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Conditional Density Matrix in the Context of Noncontextuality

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

Conditional density matrix represents a quantum state of subsystem in different schemes of quantum communication. Here we discuss some properties of conditional density matrix and its place in general scheme of quantum mechanics. Talk presented at the Eleventh Lmonosov Conference on Elementary Particle Physics, Moscow, August, 2003.

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A new quantum mechanical notion - Conditional Density Matrix - proposed by the authors, is discussed and is applied to describe some physical processes. This notion is a natural generalization of von Neumann density matrix for such processes as divisions of quantum systems into subsystems and reunifications of subsystems into new joint systems. Conditional Density Matrix assigns a quantum state to a subsystem of a composite system under condition that another part of the composition system is in some pure state. It is shown that conditional density matrix naturally arises by expanding of reduced density matrix.
Density matrix is one important tool in quantum mechanics,and it has very broad applications.However there are different definitions about the density matrix,and they describe quite different systems.There has been some misunderstanding about the density matrix in the community,and these misunderstandings hinder the right application of the density matrix.In this article,we discuss the different definitions of density matrix. We suggest to use the full density matrix,compressed density matrix and the reduced density matrix to describe the state of a complete quantum system,the state of an averaged particle in an ensemble and the state of part of a composite system.We stress that contrary to the wide accepted understanding that ensembles with the same compressed density matrix are physically indistinguishable, they are distinguishable through the so-called ensemble measurement.As an application,we suggest that the present conclusion that the present-day nuclear magnetic resonance quantum computation does not have quantum entanglement is groundless.
Density Matrix is used to unify the description of classical and quantum objects. The idea is that both states of classical object and states of quantum object can be regarded as vectors in Hilbert space, so that all the states can be represented by density matrices. After constructing the general framework, we apply this to discuss the possibility to map a quantum density matrix onto a classical density matrix, which is also the question explored in Hidden Variable Theory (HVT) of Quantum Mechanics (QM). Bell's Theorem set a boundary between HVT and QM, but the place of the real quantum world depends on experiments. However, here HVT is ruled out just by theoretical consideration. We tackle this question by explicitly constructing the most general HVT, which gives all desired results according QM, and then checking its validity and acceptability. Relation between such HVT and Bell's inequality and Kochen-Specker theorem is also discussed.