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Mixed-state twin observables

F Herbut, Milan Damnjanović

Published Jan 1, 2000• 7 citations
OpenAlex

Abstract

Abstract. Twin observables, i.e. opposite subsystem observables A+ and A− that are indistinguishable in measurement in a given mixed or pure state ρ, are investigated in detail algebraicly and geometrically. It is shown that there is a farreaching correspondence between the detectable (in ρ) spectral entities of the two operators. Twin observables are state-dependently quantum-logically equivalent, and direct subsystem measurement of one of them ipso facto gives rise to the indirect (i.e. distant) measurement of the other. Existence of nontrivial twins requires singularity of ρ. Systems in thermodynamic equilibrium do not admit subsystem twins. These observables may enable one to simplify the matrix representing ρ.

Related papers

Mixed-state twin observables

F Herbut, M Damnjanovic

Twin observables are opposite subsystem observables A + and A - if they are indistinguishable in measurement in a given mixed or pure state ρ in the sense that the measurement of one amounts to the same as the measurement of the other (direct measurement and so-called distant measurement, respectively). It is pointed out that twin observables may reveal quantum correlations that give rise to the disappearance of interference in the two-slit experiment. Twin observables in general states are investigated in detail algebraically and geometrically. It is shown that there is a far-reaching correspondence between the detectable (in ρ) spectral entities of the two operators. Twin observables are state-dependently quantum-logically equivalent, and the direct subsystem measurement of one of them ipso facto gives rise to the indirect (i.e. distant) measurement of the other. Existence of nontrivial twins requires a singularity of ρ. Systems in thermodynamic equilibrium do not admit subsystem twins. These observables may enable one to simplify the matrix representing ρ.

arXiv:quant-ph/0004085•Journal of Physics A Mathematical and General•Aug 16, 2000•12 citations

On twin observables in entangled mixed states

Fedor Herbut

It is pointed out that every mixed state statistical operator is, up to a normalization constant, a super state vector in the Hilbert space of linear Hilbert-Schmidt operators acting in the state space of the quantum system. Hence, the well understood Schmidt canonical expansion of ordinary state vectors can be carried over to mixed states. In particular, it can be utilized for evaluating all the twins, i. e., the opposite-subsystem observables the measurement of one of which is, on account of entanglement, ipso facto also a measurement of the other. This is illustrated in full detail in the case of the Horodecki two spin-one-half-particle states with maximally disordered subsystems.

arXiv:quant-ph/0106038•Jun 7, 2001•1 citation

A quantum-mechanical theory of distant correlations

Milan Vujičić, Fedor Herbut

A composite quantum system consisting of two distant subsystems and described by a correlated state vector φ12 is considered. It was shown in a previous work by the authors [Ann. Phys. 96, 382 (1976)] that such a system can be equivalently described in terms of the reduced statistical operators ρ1 and ρ2 of φ12 applying to the subsystems and a correlation operator Ua between them. It is argued that this description has a firm physical foundation for the system considered in view of the fact that, on account of the subsystems being distant, one can only measure pairs of subsystem observables A1, B2 in coincidence. The direct measurement of A1 such that [A1,ρ1]=0 on the ensemble of first subsystems performs distantly (without interaction) an orthogonal decomposition of the ensemble of second subsystems ρ2, that amounts to the measurement of the twin observable A2 (A2≡UaA1U−1aQ2, Q2 being the range projector of ρ2). A number of coincidence experiments have confirmed this claim, and have disproved all attempts (on the quantum and on the subquantum levels) to view this decomposition of ρ2 as being present also before the measurement of A1. Hence, this decomposition into subensembles comes about in the very measurement of A1, and Ua determines them in a simple way. It is demonstrated that Ua is essential for twin observables and twin symmetry operators. A detailed study of these operators is presented from a unified point of view. Puzzling features of quantum correlations described by Ua show up in composite states when the mentioned distant decompositions of ρ2 into subensembles can be incompatible with one another. A general definition of such φEPR12 states (called Einstein–Podolsky–Rosen states) is given in a few equivalent forms, and the nonuniqueness of the Schmidt canonical form of φEPR12 is investigated in order to encourage further theoretical and experimental exploration of distant quantum correlations.

