Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Bulletin of the American Physical Society

Relation of operator Schmidt decomposition and CNOT complexity

Mark W. Coffey, Ron Deiotte

Published Mar 19, 2009• 0 citations
OpenAlex

Abstract

We consider two-qubit operators and provide a correspondence between their Schmidt number and controlled-NOT (CNOT) complexity, where the CNOT complexity is up to local unitary operations. The results are obtained by complementary means, and a number of examples are given.

Related papers

Classification of nonlocal two-qubit gates using Schmidt number

S. Balakrishnan, Leona J. Felicia, R. Sankaranarayanan

It is known from Schmidt decomposition that Schmidt number of nonlocal two-qubit quantum gates is 2 or 4. We identify conditions on geometrical points of a gate to have Schmidt number 2. A simple analysis reveals that Schmidt number 2 corresponds to controlled unitary gates with CNOT being the only perfect entangler. Further, it is shown that Schmidt strength and entangling power are maximum only for CNOT in the controlled unitary family.

arXiv:0907.5519•Jul 31, 2009•0 citations

Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries

Jon E Tyson

The operator-Schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations. We construct a family of unitary operators on n ⊗ n whose operator-Schmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on 3 ⊗ 3 with operator-Schmidt number S for every S ∊ {1, ..., 9}. This corollary was unexpected, since it contradicted reasonable conjectures of Nielsen et al (2003 Phys. Rev. A 67 052301) based on intuition from a striking result in the two-qubit case. By the results of Dür et al (2002 Phys. Rev. Lett. 89 057901), who also considered the two-qubit case, our result implies that there are nine equivalence classes of unitaries on 3 ⊗ 3 which are probabilistically interconvertible by (stochastic) local operations and classical communication. As another corollary, a prescription is produced for constructing maximally-entangled unitaries from biunimodular functions. Reversing tact, we state a generalized operator-Schmidt decomposition of the quantum Fourier transform considered as an operator M 1 ⊗ M 2 → N 1 ⊗ N 2 , with M 1 M 2 = N 1 N 2 . This decomposition shows (by Nielsen's bound) that the communication cost of the QFT remains maximal when a net transfer of qudits is permitted. In an appendix, a canonical procedure is given for removing basis-dependence for results and proofs depending on the 'magic basis' introduced in S Hill and W Wootters (1997 Entanglement of a pair of quantum bits Phys Rev. Lett. 78 5022–5).

arXiv:quant-ph/0306144•Journal of Physics A Mathematical and General•Sep 15, 2003•41 citations

Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries

Tyson, J E

The operator-Schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations. We construct a family of unitary operators on C^n tensor C^n whose operator-Schmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on C^3 tensor C^3 with operator-Schmidt number S for every S in {1,...,9}. This corollary was unexpected, since it contradicted reasonable conjectures of Nielsen et al [Phys. Rev. A 67 (2003) 052301] based on intuition from a striking result in the two-qubit case. By the results of Dur, Vidal, and Cirac [Phys. Rev. Lett. 89 (2002) 057901 quant-ph/0112124], who also considered the two-qubit case, our result implies that there are nine equivalence classes of unitaries on C^3 tensor C^3 which are probabilistically interconvertible by (stochastic) local operations and classical communication. As another corollary, a prescription is produced for constructing maximally-entangled operators from biunimodular functions. Reversing tact, we state a generalized operator-Schmidt decomposition of the quantum Fourier transform considered as an operator C^M_1 tensor C^M_2 --> C^N_1 tensor C^N_2, with M_1 x M_2 = N_1 x N_2. This decomposition shows (by Nielsen's bound) that the communication cost of the QFT remains maximal when a net transfer of qudits is permitted. In an appendix, a canonical procedure is given for removing basis-dependence for results and proofs depending on the "magic basis" introduced in [S. Hill and W. Wootters, "Entanglement of a pair of quantum bits," Phys Rev. Lett 78 (1997) 5022-5025, quant-ph/9703041 (and quant-ph/9709029)].

