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arXiv:quant-ph/0306144 · Journal of Physics A Mathematical and General

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

Jon E Tyson

Published Sep 15, 2003• 41 citations
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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 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).

DOI
10.1088/0305-4470/36/39/309

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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 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)].
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].
In this paper, we study the distinguishability of multipartite quantum states by separable operations. We first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations. An analytical version of this condition is derived for the case of(D-1) pure states, whereDis the total dimension of the state space under consideration. A number of interesting consequences of this result are then carefully investigated. Remarkably, we show there exists a large class of 2 otimes 2 separable operations not being realizable by local operations and classical communication. Before our work, only a class of 3 otimes 3 nonlocal separable operations was known [Bennett , Phys. Rev. A 59, 1070 (1999)]. We also show that any basis of the orthogonal complement of a multipartite pure state is indistinguishable by separable operations if and only if this state cannot be a superposition of one or two orthogonal product states, i.e., has an orthogonal Schmidt number not less than three, thus generalize the recent work about indistinguishable bipartite subspaces [Watrous, Phys. Rev. Lett. 95, 080505 (2005)]. Notably, we obtain an explicit construction of indistinguishable subspaces of dimension 7 (or 6) by considering a composite quantum system consisting of two qutrits (resp., three qubits), which is slightly better than the previously known indistinguishable bipartite subspace with dimension 8.
An elaborated review with proofs of Schmidt canonical decomposition of any bipartite state vector is approached through general subsystem basis expansion. The upgraded forms of Schmidt decomposition in terms of correlation operator and twin observables are presented in detail. The discussion is extended to distant measurement, Einstein–Podolsky–Rosen states and Schrödinger's steering. All claims and proofs are given in standard form unlike in the previous articles of the author where all results were obtained utilizing the very rarely used antilinear Hilbert–Schmidt maps of one subsystem state space into the other. For practical reasons the formalism of partial traces with their rules and reduced density operators together with correlation operator are used. Quanta 2018; 7: 19–39.
The Schmidt number quantifies the dimensionality of entanglement in bipartite quantum states. We investigate when the spectrum of a state on Cd⊗Cd\mathbb{C}^d \otimes \mathbb{C}^dCd⊗Cd alone guarantees that its Schmidt number is not maximal. By deriving spectral bounds for (d−1)(d-1)(d−1)-block positive operators, we prove that LSp⊆ASNd−1\mathrm{LS}_p\subseteq \mathrm{ASN}_{d-1}LSp​⊆ASNd−1​ for p≤⌊d2/2⌋p\leq \lfloor d^2/2\rfloorp≤⌊d2/2⌋, where LSp\mathrm{LS}_pLSp​ denotes the set of states whose largest eigenvalue does not exceed the sum of their ppp smallest eigenvalues, and ASNd−1\mathrm{ASN}_{d-1}ASNd−1​ denotes the set of states whose Schmidt number remains at most d−1d-1d−1 under all global unitaries. This also yields a simple sufficient condition involving only the largest eigenvalue. As an application, we show that states satisfying the reduction criterion under arbitrary global unitaries belong to ASNd−1\mathrm{ASN}_{d-1}ASNd−1​ in every dimension. This subsumes the earlier result that absolutely positive partial transpose states belong to ASNd−1\mathrm{ASN}_{d-1}ASNd−1​. The global unitary requirement is nevertheless essential: for every d≥3d\geq3d≥3, we construct states of maximal Schmidt number satisfying the reduction criterion on both subsystems.