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Title: Local unitary equivalence of quantum states and simultaneous orthogonal equivalence

Abstract

The correspondence between local unitary equivalence of bipartite quantum states and simultaneous orthogonal equivalence is thoroughly investigated and strengthened. It is proved that local unitary equivalence can be studied through simultaneous similarity under projective orthogonal transformations, and four parametrization independent algorithms are proposed to judge when two density matrices on ℂ{sup d{sub 1}} ⊗ ℂ{sup d{sub 2}} are locally unitary equivalent in connection with trace identities, Kronecker pencils, Albert determinants and Smith normal forms.

Authors:
;  [1];  [2]
  1. Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695 (United States)
  2. College of Applied Science, Beijing University of Technology, Beijing 100124 (China)
Publication Date:
OSTI Identifier:
22596847
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 57; Journal Issue: 6; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; DENSITY MATRIX; ORTHOGONAL TRANSFORMATIONS; QUANTUM STATES

Citation Formats

Jing, Naihuan, E-mail: jing@ncsu.edu, Yang, Min, and Zhao, Hui, E-mail: zhaohui@bjut.edu.cn. Local unitary equivalence of quantum states and simultaneous orthogonal equivalence. United States: N. p., 2016. Web. doi:10.1063/1.4954230.
Jing, Naihuan, E-mail: jing@ncsu.edu, Yang, Min, & Zhao, Hui, E-mail: zhaohui@bjut.edu.cn. Local unitary equivalence of quantum states and simultaneous orthogonal equivalence. United States. doi:10.1063/1.4954230.
Jing, Naihuan, E-mail: jing@ncsu.edu, Yang, Min, and Zhao, Hui, E-mail: zhaohui@bjut.edu.cn. 2016. "Local unitary equivalence of quantum states and simultaneous orthogonal equivalence". United States. doi:10.1063/1.4954230.
@article{osti_22596847,
title = {Local unitary equivalence of quantum states and simultaneous orthogonal equivalence},
author = {Jing, Naihuan, E-mail: jing@ncsu.edu and Yang, Min and Zhao, Hui, E-mail: zhaohui@bjut.edu.cn},
abstractNote = {The correspondence between local unitary equivalence of bipartite quantum states and simultaneous orthogonal equivalence is thoroughly investigated and strengthened. It is proved that local unitary equivalence can be studied through simultaneous similarity under projective orthogonal transformations, and four parametrization independent algorithms are proposed to judge when two density matrices on ℂ{sup d{sub 1}} ⊗ ℂ{sup d{sub 2}} are locally unitary equivalent in connection with trace identities, Kronecker pencils, Albert determinants and Smith normal forms.},
doi = {10.1063/1.4954230},
journal = {Journal of Mathematical Physics},
number = 6,
volume = 57,
place = {United States},
year = 2016,
month = 6
}
  • We study the relation between local unitary (LU) equivalence and local Clifford (LC) equivalence of stabilizer states. We introduce a large subclass of stabilizer states, such that every two LU equivalent states in this class are necessarily LC equivalent. Together with earlier results, this shows that LC, LU, and stochastic local operation with classical communication equivalence are the same notions for this class of stabilizer states. Moreover, recognizing whether two given stabilizer states in the present subclass are locally equivalent only requires a polynomial number of operations in the number of qubits.
  • The equivalence of stabilizer states under local transformations is of fundamental interest in understanding properties and uses of entanglement. Two stabilizer states are equivalent under the usual stochastic local operations and classical communication criterion if and only if they are equivalent under local unitary (LU) operations. More surprisingly, under certain conditions, two LU-equivalent stabilizer states are also equivalent under local Clifford (LC) operations, as was shown by Van den Nest et al. [Phys. Rev. A 71, 062323 (2005)]. Here, we broaden the class of stabilizer states for which LU equivalence implies LC equivalence (LU<==>LC) to include all stabilizer states representedmore » by graphs with cycles of length neither 3 nor 4. To compare our result with Van den Nest et al.'s, we show that any stabilizer state of distance {delta}=2 is beyond their criterion. We then further prove that LU<==>LC holds for a more general class of stabilizer states of {delta}=2. We also explicitly construct graphs representing {delta}>2 stabilizer states which are beyond their criterion: we identify all 58 graphs with up to 11 vertices and construct graphs with 2{sup m}-1 (m{>=}4) vertices using quantum error-correcting codes which have non-Clifford transversal gates.« less
  • The necessary and sufficient conditions for the equivalence of arbitrary n-qubit pure quantum states under local unitary (LU) operations derived in [B. Kraus, Phys. Rev. Lett. 104, 020504 (2010)] are used to determine the different LU-equivalence classes of up to five-qubit states. Due to this classification new parameters characterizing multipartite entanglement are found and their physical interpretation is given. Moreover, the method is used to derive examples of two n-qubit states (with n>2 arbitrary) which have the properties that all the entropies of any subsystem coincide; however, the states are neither LU equivalent nor can be mapped into each othermore » by general local operations and classical communication.« less
  • Given L-qubit states with the fixed spectra of reduced one-qubit density matrices, we find a formula for the minimal number of invariant polynomials needed for solving local unitary (LU) equivalence problem, that is, problem of deciding if two states can be connected by local unitary operations. Interestingly, this number is not the same for every collection of the spectra. Some spectra require less polynomials to solve LU equivalence problem than others. The result is obtained using geometric methods, i.e., by calculating the dimensions of reduced spaces, stemming from the symplectic reduction procedure.
  • We investigate the possibility of distinguishing a set of mutually orthogonal multipartite quantum states by local operations and classical communication (LOCC). We connect this problem with generators of SU(N) and present a condition that is necessary for a set of orthogonal states to be locally distinguishable. We show that even in multipartite cases there exists a systematic way to check whether the presented condition is satisfied for a given set of orthogonal states. Based on the proposed checking method, we find that LOCC cannot distinguish three mutually orthogonal states in which two of them are Greenberger-Horne-Zeilinger-like states.