•Jun 22, 2003•17 citations

Relation of operator Schmidt decomposition and CNOT complexity

Mark W. Coffey, Ron Deiotte

Quantum Information Processing•Jun 1, 2008•2 citations

Comment on “Universal quantum circuit for two-qubit transformations with three controlled-NOT gates” and “Recognizing small-circuit structure in two-qubit operators”

Mark W. Coffey, Ron Deiotte, Torey Semi

Recent papers in Physical Review A have considered small quantum circuit decompositions in terms of a minimal number of controlled-NOT (CNOT) gates. Specifically, we point out errors in the papers by Vidal and Dawson [Phys. Rev. A 69, 010301(R) (2004)] and by Shende et al. [Phys. Rev. A 70, 012310 (2004)].

Physical Review A•Jun 26, 2008•20 citations

Decomposing the SWAP quantum gate

Yorick Hardy, W.‐H. Steeb

Quantum gates are represented by unitary operators on a Hilbert space. We consider the unitary operators on and describe the component operators on each Hilbert space in the product space. From these criteria we derive a decomposition for a specific class of unitary operators such that each operator in the product has Schmidt rank bounded by 2. This decomposition directly yields the XOR (controlled NOT) implementation of the SWAP operation.

Journal of Physics A Mathematical and General•Jan 24, 2006•9 citations
The operator-Schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations. We construct a family of unitary operators on n ⊗ n whose operator-Schmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on 3 ⊗ 3 with operator-Schmidt number S for every S ∊ {1, ..., 9}. This corollary was unexpected, since it contradicted reasonable conjectures of Nielsen et al (2003 Phys. Rev. A 67 052301) based on intuition from a striking result in the two-qubit case. By the results of Dür et al (2002 Phys. Rev. Lett. 89 057901), who also considered the two-qubit case, our result implies that there are nine equivalence classes of unitaries on 3 ⊗ 3 which are probabilistically interconvertible by (stochastic) local operations and classical communication. As another corollary, a prescription is produced for constructing maximally-entangled unitaries from biunimodular functions. Reversing tact, we state a generalized operator-Schmidt decomposition of the quantum Fourier transform considered as an operator M 1 ⊗ M 2 → N 1 ⊗ N 2 , with M 1 M 2 = N 1 N 2 . This decomposition shows (by Nielsen's bound) that the communication cost of the QFT remains maximal when a net transfer of qudits is permitted. In an appendix, a canonical procedure is given for removing basis-dependence for results and proofs depending on the 'magic basis' introduced in S Hill and W Wootters (1997 Entanglement of a pair of quantum bits Phys Rev. Lett. 78 5022–5).
The operator-Schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations. We construct a family of unitary operators on C^n tensor C^n whose operator-Schmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on C^3 tensor C^3 with operator-Schmidt number S for every S in {1,...,9}. This corollary was unexpected, since it contradicted reasonable conjectures of Nielsen et al [Phys. Rev. A 67 (2003) 052301] based on intuition from a striking result in the two-qubit case. By the results of Dur, Vidal, and Cirac [Phys. Rev. Lett. 89 (2002) 057901 quant-ph/0112124], who also considered the two-qubit case, our result implies that there are nine equivalence classes of unitaries on C^3 tensor C^3 which are probabilistically interconvertible by (stochastic) local operations and classical communication. As another corollary, a prescription is produced for constructing maximally-entangled operators from biunimodular functions. Reversing tact, we state a generalized operator-Schmidt decomposition of the quantum Fourier transform considered as an operator C^M_1 tensor C^M_2 --> C^N_1 tensor C^N_2, with M_1 x M_2 = N_1 x N_2. This decomposition shows (by Nielsen's bound) that the communication cost of the QFT remains maximal when a net transfer of qudits is permitted. In an appendix, a canonical procedure is given for removing basis-dependence for results and proofs depending on the "magic basis" introduced in [S. Hill and W. Wootters, "Entanglement of a pair of quantum bits," Phys Rev. Lett 78 (1997) 5022-5025, quant-ph/9703041 (and quant-ph/9709029)].