On the States Having Pure-State Restrictions for a Pair of Regions with a Non-Trivial Intersection ( The Proceedings of the Second Sendai Workshop on Quantum Probability and Quantum Information)

On Energy-Momentum Spectrum of Stationary States with Nonvanishing Current on 1-d Lattice Systems ( The Proceedings of the Second Sendai Workshop on Quantum Probability and Quantum Information)

Takayuki Miyadera

On one-dimensional two-way infinite quantum lattice system, a property of translationally invariant stationary states with nonvanishing current expectation is investigated. We consider GNS representation with respect to such a state, on which we have a group of space-time translation unitary operators. We show that spectrum of the unitary operators, energy-momentum spectrum with respect to the state, has a singularity at the origin.

Mar 1, 20040 citations

On the States Having Pure-State Restrictions for a Pair of Regions with a Non-Trivial Intersection ( The Proceedings of the Second Sendai Workshop on Quantum Probability and Quantum Information)

Hajime Moriya

Published Mar 1, 2004• 0 citations

OpenAlex

Abstract

Related papers

On the States Having Pure-State Restrictions for a Pair of Regions with a Non-Trivial Intersection

Hajime Moriya

For given two regions of the lattice, assume that our state has a pure-state restriction for each of them. We discuss whether its restrictions to the intersection and the union regions are pure states or not. It is immediate to see that the answer is affirmative for any tensor-product systems. (We, however, include its proof for the completeness sake.) In this note, we consider mostly CAR systems. We show that the assertion above holds for any such a state of finite-dimensional CAR systems. For infinite-dimensional CAR systems, assuming additionally the product properties between each of the regions and its complement region on the states satisfying the standing assumption, we can show the purity on the intersection region for them. However, the latter part of the assertion will not always hold for the infinite-dimensional CAR systems unlike for the finite-dimensional CAR or any dimensional tensor-product systems. We establish the criterions for the purity and the non-purity on the union region for those states of the infinite-dimensional CAR systems.For given two regions of the lattice, assume that our state has a pure-state restriction for each of them. We discuss whether its restrictions to the intersection and the union regions are pure states or not. It is immediate to see that the answer is affirmative for any tensor-product systems. (We, however, include its proof for the completeness sake.) In this note, we consider mostly CAR systems. We show that the assertion above holds for any such a state of finite-dimensional CAR systems. For infinite-dimensional CAR systems, assuming additionally the product properties between each of the regions and its complement region on the states satisfying the standing assumption, we can show the purity on the intersection region for them. However, the latter part of the assertion will not always hold for the infinite-dimensional CAR systems unlike for the finite-dimensional CAR or any dimensional tensor-product systems. We establish the criterions for the purity and the non-purity on the union region for those states of the infinite-dimensional CAR systems.

Interdisciplinary Information SciencesJan 1, 20040 citations

ON THE SEPARABILITY OF PURE STATES

Horia Scutaru

It is well know that the problem of the description of separable states of finite dimensional quantum system is central for the quantum information theory. The aim of the present paper is to point out that the conjecture of An Min Wang [1] concerning the separability of pure states is valid for pure states of bipartite quantum systems. A general condition for separability of a quantum state could be in principle obtained from the measure of entanglement. For a bipartite pure state the natural measure of entanglement is given by the von Neumann entropy of the reduced density matrices. Then, a bipartite state is separable if and only if this measure of entanglement is equal to zero, i.e., if and only if the reduced states are pure states. The purity of any state, described by a density matrix, can be verified directly using the Bloch vector, associated with that state. One of the restrictions is on the value of the squared Euclidean norm of the Bloch vector. Exactly on this restriction is based the An Min Wang criterion for the separability of a bipartite quantum state. The other restrictions concern the symmetric product of the Bloch vector with itself. We shall prove that the equations of the second set of restrictions follows from the An Min Wang restrictions and from the equations fulfilled by the Fano parameters of any pure bipartite state. In order to prove the An Min Wang conjecture we shall use three different parametrizations for the pure states of bipartite quantum systems: the generalized Bloch vector parametrization [2-10]; the Fano parametrization [3-10]; the Schmidt parametrization [11,12], and the relations between them.It is well know that the problem of the description of separable states of finite dimensional quantum system is central for the quantum information theory. The aim of the present paper is to point out that the conjecture of An Min Wang [1] concerning the separability of pure states is valid for pure states of bipartite quantum systems. A general condition for separability of a quantum state could be in principle obtained from the measure of entanglement. For a bipartite pure state the natural measure of entanglement is given by the von Neumann entropy of the reduced density matrices. Then, a bipartite state is separable if and only if this measure of entanglement is equal to zero, i.e., if and only if the reduced states are pure states. The purity of any state, described by a density matrix, can be verified directly using the Bloch vector, associated with that state. One of the restrictions is on the value of the squared Euclidean norm of the Bloch vector. Exactly on this restriction is based the An Min Wang criterion for the separability of a bipartite quantum state. The other restrictions concern the symmetric product of the Bloch vector with itself. We shall prove that the equations of the second set of restrictions follows from the An Min Wang restrictions and from the equations fulfilled by the Fano parameters of any pure bipartite state. In order to prove the An Min Wang conjecture we shall use three different parametrizations for the pure states of bipartite quantum systems: the generalized Bloch vector parametrization [2-10]; the Fano parametrization [3-10]; the Schmidt parametrization [11,12], and the relations between them.

