Wigner function and the successive measurement of position and momentum

Mello, Pier A.; Revzen, M.

doi:10.1103/physreva.89.012106

Cited by 11 publications

(8 citation statements)

References 34 publications

(88 reference statements)

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“…[9,10]) that, for continuous variables, partial transposition of one particle of a bipartite state amounts to a change in sign of the momentum of that particle in the Wigner function (WF) of the state. For the case of discrete variables one can define a "coordinatelike" and a "momentum-like" variable [11,12]. Here we prove that for the discrete variables case, PT can be interpreted in terms of a change in sign of a momentum-like variable of one of the particles in the Wigner function of the state.…”

mentioning

confidence: 68%

“…Recall that the definition of the Wigner function of Refs. [11,12] requires N to be a prime number larger than 2 (see discussion in the previous subsection for one particle).…”

Section: Two-particlesmentioning

confidence: 99%

“…(2) and (4) We define the Wigner function for the density operatorρ as in Refs. [11,12,20], as Wρ(q 1 , q 2 , p 1 , p 2 ) = Tr ρ(Pq 1 p 1 ⊗Pq 2 p 2 ) ,…”

Section: B Proof Of Eq (3)mentioning

confidence: 99%

“…as demonstrated in Appendix C, thus again exhibiting a change in sign of p. The definition of the Wigner function in Refs. [11,12] requires N to be a prime number larger than 2. It turns out that this is the simplest extension of the continuous case to the discrete one that one can study, which can then be extended to the case where N is not prime (see, e.g., Ref.…”

Section: Schwinger Operators Partial Transposition and Change In Sign...mentioning

confidence: 99%

“…This generalization has been widely studied (see, e.g., Ref. [11] and references cited therein); here we use the formulation developed in Refs. [11,12].…”

mentioning

confidence: 99%

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

Band

Mello

2017

Quantum Stud.: Math. Found.

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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 r, the original states are entangled. Keywords 1 IntroductionQuantum entanglement in multipartite qubit (and qunits, i.e., N -dimensional quantum bits) states is a powerful computation and information resource [1,2]. Entanglement of pure quantum states is well understood, but entanglement of mixed quantum states, i.e., states that cannot be represented using a wave

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mentioning

confidence: 68%

“…Recall that the definition of the Wigner function of Refs. [11,12] requires N to be a prime number larger than 2 (see discussion in the previous subsection for one particle).…”

Section: Two-particlesmentioning

confidence: 99%

“…(2) and (4) We define the Wigner function for the density operatorρ as in Refs. [11,12,20], as Wρ(q 1 , q 2 , p 1 , p 2 ) = Tr ρ(Pq 1 p 1 ⊗Pq 2 p 2 ) ,…”

Section: B Proof Of Eq (3)mentioning

confidence: 99%

Section: Schwinger Operators Partial Transposition and Change In Sign...mentioning

confidence: 99%

“…This generalization has been widely studied (see, e.g., Ref. [11] and references cited therein); here we use the formulation developed in Refs. [11,12].…”

mentioning

confidence: 99%

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

Band

Mello

2017

Quantum Stud.: Math. Found.

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Kirkwood and Wigner Quantum Densities, Their Properties, and Applications in Radiophysics

Горбунов

Koval

2021

Radiophys Quantum El

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Finite linear spaces, plane geometries, Hilbert spaces and finite phase space

Revzen

Mann

2016

Quantum Stud.: Math. Found.

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Finite plane geometry is associated with finite dimensional Hilbert space. The association allows mapping of q-number Hilbert space observables to the c-number formalism of quantum mechanics in phase space. The mapped entities reflect geometrically based line-point interrelation. Particularly simple formulas are involved when use is made of mutually unbiased bases (MUB) representations for the Hilbert space entries.The geometry specifies a point-line interrelation. Thus underpinning d-dimensional Hilbert space operators (resp. states) with geometrical points leads to operators termed "line operators" underpinned by the geometrical lines. These "line operators",Lj ; (j designates the line) form a complete orthogonal basis for Hilbert space operators. The representation of Hilbert space operators in terms of these operators form the phase space representation of the d-dimensional Hilbert space.Examples for the use of the "line operators" in mapping (finite dimensional) Hilbert space operators onto finite dimensional phase space functions are considered. These include finite dimensional Wigner function and Radon transform and a geometrical interpretation for the involvement of parity in the mappings of Hilbert space onto phase space.Two d-dimensional particles product states are underpinned with geometrical points. The states, |Lj underpinned with the corresponding geometrical lines are maximally entangled states (MES). These "line states" provide a complete d 2 dimensional orthogonal MES basis for for the two d-dimensional particles. The complete d2 dimensional MES i.e. the "line states" are shown to provide a transparent geometrical interpretation to the so called Mean King Problem and its variant.The "line operators" (resp. "line states") are studied in detail.The paper aims at self sufficiency and to this end all relevant notions are explained herewith.

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Wigner function and the successive measurement of position and momentum

Cited by 11 publications

References 34 publications

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

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

Kirkwood and Wigner Quantum Densities, Their Properties, and Applications in Radiophysics

Finite linear spaces, plane geometries, Hilbert spaces and finite phase space

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