Journal of Mathematical Physics•Jul 1, 1984•28 citations

Strong twin events in mixed-state entanglement

Fedor Herbut

Continuing the study of mixed-state entanglement in terms of opposite-subsystem observables the measurement of one of which amounts to the same as that of the other (so-called twins), begun in a recent article, so-called strong twin events, which imply biorthogonal mixing of states, are defined and studied. It is shown that for each mixed state there exists a Schmidt canonical (super state vector) expansion in terms of Hermitian operators, and that it can be the continuation of the mentioned biorthogonal mixing due to strong twins. The case of weak twins and nonhermitian Schmidt canonical expansion is also investigated. A necessary and sufficient condition for the existence of nontrivial twins for separable states is derived.

arXiv:quant-ph/0106101•Jun 18, 2001•1 citation

Delayed Twin Observables Are They a Fundamental Concept in Quantum Mechanics?

Fedor Herbut

Opposite-subsystem twin events and twin observables, studied previously in the context of distant correlations, are first generalized to pure states of not-necessarily-composite systems, and afterwards they are further generalized to delayed twins that are due to unitary evolution of the quantum system. The versatile aspects of delayed twin observables are studied in terms of necessary and sufficient conditions to make possible various applications. Three of these are sketched: Preparation of some quantum experiments, easy solution of a puzzle in an important Scully et al. real experiment, and, finally, it is shown that exact measurement in quantum mechanics is an example of opposite-subsystem delayed twins in bipartite pure states.

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Hermitian Schmidt decomposition and twin observables of bipartite mixed states

F Herbut

The study of mixed-state quantum correlations in terms of opposite-subsystem observables - the measurement of one of which amounts to the same as that of the other - of so-called twins is continued. Twin events that imply biorthogonal mixing of states, called `strong twin events', are studied. It is shown that for each mixed state there exists a Schmidt (super-state-vector) decomposition in terms of Hermitian operators, and that it can be the continuation of the above-mentioned biorthogonal mixing due to strong twins. The case of weak twins and non-Hermitian Schmidt decomposition is also investigated. For separable states a necessary and sufficient condition for the existence of nontrivial twins is derived. Utilization of the Hermitian Schmidt decomposition for finding all twins is illustrated in full detail for the case of the two spin-half-particle states with maximally disordered subsystems (mixtures of Bell states). It is shown that only rank-two mixtures have nontrivial twins.

