Partial transposition in a finite-dimensional Hilbert space: physical interpretation, measurement of observables, and entanglement

Yehuda B. Band, Pier A. Mello

Research output: Contribution to journalArticlepeer-review

Abstract

We show that partial transposition for pure and mixed two-particle states in a discrete N-dimensional Hilbert space is equivalent to a change in sign of a “momentum-like” variable of one of the particles in the Wigner function for the state. This generalizes a result obtained for continuous-variable systems to the discrete-variable system case. We show that, in principle, quantum mechanics allows measuring the expectation value of an observable in a partially transposed state, in spite of the fact that the latter may not be a physical state. We illustrate this result with the example of an “isotropic state”, which is dependent on a parameter r, and an operator whose variance becomes negative for the partially transposed state for certain values of r; for such rs, the original states are entangled.

Original languageEnglish
Pages (from-to)177-188
Number of pages12
JournalQuantum Studies: Mathematics and Foundations
Volume5
Issue number2
DOIs
StatePublished - 1 Jun 2018

Keywords

  • Entanglement
  • Finite-dimensional Hilbert space
  • Partial transposition

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics
  • Mathematical Physics

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