On the Schmidt-rank-three bipartite and multipartite unitary operator

Abstract

Unitary operations are physically implementable. We further the understanding of such operations by studying the possible forms of nonlocal unitary operators, which are bipartite or multipartite unitary operators that are not tensor product operators. They are of broad relevance in quantum information processing. We prove that any nonlocal unitary operator of Schmidt rank three on a $d_A\times d_B$ bipartite system is locally equivalent to a controlled unitary. This operator can be locally implemented assisted by a maximally entangled state of Schmidt rank $\min \{d_A^2,d_B\}$ when $d_A\le d_B$. We further show that any multipartite unitary operator $U$ of Schmidt rank three can be controlled by one system or collectively controlled by two systems, regardless of the number of systems of $U$. In the scenario of $n$-qubit, we construct non-controlled $U$ for any odd $n\ge 5$, and prove that $U$ is a controlled unitary for any even $n\ge 4$.

arXiv comment25 pages, 2 figures, almost published version

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Cited by in corpus (13)

• Decomposition of bipartite and multipartite unitary gates into the product of controlled unitary gates
• Entanglement cost and entangling power of bipartite unitary and permutation operators
• Mutually unbiased bases in dimension six containing a product-vector basis
• Entangling and assisted entangling power of bipartite unitary operations
• Product states and Schmidt rank of mutually unbiased bases in dimension six
• Orthogonal product bases of four qubits
• Restrictions on the Schmidt rank of bipartite unitary operators beyond dimension two
• Distributed Encoding and Decoding of Quantum Information over Networks
• Delocalized and Dynamical Catalytic Randomness and Information Flow
• Classification of Schmidt-rank-two multipartite unitary gates by singular number
• Constructing three-qubit unitary gates in terms of Schmidt rank and CNOT gates
• Synthesis and upper bound of Schmidt rank of the bipartite controlled-unitary gates
• Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three