arXiv:quant-ph/0305181•Journal of Physics A Mathematical and General•Feb 11, 2002•4 citations
Abstract. Twin observables, i.e. opposite subsystem observables A+ and A− that are indistinguishable in measurement in a given mixed or pure state ρ, are investigated in detail algebraicly and geometrically. It is shown that there is a farreaching correspondence between the detectable (in ρ) spectral entities of the two operators. Twin observables are state-dependently quantum-logically equivalent, and direct subsystem measurement of one of them ipso facto gives rise to the indirect (i.e. distant) measurement of the other. Existence of nontrivial twins requires singularity of ρ. Systems in thermodynamic equilibrium do not admit subsystem twins. These observables may enable one to simplify the matrix representing ρ.
Twin observables are opposite subsystem observables A + and A - if they are indistinguishable in measurement in a given mixed or pure state ρ in the sense that the measurement of one amounts to the same as the measurement of the other (direct measurement and so-called distant measurement, respectively). It is pointed out that twin observables may reveal quantum correlations that give rise to the disappearance of interference in the two-slit experiment. Twin observables in general states are investigated in detail algebraically and geometrically. It is shown that there is a far-reaching correspondence between the detectable (in ρ) spectral entities of the two operators. Twin observables are state-dependently quantum-logically equivalent, and the direct subsystem measurement of one of them ipso facto gives rise to the indirect (i.e. distant) measurement of the other. Existence of nontrivial twins requires a singularity of ρ. Systems in thermodynamic equilibrium do not admit subsystem twins. These observables may enable one to simplify the matrix representing ρ.
It is pointed out that every mixed state statistical operator is, up to a normalization constant, a super state vector in the Hilbert space of linear Hilbert-Schmidt operators acting in the state space of the quantum system. Hence, the well understood Schmidt canonical expansion of ordinary state vectors can be carried over to mixed states. In particular, it can be utilized for evaluating all the twins, i. e., the opposite-subsystem observables the measurement of one of which is, on account of entanglement, ipso facto also a measurement of the other. This is illustrated in full detail in the case of the Horodecki two spin-one-half-particle states with maximally disordered subsystems.
A composite quantum system consisting of two distant subsystems and described by a correlated state vector φ12 is considered. It was shown in a previous work by the authors [Ann. Phys. 96, 382 (1976)] that such a system can be equivalently described in terms of the reduced statistical operators ρ1 and ρ2 of φ12 applying to the subsystems and a correlation operator Ua between them. It is argued that this description has a firm physical foundation for the system considered in view of the fact that, on account of the subsystems being distant, one can only measure pairs of subsystem observables A1, B2 in coincidence. The direct measurement of A1 such that [A1,ρ1]=0 on the ensemble of first subsystems performs distantly (without interaction) an orthogonal decomposition of the ensemble of second subsystems ρ2, that amounts to the measurement of the twin observable A2 (A2≡UaA1U−1aQ2, Q2 being the range projector of ρ2). A number of coincidence experiments have confirmed this claim, and have disproved all attempts (on the quantum and on the subquantum levels) to view this decomposition of ρ2 as being present also before the measurement of A1. Hence, this decomposition into subensembles comes about in the very measurement of A1, and Ua determines them in a simple way. It is demonstrated that Ua is essential for twin observables and twin symmetry operators. A detailed study of these operators is presented from a unified point of view. Puzzling features of quantum correlations described by Ua show up in composite states when the mentioned distant decompositions of ρ2 into subensembles can be incompatible with one another. A general definition of such φEPR12 states (called Einstein–Podolsky–Rosen states) is given in a few equivalent forms, and the nonuniqueness of the Schmidt canonical form of φEPR12 is investigated in order to encourage further theoretical and experimental exploration of distant quantum correlations.
Continuing the study of mixed-state entanglement in terms of opposite-subsystem observables the measurement of one of which amounts to the same as that of the other (so-called twins), begun in a recent article, so-called strong twin events, which imply biorthogonal mixing of states, are defined and studied. It is shown that for each mixed state there exists a Schmidt canonical (super state vector) expansion in terms of Hermitian operators, and that it can be the continuation of the mentioned biorthogonal mixing due to strong twins. The case of weak twins and nonhermitian Schmidt canonical expansion is also investigated. A necessary and sufficient condition for the existence of nontrivial twins for separable states is derived.
Opposite-subsystem twin events and twin observables, studied previously in the context of distant correlations, are first generalized to pure states of not-necessarily-composite systems, and afterwards they are further generalized to delayed twins that are due to unitary evolution of the quantum system. The versatile aspects of delayed twin observables are studied in terms of necessary and sufficient conditions to make possible various applications. Three of these are sketched: Preparation of some quantum experiments, easy solution of a puzzle in an important Scully et al. real experiment, and, finally, it is shown that exact measurement in quantum mechanics is an example of opposite-subsystem delayed twins in bipartite pure states.
The study of mixed-state quantum correlations in terms of opposite-subsystem observables - the measurement of one of which amounts to the same as that of the other - of so-called twins is continued. Twin events that imply biorthogonal mixing of states, called `strong twin events', are studied. It is shown that for each mixed state there exists a Schmidt (super-state-vector) decomposition in terms of Hermitian operators, and that it can be the continuation of the above-mentioned biorthogonal mixing due to strong twins. The case of weak twins and non-Hermitian Schmidt decomposition is also investigated. For separable states a necessary and sufficient condition for the existence of nontrivial twins is derived. Utilization of the Hermitian Schmidt decomposition for finding all twins is illustrated in full detail for the case of the two spin-half-particle states with maximally disordered subsystems (mixtures of Bell states). It is shown that only rank-two mixtures have nontrivial twins.