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arXiv:1301.7551 · Journal of Physics A Mathematical and Theoretical

Quantum measurements and maps preserving strict convex combinations and pure states

Lihua Yang, Jinchuan Hou

Published Apr 29, 2013• 1 citation
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Abstract

In this paper, a characterization of maps between quantum states that preserve pure states and strict convex combinations is obtained. Based on this characterization, a structure theorem for maps between multipartite quantum states that preserve separable pure states and strict convex combinations is established. Then, these results are applied to characterize injective (local) quantum measurements and answer some conjectures proposed in Hou and Liu (2012 J. Phys. A: Math. Theor. 45 205305).

DOI
10.1088/1751-8113/46/19/195304

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In this paper, a characterization of maps between quantum states that preserve pure states and strict convex combinations is obtained. Based on this characterization, a structure theorem for maps between multipartite quantum states that preserve separable pure states and strict convex combinations is established. Then, these results are applied to characterize injective (local) quantum measurements and answer some conjectures proposed in Hou and Liu (2012 J. Phys. A: Math. Theor. 45 205305).
A map ϕ between convex sets is said to be convex combination preserving if, for any elements x , y in its domain and any number 0 < t < 1, there is some s with 0 < s < 1, such that ϕ( tx + (1 − t ) y ) = s ϕ( x ) + (1 − s )ϕ( y ). Every quantum measurement described by a measurement operator M introduces a map (i.e. if ) that is convex combination preserving. In this paper, we give a characterization of convex combination preserving bijective maps on the set of all separable states in a bipartite quantum system H 1 ⊗ H 2 and show that, under a mild additional condition, such a map is a composition of an invertible local quantum measurement (i.e. the map of the form with S , T invertible) and some of the following maps: the transpose, the partial transpose and the swap.
Let H and K be the complex Hilbert spaces and mathcal SH) theconvex set of all quantum states (i.e., the positive operators withtrace one) on H. In this paper, the convex combination preservingmaps from mathcal SH) into mathcal SK)∪0and theirconnection to quantum measurements are discussed. It is shown that amap Φ:mathcal SH )→ mathcal SK) with 2≤ dimH≤∞ preserves pure states and strict convex combinations ifand only if one of the following holds: (1) there exists a purestate σ0 on K such that Φ(ρ)= σ0 for allρ∈ mathcal SH); (2) Φ is an injective selective quantummeasurement map up to the transpose, i.e., there exists an injective bounded linear operator M: H→ K such thatΦ(ρ)=fracMρ M^*Tr(Mρ M^*)
We use classical results from the theory of linear preserver problems to characterize operators that send the set of pure states with Schmidt rank no greater than k back into itself, extending known results characterizing operators that send separable pure states to separable pure states. We also provide a new proof of an analogous statement in the multipartite setting. We use these results to develop a bipartite version of a classical result about the structure of maps that preserve rank-1 operators and then characterize the isometries for two families of norms that have recently been studied in quantum information theory. We see, in particular, that for k ≥ 2 the operator norms induced by states with Schmidt rank k are invariant only under local unitaries, the swap operator and the transpose map. However, in the k = 1 case there is an additional isometry: the partial transpose map.
Abstract In the study of optical quantum information, quantum teleportation can be realized by preparing multiphoton entangled states. Entanglement is a special quantum correlation, and there is also a quantum correlation in separable quantum States, which comes from the measurement of quantum states. Convex combinations of arbitrary quantum states are still quantum states. From the spectral decomposition of a quantum state, an ensemble formed by a complete eigenvector system can be determined, which corresponds to an orthogonal projection operator family with rank 1, and the convex combination of the operator family will correspond to a quantum state. The characterization of quantum correlation is an important issue in the study of quantum information theory. A Quantum state that remains unchanged under local orthogonal projection is a classical correlation state, which is not affected by decoherence. Based on the standard orthogonal basis, this paper considers the correlation of the quantum states formed by the convex combination of the operator family. According to the separability and entanglement of the ground state and the mutual dissimilarity of the corresponding non-zero eigenvalues, it is concluded that the convex combination of the operator family formed by the separable standard orthogonal basis in the composite system must be a classical correlation state. When the entangled pure state is an eigenvector, it is a quantum correlation state if the corresponding eigenvalue is a non-zero single value. In the case of multiple roots, we can illustrate that both are possible. For the correlation of unitary evolution of quantum states under different standard orthogonal bases, the evolution law is revealed by considering important quantum gates.