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Hermitian Schmidt Decomposition and Twin Observables of Bipartite Mixed States

Fedor Herbut

Published May 29, 2003• 7 citations
OpenAlex

Abstract

It is shown that for each mixed state there exists a Schmidt (super state vector) decomposition in terms of Hermitian operators. Its utilization for finding all twins is illustrated in full detail in the case of the two spin-one-half-particle states with maximally disordered subsystems (mixtures of Bell 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

On twin observables in entangled mixed states

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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.

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Existence of the Schmidt decomposition for tripartite systems

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For any bipartite quantum system the Schmidt decomposition allows us to express the state vector in terms of a single sum instead of double sums. We show the existence of the Schmidt decomposition for tripartite system under certain condition. If the partial inner product of a basis (belonging to a Hilbert space of smaller dimension) with the state of the composite system gives a disentangled basis, then the Schmidt decomposition for a tripartite system exists. In this case the reduced density matrix of each of the subsystem has equal spectrum in the Schmidt basis.

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Finding a maximally correlated state: Simultaneous Schmidt decomposition of bipartite pure states

Tohya Hiroshima, Masahito Hayashi

We consider a bipartite mixed state of the form $\ensuremath{\rho}={\ensuremath{\sum}}_{\ensuremath{\alpha},\ensuremath{\beta}=1}^{l}{a}_{\ensuremath{\alpha}\ensuremath{\beta}}\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩⟨{\ensuremath{\psi}}_{\ensuremath{\beta}}⟩$, where $\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩$ are normalized bipartite state vectors, and matrix $({a}_{\ensuremath{\alpha}\ensuremath{\beta}})$ is positive semidefinite. We provide a necessary and sufficient condition for the state $\ensuremath{\rho}$ taking the form of maximally correlated states by a local unitary transformation. More precisely, we give a criterion for simultaneous Schmidt decomposability of $\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩$ for $\ensuremath{\alpha}=1,2,\dots{},l$. Using this criterion, we can judge completely whether or not the state $\ensuremath{\rho}$ is equivalent to the maximally correlated state, in which the distillable entanglement is given by a simple formula. For generalized Bell states, this criterion is written as a simple algebraic relation between indices of the states. We also discuss the local distinguishability of the generalized Bell states that are simultaneously Schmidt decomposable.

arXiv:quant-ph/0405107•Physical Review A•Sep 22, 2004•30 citations
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.
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.
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.
An elaborated review with proofs of Schmidt canonical decomposition of any bipartite state vector is approached through general subsystem basis expansion. The upgraded forms of Schmidt decomposition in terms of correlation operator and twin observables are presented in detail. The discussion is extended to distant measurement, Einstein–Podolsky–Rosen states and Schrödinger's steering. All claims and proofs are given in standard form unlike in the previous articles of the author where all results were obtained utilizing the very rarely used antilinear Hilbert–Schmidt maps of one subsystem state space into the other. For practical reasons the formalism of partial traces with their rules and reduced density operators together with correlation operator are used. Quanta 2018; 7: 19–39.
For any bipartite quantum system the Schmidt decomposition allows us to express the state vector in terms of a single sum instead of double sums. We show the existence of the Schmidt decomposition for tripartite system under certain condition. If the partial inner product of a basis (belonging to a Hilbert space of smaller dimension) with the state of the composite system gives a disentangled basis, then the Schmidt decomposition for a tripartite system exists. In this case the reduced density matrix of each of the subsystem has equal spectrum in the Schmidt basis.
We consider a bipartite mixed state of the form $\ensuremath{\rho}={\ensuremath{\sum}}_{\ensuremath{\alpha},\ensuremath{\beta}=1}^{l}{a}_{\ensuremath{\alpha}\ensuremath{\beta}}\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩⟨{\ensuremath{\psi}}_{\ensuremath{\beta}}⟩$, where $\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩$ are normalized bipartite state vectors, and matrix $({a}_{\ensuremath{\alpha}\ensuremath{\beta}})$ is positive semidefinite. We provide a necessary and sufficient condition for the state $\ensuremath{\rho}$ taking the form of maximally correlated states by a local unitary transformation. More precisely, we give a criterion for simultaneous Schmidt decomposability of $\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩$ for $\ensuremath{\alpha}=1,2,\dots{},l$. Using this criterion, we can judge completely whether or not the state $\ensuremath{\rho}$ is equivalent to the maximally correlated state, in which the distillable entanglement is given by a simple formula. For generalized Bell states, this criterion is written as a simple algebraic relation between indices of the states. We also discuss the local distinguishability of the generalized Bell states that are simultaneously Schmidt decomposable.