Jan 1, 20030 citations

Restrictions of holomorphic sections to products

Manimugdha Saikia

We associate quantum states with subsets of a product of two compact connected Kähler manifolds $M_1$ and $M_2$ . To associate the quantum state with the subset, we use the map that restricts holomorphic sections of the quantum line bundle over the product of the two Kähler manifolds to the subset. We present a description of the kernel of this restriction map when the subset is a finite union of products. This in turn shows that the quantum states associated with the finite union of products are separable. Finally, for every pure state and certain mixed state, we construct subsets of $M_1\times M_2$ such that the states associated with these subsets are the original states, to begin with.We associate quantum states with subsets of a product of two compact connected Kähler manifolds $M_1$ and $M_2$. To associate the quantum state with the subset, we use the map that restricts holomorphic sections of the quantum line bundle over the product of the two Kähler manifolds to the subset. We present a description of the kernel of this restriction map when the subset is a finite union of products. This in turn shows that the quantum states associated with the finite union of products are separable. Finally, for every pure state and certain mixed state, we construct subsets of $M_1\times M_2$ such that the states associated with these subsets are the original states, to begin with.

arXiv:2403.17435Mar 26, 20240 citations

Antidistinguishability of pure quantum states

Teiko Heinosaari, Oskari Kerppo

Abstract The Pusey–Barrett–Rudolph theorem has recently provoked a lot of discussion regarding the reality of the quantum state. In this article we focus on a property called antidistinguishability, which is a main component in constructing the proof for the PBR theorem. In particular we study algebraic conditions for a set of pure quantum states to be antidistinguishable, and a novel sufficient condition is presented. We also discuss a more general criterion which can be used to show that the sufficient condition is not necessary. Lastly, we consider how many quantum states needs to be added into a set of pure quantum states in order to make the set antidistinguishable. It is shown that in the case of qubit states the answer is one, while in the general but finite dimensional case the answer is at most n , where n is the size of the original set.Abstract The Pusey–Barrett–Rudolph theorem has recently provoked a lot of discussion regarding the reality of the quantum state. In this article we focus on a property called antidistinguishability, which is a main component in constructing the proof for the PBR theorem. In particular we study algebraic conditions for a set of pure quantum states to be antidistinguishable, and a novel sufficient condition is presented. We also discuss a more general criterion which can be used to show that the sufficient condition is not necessary. Lastly, we consider how many quantum states needs to be added into a set of pure quantum states in order to make the set antidistinguishable. It is shown that in the case of qubit states the answer is one, while in the general but finite dimensional case the answer is at most n , where n is the size of the original set.

arXiv:1804.10457Journal of Physics A Mathematical and TheoreticalJul 9, 201829 citations

SEPARABILITY OF PURE N-QUBIT STATES: TWO CHARACTERIZATIONS

Philippe Jorrand, Mehdi Mhalla

Given a pure state |ψN>∈ℋN of a quantum system composed of n qubits, where ℋN is the Hilbert space of dimension N=2n, this paper answers two questions: what conditions should the amplitudes in |ψN> satisfy for this state to be separable (i) into a tensor product of n qubit states |ψ2>0⊗ |ψ2>1 ⊗⋯⊗ |ψ2>n-1, and (ii), into a tensor product of two subsystems states |ψP> ⊗ |ψQ> with P=2p and Q=2q such that p+q=n? For both questions, necessary and sufficient conditions are proved, thus characterizing at the same time families of separable and entangled states of n qubit systems. A number of more refined questions about separability in n qubit systems can be studied on the basis of these results.Given a pure state |ψN>∈ℋN of a quantum system composed of n qubits, where ℋN is the Hilbert space of dimension N=2n, this paper answers two questions: what conditions should the amplitudes in |ψN> satisfy for this state to be separable (i) into a tensor product of n qubit states |ψ2>0⊗ |ψ2>1 ⊗⋯⊗ |ψ2>n-1, and (ii), into a tensor product of two subsystems states |ψP> ⊗ |ψQ> with P=2p and Q=2q such that p+q=n? For both questions, necessary and sufficient conditions are proved, thus characterizing at the same time families of separable and entangled states of n qubit systems. A number of more refined questions about separability in n qubit systems can be studied on the basis of these results.

arXiv:quant-ph/0209125International Journal of Foundations of Computer ScienceOct 1, 200310 citations

On Energy-Momentum Spectrum of Stationary States with Nonvanishing Current on 1-d Lattice Systems ( The Proceedings of the Second Sendai Workshop on Quantum Probability and Quantum Information)

Takayuki Miyadera

Mar 1, 20040 citations

We associate quantum states with subsets of a product of two compact connected Kähler manifolds

M_1

and

M_2

. To associate the quantum state with the subset, we use the map that restricts holomorphic sections of the quantum line bundle over the product of the two Kähler manifolds to the subset. We present a description of the kernel of this restriction map when the subset is a finite union of products. This in turn shows that the quantum states associated with the finite union of products are separable. Finally, for every pure state and certain mixed state, we construct subsets of

M_1\times M_2

such that the states associated with these subsets are the original states, to begin with.