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

Abstract

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)].

More formal version of my talk at the Simons Conference on Quantum and Reversible Computation at Stony Brook May 31, 2003. The talk slides and audio are available at http://www.physics.sunysb.edu/itp/conf/simons-qcomputation.html. Fixed typos and minor cosmetics

References in corpus (8)

• Entanglement of Formation of an Arbitrary State of Two Qubits
• Entanglement of a Pair of Quantum Bits
• Time Optimal Control in Spin Systems
• Optimal Creation of Entanglement Using a Two--Qubit Gate
• Computing Local Invariants of Qubit Systems
• Quantum dynamics as a physical resource
• Bell Measurements and Observables
• Why Two Qubits Are Special

Cited by in corpus (32)

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• Decomposition of bipartite and multipartite unitary gates into the product of controlled unitary gates
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• Dual unitaries as maximizers of the distance to local product gates
• Quantum Tensor Product Decomposition from Choi State Tomography
• Dynamical Signatures of Liouvillian Flat Band
• Construction of perfect tensors using biunimodular vectors
• Conditions for uniqueness of product representations for separable quantum channels and separable quantum states
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