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.

math.RACross-listedarXiv:2511.19439Nov 11, 20250 citations

A geometrical approach to quantum estimation theory

啓史松本

Jan 1, 19986 citations

Weak Schmidt decomposition and generalized Bell basis related to Hadamard matrices

Bobo Hua, Shao-Ming Fei, Jost, Juergen +1 more

We study the mathematical structures and relations among some quantities in the theory of quantum entanglement, such as separability, weak Schmidt decompositions, Hadamard matrices etc.. We provide an operational method to identify the Schmidt-correlated states by using weak Schmidt decomposition. We show that a mixed state is Schmidt-correlated if and only if its spectral decomposition consists of a set of pure eigenstates which can be simultaneously diagonalized in weak Schmidt decomposition, i.e. allowing for complex-valued diagonal entries. For such states, the separability is reduced to the orthogonality conditions of the vectors consisting of diagonal entries associated to the eigenstates, which is surprisingly related to the so-called complex Hadamard matrices. Using the Hadamard matrices, we provide a variety of generalized maximal entangled Bell bases.We study the mathematical structures and relations among some quantities in the theory of quantum entanglement, such as separability, weak Schmidt decompositions, Hadamard matrices etc.. We provide an operational method to identify the Schmidt-correlated states by using weak Schmidt decomposition. We show that a mixed state is Schmidt-correlated if and only if its spectral decomposition consists of a set of pure eigenstates which can be simultaneously diagonalized in weak Schmidt decomposition, i.e. allowing for complex-valued diagonal entries. For such states, the separability is reduced to the orthogonality conditions of the vectors consisting of diagonal entries associated to the eigenstates, which is surprisingly related to the so-called complex Hadamard matrices. Using the Hadamard matrices, we provide a variety of generalized maximal entangled Bell bases.

arXiv:1308.1505Aug 7, 20130 citations

A Geometrical Approach to Quantum Estimation Theory

Keiji Matsumoto

WORLD SCIENTIFIC eBooksFeb 1, 20056 citations

Generation of Matrices with Specified Eigenvalues and their Diagonalization

Habatwa Vincent Mweene

We present a prescription for forming matrices with specified eigenvalues and known eigenvectors. With this method, we can form Hermitian, anti-Hermitian, symmetric and general matrices with arbitrary eigenvalues. In addition we propose an algorithm for diagonalizing such matrices. The functions required for the realization of this are probability amplitudes connecting observables with discrete eigenvalue spectra and can be obtained from spin theory. For the example case of $5\times 5$ matrices, these functions are given.We present a prescription for forming matrices with specified eigenvalues and known eigenvectors. With this method, we can form Hermitian, anti-Hermitian, symmetric and general matrices with arbitrary eigenvalues. In addition we propose an algorithm for diagonalizing such matrices. The functions required for the realization of this are probability amplitudes connecting observables with discrete eigenvalue spectra and can be obtained from spin theory. For the example case of $5\times 5$ matrices, these functions are given.

arXiv:quant-ph/0506074Jun 9, 20050 citations

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.

5\times 5

matrices, these functions are given.