Density Matrix in Quantum Mechanics and Distinctness of Ensembles Having the Same Compressed Density Matrix

Abstract

We clarify different definitions of the density matrix by proposing the use of different names, the full density matrix for a single-closed quantum system, the compressed density matrix for the averaged single molecule state from an ensemble of molecules, and the reduced density matrix for a part of an entangled quantum system, respectively. We show that ensembles with the same compressed density matrix can be physically distinguished by observing fluctuations of various observables. This is in contrast to a general belief that ensembles with the same compressed density matrix are identical. Explicit expression for the fluctuation of an observable in a specified ensemble is given. We have discussed the nature of nuclear magnetic resonance quantum computing. We show that the conclusion that there is no quantum entanglement in the current nuclear magnetic resonance quantum computing experiment is based on the unjustified belief that ensembles having the same compressed density matrix are identical physically. Related issues in quantum communication are also discussed.

26 pages. To appear in Foundations of Physics, 36 (8), 2006

References in corpus (5)

• Quantum cryptography: Public key distribution and coin tossing
• Entanglement in a two-identical-particle system
• Experimental realization of a fetching algorithm in a 7 qubit NMR quantum computer
• No-relationship between impossibility of faster-than-light quantum communication and distinction of ensembles with the same density matrix
• Can differently prepared mixed states be distinguished?

Cited by in corpus (6)

• When Entanglement meets Classical Communications: Quantum Teleportation for the Quantum Internet (Invited Paper)
• Polarimetric characterization of light and media
• Role of weak measurements on states ordering and monogamy of quantum correlation
• Quantum dissonance provide power to deterministic quantum computation with single qubit
• Quantumness of pure-state ensembles via coherence of Gram matrix based on generalized $\alpha$-$z$-relative Rényi entropy
• Distribution of Standard deviation of an observable among superposed states