We investigate the number of zero entries in a unitary matrix. We show that the sets of numbers of zero entries for n×n unitary and orthogonal matrices are the same. They are both the set {0,1,…,n2−n−4,n2−n−2,n2−n} for n>4. We explicitly construct examples of orthogonal matrices with the numbers in the set. We apply our results to construct a necessary condition by which a multipartite unitary operation is a product operation. The latter is a fundamental problem in quantum information. We also construct an n×n orthogonal matrix of Schmidt rank n2−1 with many zero entries, and it solves an open problem in Muller-Hermes and Nechita [Operator Schmidt ranks of bipartite unitary matrices. Linear Algebra Appl. 2018;557:174—187].We investigate the number of zero entries in a unitary matrix. We show that the sets of numbers of zero entries for n×n unitary and orthogonal matrices are the same. They are both the set {0,1,…,n2−n−4,n2−n−2,n2−n} for n>4. We explicitly construct examples of orthogonal matrices with the numbers in the set. We apply our results to construct a necessary condition by which a multipartite unitary operation is a product operation. The latter is a fundamental problem in quantum information. We also construct an n×n orthogonal matrix of Schmidt rank n2−1 with many zero entries, and it solves an open problem in Muller-Hermes and Nechita [Operator Schmidt ranks of bipartite unitary matrices. Linear Algebra Appl. 2018;557:174—187].

Linear and Multilinear AlgebraMay 5, 20203 citations

Inequalities for the Schmidt number of bipartite states

Daniel Cariello

arXiv:1902.11069Letters in Mathematical PhysicsNov 25, 201911 citations

Entanglement cost and entangling power of bipartite unitary operators of Schmidt rank three

Lin Chen, Li Zhi Yu

It is known that any bipartite unitary operator of Schmidt rank three is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank three can be implemented using LOCC and two ebits. We give two protocols for implementing bipartite permutation unitaries of any Schmidt rank $r$ , and showed that one of the protocol uses $O(r)$ ebits of entanglement and $O(r)$ bits of classical communication, while these two types of costs for the other protocol scale as $O(r\log r)$ but the actual values are smaller for all $r<1100$ . Based on this we obtain upper bounds of the number of nonlocal CNOT gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt rank two and three. We show that they are respectively $1$ ebit and some value between $\log_2 9 - 16/9$ and $\log_2 3$ ebits.It is known that any bipartite unitary operator of Schmidt rank three is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank three can be implemented using LOCC and two ebits. We give two protocols for implementing bipartite permutation unitaries of any Schmidt rank $r$, and showed that one of the protocol uses $O(r)$ ebits of entanglement and $O(r)$ bits of classical communication, while these two types of costs for the other protocol scale as $O(r\log r)$ but the actual values are smaller for all $r<1100$. Based on this we obtain upper bounds of the number of nonlocal CNOT gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt rank two and three. We show that they are respectively $1$ ebit and some value between $\log_2 9 - 16/9$ and $\log_2 3$ ebits.

Jul 19, 20150 citations

Entanglement cost and entangling power of bipartite unitary and permutation operators

Lin Chen, Li Zhi Yu

It is known that any bipartite unitary operator of Schmidt rank 3 is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank 3 can be implemented using LOCC and two ebits. We give two protocols for implementing bipartite permutation unitaries of any Schmidt rank $r$ and showed that one of the protocol uses $O(r)$ ebits of entanglement and $O(r)$ bits of classical communication, while these two types of costs for the other protocol scale as $O(rlogr)$ but the actual values are smaller for all $r<1100$ . Based on this we obtain upper bounds of the number of nonlocal controlled-not gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt ranks 2 and 3. We show that they are respectively 1 ebit and some value between ${log}_{2}9--16/9$ and ${log}_{2}3$ ebits.It is known that any bipartite unitary operator of Schmidt rank 3 is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank 3 can be implemented using LOCC and two ebits. We give two protocols for implementing bipartite permutation unitaries of any Schmidt rank $r$ and showed that one of the protocol uses $O(r)$ ebits of entanglement and $O(r)$ bits of classical communication, while these two types of costs for the other protocol scale as $O(rlogr)$ but the actual values are smaller for all $r<1100$. Based on this we obtain upper bounds of the number of nonlocal controlled-not gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt ranks 2 and 3. We show that they are respectively 1 ebit and some value between ${log}_{2}9--16/9$ and ${log}_{2}3$ ebits.

arXiv:1507.05260Physical Review AApr 19, 201619 citations

Local Unitary Invariants for Multipartite States

Ting-Gui Zhang, Ming‐Jing Zhao, Xianqing Li‐Jost +1 more

arXiv:1307.6659International Journal of Theoretical PhysicsApr 19, 20138 citations

r

, and showed that one of the protocol uses

O(r)

ebits of entanglement and

O(r)

bits of classical communication, while these two types of costs for the other protocol scale as

O(r\log r)

but the actual values are smaller for all

r<1100

. Based on this we obtain upper bounds of the number of nonlocal CNOT gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt rank two and three. We show that they are respectively

1

ebit and some value between

\log_2 9 - 16/9

and

\log_2 3

ebits.

r

and showed that one of the protocol uses

O(r)

ebits of entanglement and

O(r)

bits of classical communication, while these two types of costs for the other protocol scale as

O(rlogr)

but the actual values are smaller for all

r<1100

. Based on this we obtain upper bounds of the number of nonlocal controlled-not gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt ranks 2 and 3. We show that they are respectively 1 ebit and some value between

{log}_{2}9--16/9

and

{log}_{2}3

ebits.