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arXiv:0807.2360 · Physical Review A

Separable operations on pure states

Vlad Gheorghiu, Robert B. Griffiths

Published Aug 29, 2008• 43 citations
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Abstract

We show that the possible ensembles produced when a separable operation acts on a single pure bipartite entangled state are completely characterized by a majorization condition, a collection of inequalities for Schmidt coefficients, which is identical to that already known for the particular case of local operations and classical communication (LOCC). As a consequence, various known results for LOCC, including some involving monotonicity of entanglement, can be extended to the class of all separable operations.

DOI
10.1103/physreva.78.020304

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We show that the possible ensembles produced when a separable operation acts on a single pure bipartite entangled state are completely characterized by a majorization condition, a collection of inequalities for Schmidt coefficients, which is identical to that already known for the particular case of local operations and classical communication (LOCC). As a consequence, various known results for LOCC, including some involving monotonicity of entanglement, can be extended to the class of all separable operations.
We study conditions for the deterministic transformation $\ensuremath{\mid}\ensuremath{\psi}⟩\ensuremath{\rightarrow}\ensuremath{\mid}\ensuremath{\phi}⟩$ of a bipartite entangled state by a separable operation. If the separable operation is a local operation with classical communication (LOCC), Nielsen's majorization theorem provides necessary and sufficient conditions. For the general case, we derive a necessary condition in terms of products of Schmidt coefficients, which is equivalent to the Nielsen condition when either of the two factor spaces is of dimension 2, but is otherwise weaker. One implication is that no separable operation can reverse a deterministic map produced by another separable operation, if one excludes the case where the Schmidt coefficients of $\ensuremath{\mid}\ensuremath{\psi}⟩$ are the same as those of $\ensuremath{\mid}\ensuremath{\phi}⟩$. The question of sufficient conditions in the general separable case remains open. When the Schmidt coefficients of $\ensuremath{\mid}\ensuremath{\psi}⟩$ are the same as those of $\ensuremath{\mid}\ensuremath{\phi}⟩$, we show that the Kraus operators of the separable transformation restricted to the supports of $\ensuremath{\mid}\ensuremath{\psi}⟩$ on the factor spaces are proportional to unitaries. When that proportionality holds and the factor spaces have equal dimension, we find conditions for the deterministic transformation of a collection of several full Schmidt rank pure states $\ensuremath{\mid}{\ensuremath{\psi}}_{j}⟩$ to pure states $\ensuremath{\mid}{\ensuremath{\phi}}_{j}⟩$.
In this study, we enhance the understanding of entanglement transformations and their quantification by extending the concept of Schmidt vector from pure to mixed bipartite states, exploiting the lattice structure of majorization. The Schmidt vector of a bipartite mixed state is defined using two distinct methods: as a concave roof extension of Schmidt vectors of pure states, or equivalently, from the set of pure states that can be transformed into the mixed state through local operations and classical communication (LOCC). We demonstrate that the Schmidt vector fully characterized separable and maximally entangled states. Furthermore, we prove that the Schmidt vector is monotonic and strongly monotonic under LOCC, giving necessary conditions for conversions between mixed states. Additionally, we extend the definition of the Schmidt rank from pure states to mixed states as the cardinality of the support of the Schmidt vector and show that it is equal to the Schmidt number introduced in previous work [Terhal and Horodecki, Phys. Rev. A 61, 040301(R) (2000)]. Finally, we introduce a family of entanglement monotones by considering concave and symmetric functions applied to the Schmidt vector.
Entanglement plays an important role in quantum computation and information. We can only manipulate an entangled quantum state shared among several distant parties by local operations with classical communication (LOCC). Thus, it is of great importance to characterize the conditions by which we can determine whether one pure quantum state can be transformed to another pure state by LOCC. It is well known that separable operations and LOCC are not equivalent. Dozens of results have illustrated this difference. Little do we know about this difference despite these recent results. It is shown in this paper that if the initial state $|\ensuremath{\psi}\ensuremath{\rangle}$ and final state $|\ensuremath{\phi}\ensuremath{\rangle}$ are genuine tripartite pure states in the Greenberger-Horne-Zeilinger (GHZ) class then $|\ensuremath{\psi}\ensuremath{\rangle}$ can be transformed to $|\ensuremath{\phi}\ensuremath{\rangle}$ by separable operations if and only if $|\ensuremath{\psi}\ensuremath{\rangle}$ can be transformed to $|\ensuremath{\phi}\ensuremath{\rangle}$ by deterministic LOCC. That is, SEP equals LOCC on tripartite GHZ states.
We prove, using a method based on map-state duality, lower bounds on entanglement resources needed to deterministically implement a bipartite unitary using separable (SEP) operations, which include LOCC (local operations and classical communication) as a particular case. It is known that the Schmidt rank of an entangled pure state resource cannot be less than the Schmidt rank of the unitary. We prove that if these ranks are equal the resource must be uniformly (maximally) entangled: equal nonzero Schmidt coefficients. Higher rank resources can have less entanglement: we have found numerical examples of Schmidt rank 2 unitaries, which can be deterministically implemented, by either SEP or LOCC, using an entangled resource of two qutrits with less than one ebit of entanglement.
The conditions for transforming pure entangled states under local operations and classical communication (LOCC) are well understood. A natural question then arises: Can we determine the transformation conditions for mixed entangled states under LOCC based on the properties of their pure-state ensembles? While much effort has been devoted to this issue, in this paper, we rule out this possibility. Our findings address several open questions, including: (i) The conditions Efcr(ρ)≥Efcr(σ) E_f^{cr}(ρ) \geq E_f^{cr}(σ) Efcr​(ρ)≥Efcr​(σ) for all convex roof entanglement measures EfcrE_f^{cr}Efcr​ is insufficient to guarantee the existence of an LOCC transformation ΛL(⋅)Λ^L(\cdot)ΛL(⋅) from ρρρ to σσσ; and (ii) The inequalities ∑jpjE(φj)≥∑lqlE(ψl)\sum_j p_j E(\varphi_j) \geq \sum_l q_l E(ψ_l)∑j​pj​E(φj​)≥∑l​ql​E(ψl​) for all entanglement monotones EEE are not sufficient to ensure the existence of an LOCC transformation from {pj,∣φj⟩}\{p_j, \ket{\varphi_j}\}{pj​,∣φj​⟩} to {ql,∣ψl⟩}\{q_l, \ket{ψ_l}\}{ql​,∣ψl​⟩}.
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.