Quantum Information and Computation

Deciding unitary equivalence between matrix polynomials and sets of bipartite quantum states

Eric Chitambar, Carl W. Miller, Yaoyun Shi

Published Sep 1, 2011• 0 citations

Publisher

Abstract

DOI: 10.26421/qic11.9-10-6

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Quantum Information and Computation

Deciding unitary equivalence between matrix polynomials and sets of bipartite quantum states

Eric Chitambar, Carl W. Miller, Yaoyun Shi

Published Sep 1, 2011• 0 citations

Publisher

Abstract

In this brief report, we consider the equivalence between two sets of

m+1

bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree

m

matrix polynomials are unitarily equivalent; i.e.

UA_iV^\dagger=B_i

for

0\leq i\leq m

where

U

and

V

are unitary and

(A_i, B_i)

are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices

U

and

V

DOI: 10.26421/qic11.9-10-6

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We present an algorithm to solve the Simultaneous Unitary Similarity(S.U.S) problem which is to check if there exists a Similarity transformation determined by a Unitary $U$ s.t $UA_lU^*=B_l$ , $l \in \{1,...,p\}$ , where $A_l$ and $B_l$ are $nxn$ complex matrices. We observe that the problem is simplest when $U$ is diagonal, where we see that the `paths' in the graph defined by non-zero elements of $A_l$ and $B_l$ determine the solution. Inspired by this we generalize this to the case when $U$ is block-diagonal to identify a form refered to as the `Solution-form' using `paths' determined by non-zero sub-matrices of $A_l,B_l$ which are non-zero multiples of Unitary. When not in Solution form we find an equivalent problem to solve by diagonalizing a Hermitian or a Normal matrix related to the sub-matrices. The problem is solved in a maximum of $n$ steps. The same idea can be extended to solve the Simultaneous Unitary Equivalence (S $.$ U $.$ Eq) problem where we solve for $U,V$ in $UA_lV^*=B_l$ , $A_l,B_l$ being $mxn$ Complex rectangular matrices. Here we work with the 'paths' in the related bi-graph to define the Solution-form. The algorithms have a complexity of $O(pn^4)$ . This work finds application in Quantum Evolution, Quantum gate design and Simulation. The salient features of each step of the algorithm can be retained as Canonical features to classify a given collection of complex matrices up to Unitary Similarity.We present an algorithm to solve the Simultaneous Unitary Similarity(S.U.S) problem which is to check if there exists a Similarity transformation determined by a Unitary $U$ s.t $UA_lU^*=B_l$, $l \in \{1,...,p\}$, where $A_l$ and $B_l$ are $nxn$ complex matrices. We observe that the problem is simplest when $U$ is diagonal, where we see that the `paths' in the graph defined by non-zero elements of $A_l$ and $B_l$ determine the solution. Inspired by this we generalize this to the case when $U$ is block-diagonal to identify a form refered to as the `Solution-form' using `paths' determined by non-zero sub-matrices of $A_l,B_l$ which are non-zero multiples of Unitary. When not in Solution form we find an equivalent problem to solve by diagonalizing a Hermitian or a Normal matrix related to the sub-matrices. The problem is solved in a maximum of $n$ steps. The same idea can be extended to solve the Simultaneous Unitary Equivalence (S$.$U$.$Eq) problem where we solve for $U,V$ in $UA_lV^*=B_l$, $A_l,B_l$ being $mxn$ Complex rectangular matrices. Here we work with the 'paths' in the related bi-graph to define the Solution-form. The algorithms have a complexity of $O(pn^4)$. This work finds application in Quantum Evolution, Quantum gate design and Simulation. The salient features of each step of the algorithm can be retained as Canonical features to classify a given collection of complex matrices up to Unitary Similarity.

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Qing Fa Zhou, Yi-Zheng Zhen, Xin-Yu Xu +6 more

Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. Verifying whether or not two quantum states are local unitary equivalent is a crucial problem, where only the case of multipartite pure states is solved. For mixed states, however, the verification of local unitary equivalence is still a challenging problem. In this paper, based on the coefficient matrices of generalized Bloch representations of quantum states, we find a variety of local unitary invariants for arbitrary-dimensional bipartite quantum states. These invariants are operational and can be used as necessary conditions for verifying the local unitary equivalence of two quantum states. Furthermore, we extend the construction to the arbitrary-dimensional multipartite case. We finally apply these invariants to estimate concurrence, a vital entanglement measure, showing the practicability of local unitary invariants in characterizing entanglement.Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. Verifying whether or not two quantum states are local unitary equivalent is a crucial problem, where only the case of multipartite pure states is solved. For mixed states, however, the verification of local unitary equivalence is still a challenging problem. In this paper, based on the coefficient matrices of generalized Bloch representations of quantum states, we find a variety of local unitary invariants for arbitrary-dimensional bipartite quantum states. These invariants are operational and can be used as necessary conditions for verifying the local unitary equivalence of two quantum states. Furthermore, we extend the construction to the arbitrary-dimensional multipartite case. We finally apply these invariants to estimate concurrence, a vital entanglement measure, showing the practicability of local unitary invariants in characterizing entanglement.

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Tomasz Maciążek, Michał Oszmaniec, Adam Sawicki

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.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.

arXiv:1305.3894Journal of Mathematical PhysicsSep 1, 201326 citations

In this brief report, we consider the equivalence between two sets of

m+1

bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree

m

matrix polynomials are unitarily equivalent; i.e.

UA_iV^\dagger=B_i

for

0\leq i\leq m

where

U

and

V

are unitary and

(A_i, B_i)

are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices

U

and

V

In this brief report, we consider the equivalence between two sets of

m+1

bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree

m

matrix polynomials are unitarily equivalent; i.e.

UA_iV^\dagger=B_i

for

0\leq i\leq m

where

U

and

V

are unitary and

(A_i, B_i)

are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices

U

and

V

We present an algorithm to solve the Simultaneous Unitary Similarity(S.U.S) problem which is to check if there exists a Similarity transformation determined by a Unitary

U

s.t

UA_lU^*=B_l

l \in \{1,...,p\}

, where

A_l

and

B_l

are

nxn

complex matrices. We observe that the problem is simplest when

U

is diagonal, where we see that the `paths' in the graph defined by non-zero elements of

A_l

and

B_l

determine the solution. Inspired by this we generalize this to the case when

U

is block-diagonal to identify a form refered to as the `Solution-form' using `paths' determined by non-zero sub-matrices of

A_l,B_l

which are non-zero multiples of Unitary. When not in Solution form we find an equivalent problem to solve by diagonalizing a Hermitian or a Normal matrix related to the sub-matrices. The problem is solved in a maximum of

n

steps. The same idea can be extended to solve the Simultaneous Unitary Equivalence (S

.

.

Eq) problem where we solve for

U,V

UA_lV^*=B_l

A_l,B_l

being

mxn

Complex rectangular matrices. Here we work with the 'paths' in the related bi-graph to define the Solution-form. The algorithms have a complexity of

O(pn^4)

. This work finds application in Quantum Evolution, Quantum gate design and Simulation. The salient features of each step of the algorithm can be retained as Canonical features to classify a given collection of complex matrices up to Unitary Similarity.