensuremath{\rho}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\stackrel{^}{\ensuremath{\varrho}}{\stackrel{^}{\ensuremath{\varrho}}}^{\ifmmode\dagger\else\textdagger\fi{}}. This formalism allows a (generalized) unitary evolution between pure and mixed states. It also preserves much of the connection between symmetries and conservation laws. The new formalism is illustrated for the case of a two-level system.
In the formulation of Banks, Peskin and Susskind, we show that one can construct evolution equations for the quantum mechanical density matrix ρρρ with operators which do not commute with hamiltonian which evolve pure states into mixed states, preserve the normalization and positivity of ρρρ and conserve energy. Furthermore, it seems to be different from a quantum mechanical system with random sources.
Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics, are handled through master equations for the density matrix. By embedding elements of this matrix in a higher-dimensional Liouville-Bloch equation, the methods of unitary integration are adapted to solve for the density matrix as a function of time, including the nonunitary effects of dissipation and decoherence. The input requires only solutions of classical, initial value time-dependent equations. Results are illustrated for a damped, driven two-level system.
Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics, are handled through master equations for the density matrix. Master equations such as the Lindblad equation preserve the trace of this matrix. Viewing them as first-order time-dependent operator equations for the elements of the density matrix, a unitary integration procedure can be adapted to solve for these elements. A simple model for decoherence preserves the Hermiticity of the density matrix. A single, classical Riccati equation is the only one requiring numerical handling to obtain a full solution of the quantum evolution. The procedure is general, valid for any number of levels, but is illustrated here for a three-level system with two driving fields. For various choices of the initial state, we study the evolution of the system as a function of the amplitudes, relative frequencies, and phases of the driven fields and of the strength of the decoherence. The monotonic growth of the entropy is followed as the system evolves from a pure to a mixed state. An example is provided by the n=3n=3n=3 states of the hydrogen atom in a time-dependent electric field, such degenerate manifolds affording an analytical solution.
The mixed states are important in quantum optics since they frequently appear in the decoherence problems. When one of the components of the system is prepared in the mixed state and the evolution operator of this system is not available, one cannot deduce the density matrix. We present analytical approach to accurately solve this problem. The approach can be applied on the condition that the Schrödinger's equation of the system is solvable with any arbitrary initial state. In deriving the solution we exploit the fact that any mixed state can be expressed in terms of a phase state. The approach is illustrated by deriving the density matrix of a single-mode heat environment interacting asymmetrically with two qubits. Our results are in good agreement with the available results in the literature. This approach opens new perspectives for treating complicated systems and may impact other applications in the quantum theory.
Usually models for quantum computations deal with unitary gates on pure states. In this paper we generalize the usual model. We consider a model of quantum computations in which the state is an operator of density matrix and the gates are quantum operations, not necessarily unitary. A mixed state (operator of density matrix) of n two-level quantum systems is considered as an element of 4n4^{n}4n-dimensional operator Hilbert space. Unitary quantum gates and nonunitary quantum operations for n-qubit system are considered as generalized quantum gates acting on mixed state. In this paper we study universality for quantum computations by quantum operations on mixed states.
Students in quantum mechanics class are taught that the wave function contains all knowable information about an isolated system. Later in the course, this view seems to be contradicted by the mysterious density matrix, which introduces a new set of probabilities in addition to those that are built into the wave function. This paper brings attention to the fact that the density matrix can be reconciled with the underlying quantum-mechanical description using a two-particle entangled state with a one-particle subsystem as the simplest illustration of the basic principle. The extra-quantum probabilities are traced to the coefficients of superposition of the quantum state vector and the seemingly irreversible exponential population decay is shown to be compatible with the unitary time evolution of a pure state when the two particles interact. The two-particle universe thus provides the student with a tool for understanding how the density operator, with all its richness, emerges from quantum mechanics.