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A semiclassically entangled puzzle. (arXiv:math-ph/0212014v1 CROSS LISTED)

Pedro de M. Rios

Published Aug 17, 2020• 0 citations
OpenAlex

Abstract

For a maximally entangled eigenstate of a system of two non-interacting identical one dimensional harmonic oscilators, at the semiclassical level, it is not obviously true that a nonlinear interaction with one of the subsystems leaves the reduced semiclassical Wigner function of the other subsystem unaffected. Once stated, we advance some wild speculations regarding this seeming puzzle.

Related papers

A semiclassically entangled puzzle

Pedro de M Rios

For a maximally entangled eigenstate of a system of two non-interacting identical one dimensional harmonic oscilators, at the semiclassical level, it is not obviously true that a nonlinear interaction with one of the subsystems leaves the reduced semiclassical Wigner function of the other subsystem unaffected. Once stated, we advance some wild speculations regarding this seeming puzzle.

arXiv:math-ph/0212014•Journal of Physics A Mathematical and Theoretical•Nov 21, 2007•3 citations

A SEMICLASSICALLY ENTANGLED PUZZLE

Pedro de M. Rios, M. Rios

Abstract. For a maximally entangled eigenstate of a system of two non-interacting identical one dimensional harmonic oscilators, at the semiclassical level, it is not obviously true that a nonlinear interaction with one of the subsystems leaves the reduced semiclassical Wigner function of the other subsystem unaffected. Once stated, we advance some wild speculations regarding this seeming puzzle. 1. introduction When Einstein, Podolsky and Rosen [6] explored the “strange ” behaviour of entangled states, their main goal was to set forth an argument concerning the ontology of quantum mechanics. Only quite later, did Bell [2] show that some of the discussion had a definite empirical bearing, namely, on matters related to the nonlocal character of quantum mechanics. After the empirical confirmation of this latter [1], emphasis has been placed on the “peaceful coexistence ” [15] of this kind of nonlocality and the partial order of spacetime events derived from the principle of relativity [5]. In terms of the Wigner [17] function W, the cartesian Weyl [16] representation of the density operator ˆρ as a real function on euclidean phase space, this question is addressed in the following way: let x1 and x2 denote the phase space points of two subsystems whose total system is in an entangled eigenstate of the hamiltonian ˆ H1 + ˆ H2, with corresponding Wigner function W0(x1, x2).

•Jan 1, 2002•3 citations

Off-diagonal matrix elements in the semiclassical limit

Stephan De Bièvre, Jacques Renaud

We give a very simple proof of the convergence of the non-diagonal Wigner functions of the harmonic oscillator eigenstates to in the semiclassical limit , and .

Journal of Physics A Mathematical and General•Dec 7, 1996•1 citation

Nonlinearity without Superluminality. (arXiv:quant-ph/0204106v4 CROSS LISTED)

Adrian Kent

Quantum theory is compatible with special relativity. In particular, though measurements on entangled systems are correlated in a way that cannot be reproduced by local hidden variables, they cannot be used for superluminal signalling. As Czachor, Gisin and Polchinski pointed out, this is not true for general nonlinear modifications of the Schroedinger equation. Excluding superluminal signalling has thus been taken to rule out most nonlinear versions of quantum theory. The no superluminal signalling constraint has also been used for alternative derivations of the optimal fidelities attainable for imperfect quantum cloning and other operations. These results apply to theories satisfying the rule that their predictions for widely separated and slowly moving entangled systems can be approximated by non-relativistic equations of motion with respect to a preferred time coordinate. This paper describes a natural way in which this rule might fail to hold. In particular, it is shown that quantum readout devices which display the values of localised pure states need not allow superluminal signalling, provided that the devices display the values of the states of entangled subsystems as defined in a non-standard, but natural, way. It follows that any locally defined nonlinear evolution of pure states can be made consistent with Minkowski causality.

