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

Tyson, J E

Published Jun 22, 2003• 17 citations
OpenAlex

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)].

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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)].
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).
Operator-Schmidt decompositions of the quantum Fourier transform on ℂ N 1 ⊗ ℂ N 2 are computed for all N 1 , N 2 ≥ 2. The decomposition is shown to be completely degenerate when N 1 is a factor of N 2 and when N 1 > N 2 . The first known special case, N 1 = N 2 = 2 n , was computed by Nielsen in his study of the communication cost of computing the quantum Fourier transform of a collection of qubits equally distributed between two parties (M A Nielsen 1998 PhD Thesis University of New Mexico ch 6 Preprint quant-ph/0011036). More generally, the special case N 1 = 2 n 1 ⩽ 2 n 2 = N 2 was computed by Nielsen et al in their study of strength measures of quantum operations (M A Nielsen et al 2002 Preprint quant-ph/0208077 (2003 Phys. Rev. A at press)). Given the Schmidt decompositions presented here, it follows that in all cases the bipartite communication cost of exact computation of the quantum Fourier transform is maximal.
Entanglement between three or more parties exhibits a realm of properties unknown to two-party states. Bipartite states are easily classified using the Schmidt decomposition. The Schmidt coefficients of a bipartite pure state encompass all the nonlocal properties of the state and can be ``seen'' by looking at one party's density matrix only. Pure states of three and more parties, however, lack such a simple form. They have more invariants under local unitary transformations than any one party can ``see'' on their subsystem. These ``hidden nonlocalities'' will allow us to exhibit a class of multipartite states that cannot be distinguished from each other by any party. Generalizing a result of Bennett, Popescu, Rohrlich, Smolin, and Thapliyal, and using a recent result by Nielsen, we will show that these states cannot be transformed into each other by local actions and classical communication. Furthermore, we will use an orthogonal subset of such states to hint at applications to cryptography and illustrate an extension to quantum secret sharing [using recently suggested ((n,k))((n,k))((n,k))-threshold schemes].
The usefulness of entanglement as a resource in quantum technologies increases for larger systems, that is, if more particles or higher-dimensional quantum systems are considered. Yet, the interplay between dimensionality and multiparticle entanglement is not well understood. Only for two-particle systems an unambiguous and coherent notion of entanglement dimensionality, based on the Schmidt decomposition, is known. We introduce a concept to characterize the entanglement dimensionality of multiparticle states based on decompositions of pure states into superpositions of states without genuine multiparticle entanglement. We provide constructive methods to characterize the resulting partition rank for pure and mixed states. This allows the identification of novel maximally correlated states as well as a discrete classification of quantum states under stochastic local operations and classical communication. From a mathematical perspective, our approach can be formulated in terms of the slice rank and partition rank of tensors and our results allow to characterize these by connecting them to a generalized injective tensor norm.
Implementing nonlocal unitary operators is an important and hard question in quantum computing and cryptography. We show that any bipartite nonlocal unitary operator of Schmidt rank three on the $({d}_{A}\ifmmode\times\else\texttimes\fi{}{d}_{B})$-dimensional system is locally equivalent to a controlled unitary when dA{d}_{A}dA​ is at most three. This operator can be locally implemented assisted by a maximally entangled state of Schmidt rank r=mindA2,dBr=\mathrm{min}{{d}_{A}^{2},{d}_{B}}r=mindA2​,dB​. We further show that stochastic-equivalent nonlocal unitary operators are indeed locally equivalent, and propose a sufficient condition on which nonlocal and controlled unitary operators are locally equivalent. We also provide the solution to a special case of a conjecture on the ranks of multipartite quantum states.
The Schmidt decomposition is the go-to tool for measuring bipartite entanglement of pure quantum states. Similarly, it is possible to study the entangling features of a quantum operation using its operator-Schmidt or tensor-product decomposition. While quantum technological implementations of the former are thoroughly studied, entangling properties on the operator level are harder to extract in the quantum computational framework because of the exponential nature of sample complexity. Here, we present an algorithm for unbalanced partitions into a small subsystem and a large one (the environment) to compute the tensor-product decomposition of a unitary the effect of which on the small subsystem is captured in classical memory, while the effect on the environment is accessible as a quantum resource. This quantum algorithm may be used to make predictions about operator nonlocality and effective open quantum dynamics on a subsystem, as well as for finding low-rank approximations and low-depth compilations of quantum circuit unitaries. We demonstrate the method and its applications on a time-evolution unitary of an isotropic Heisenberg model in two dimensions. Published by the American Physical Society 2024