TY - JOUR

T1 - Partial transposition in a finite-dimensional Hilbert space

T2 - physical interpretation, measurement of observables, and entanglement

AU - Band, Yehuda B.

AU - Mello, Pier A.

N1 - Funding Information:
PAM acknowledges support by DGAPA, under Contract No. IN109014; he is also grateful to the Ben-Gurion University, Beer Sheva, Israel, where this research was started, for its kind hospitality. YBB acknowledges support from the DFG through the DIP program (FO703/2-1).
Publisher Copyright:
© 2017, Chapman University.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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.

AB - 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.

KW - Entanglement

KW - Finite-dimensional Hilbert space

KW - Partial transposition

UR - http://www.scopus.com/inward/record.url?scp=85091755962&partnerID=8YFLogxK

U2 - 10.1007/s40509-017-0120-3

DO - 10.1007/s40509-017-0120-3

M3 - Article

AN - SCOPUS:85091755962

VL - 5

SP - 177

EP - 188

JO - Quantum Studies: Mathematics and Foundations

JF - Quantum Studies: Mathematics and Foundations

SN - 2196-5609

IS - 2

ER -