•Jul 16, 2018•0 citations

Quantum entanglement and squeezing in coupled harmonic and anharmonic oscillator systems

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Journal of Russian Laser Research•Jul 1, 2011•7 citations

Moyal star product approach to the Bohr-Sommerfeld approximation. (arXiv:math-ph/0409039v1 CROSS LISTED)

Matthew Cargo, Alfonso Gracia-Saz, Robert G. Littlejohn +2 more

The Bohr-Sommerfeld approximation to the eigenvalues of a one-dimensional quantum Hamiltonian is derived through order $\hbar^2$ (i.e., including the first correction term beyond the usual result) by means of the Moyal star product. The Hamiltonian need only have a Weyl transform (or symbol) that is a power series in $\hbar$, starting with $\hbar^0$, with a generic fixed point in phase space. The Hamiltonian is not restricted to the kinetic-plus-potential form. The method involves transforming the Hamiltonian to a normal form, in which it becomes a function of the harmonic oscillator Hamiltonian. Diagrammatic and other techniques with potential applications to other normal form problems are presented for manipulating higher order terms in the Moyal series.

•Aug 17, 2020•0 citations
For a maximally entangled eigenstate of a system of two non-interacting identical one dimensional harmonic oscilators, at the semiclassical level, it is not obviously true that a nonlinear interaction with one of the subsystems leaves the reduced semiclassical Wigner function of the other subsystem unaffected. Once stated, we advance some wild speculations regarding this seeming puzzle.
Abstract. For a maximally entangled eigenstate of a system of two non-interacting identical one dimensional harmonic oscilators, at the semiclassical level, it is not obviously true that a nonlinear interaction with one of the subsystems leaves the reduced semiclassical Wigner function of the other subsystem unaffected. Once stated, we advance some wild speculations regarding this seeming puzzle. 1. introduction When Einstein, Podolsky and Rosen [6] explored the “strange ” behaviour of entangled states, their main goal was to set forth an argument concerning the ontology of quantum mechanics. Only quite later, did Bell [2] show that some of the discussion had a definite empirical bearing, namely, on matters related to the nonlocal character of quantum mechanics. After the empirical confirmation of this latter [1], emphasis has been placed on the “peaceful coexistence ” [15] of this kind of nonlocality and the partial order of spacetime events derived from the principle of relativity [5]. In terms of the Wigner [17] function W, the cartesian Weyl [16] representation of the density operator ˆρ as a real function on euclidean phase space, this question is addressed in the following way: let x1 and x2 denote the phase space points of two subsystems whose total system is in an entangled eigenstate of the hamiltonian ˆ H1 + ˆ H2, with corresponding Wigner function W0(x1, x2).
Quantum theory is compatible with special relativity. In particular, though measurements on entangled systems are correlated in a way that cannot be reproduced by local hidden variables, they cannot be used for superluminal signalling. As Czachor, Gisin and Polchinski pointed out, this is not true for general nonlinear modifications of the Schroedinger equation. Excluding superluminal signalling has thus been taken to rule out most nonlinear versions of quantum theory. The no superluminal signalling constraint has also been used for alternative derivations of the optimal fidelities attainable for imperfect quantum cloning and other operations. These results apply to theories satisfying the rule that their predictions for widely separated and slowly moving entangled systems can be approximated by non-relativistic equations of motion with respect to a preferred time coordinate. This paper describes a natural way in which this rule might fail to hold. In particular, it is shown that quantum readout devices which display the values of localised pure states need not allow superluminal signalling, provided that the devices display the values of the states of entangled subsystems as defined in a non-standard, but natural, way. It follows that any locally defined nonlinear evolution of pure states can be made consistent with Minkowski causality.
The Bohr-Sommerfeld approximation to the eigenvalues of a one-dimensional quantum Hamiltonian is derived through order ℏ2\hbar^2ℏ2 (i.e., including the first correction term beyond the usual result) by means of the Moyal star product. The Hamiltonian need only have a Weyl transform (or symbol) that is a power series in ℏ\hbarℏ, starting with ℏ0\hbar^0ℏ0, with a generic fixed point in phase space. The Hamiltonian is not restricted to the kinetic-plus-potential form. The method involves transforming the Hamiltonian to a normal form, in which it becomes a function of the harmonic oscillator Hamiltonian. Diagrammatic and other techniques with potential applications to other normal form problems are presented for manipulating higher order terms in the Moyal series.