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arXiv:1606.05061 · Journal of Mathematical Physics

Perfect embezzlement of entanglement

Richard Cleve, Li Liu, Vern I. Paulsen

Published Jan 1, 2017• 27 citations
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Abstract

Van Dam and Hayden introduced a concept commonly referred to as embezzlement, where, for any entangled quantum state ϕ, there is an entangled catalyst state ψ, from which a high fidelity approximation of ϕ ⊗ ψ can be produced using only local operations. We investigate a version of this where the embezzlement is perfect (i.e., the fidelity is 1). We prove that perfect embezzlement is impossible in a tensor product framework, even with infinite-dimensional Hilbert spaces and infinite entanglement entropy. Then we prove that perfect embezzlement is possible in a commuting operator framework. We prove this using the theory of C*-algebras and we also provide an explicit construction. Next, we apply our results to analyze perfect versions of a nonlocal game introduced by Regev and Vidick. Finally, we analyze the structure of perfect embezzlement protocols in the commuting operator model, showing that they require infinite-dimensional Hilbert spaces.

DOI
10.1063/1.4974818

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Van Dam and Hayden introduced a concept commonly referred to as embezzlement, where, for any entangled quantum state ϕ, there is an entangled catalyst state ψ, from which a high fidelity approximation of ϕ ⊗ ψ can be produced using only local operations. We investigate a version of this where the embezzlement is perfect (i.e., the fidelity is 1). We prove that perfect embezzlement is impossible in a tensor product framework, even with infinite-dimensional Hilbert spaces and infinite entanglement entropy. Then we prove that perfect embezzlement is possible in a commuting operator framework. We prove this using the theory of C*-algebras and we also provide an explicit construction. Next, we apply our results to analyze perfect versions of a nonlocal game introduced by Regev and Vidick. Finally, we analyze the structure of perfect embezzlement protocols in the commuting operator model, showing that they require infinite-dimensional Hilbert spaces.
We present an explicit construction of a universal embezzlement protocol in the C*-algebraic model of quantum information, that is equivalent to the commuting operator model. Our protocol enables exact embezzlement of arbitrary bipartite pure states using a single, fixed catalyst state. Unlike prior constructions that achieve only approximate embezzlement or require state-dependent catalysts, our approach is both exact and state-independent. The construction is explicit, based on simple *-automorphisms acting locally on infinite tensor products of CAR algebras with the underlying idea of the Hilbert hotel. In the dense-state case, the protocol naturally recovers the Type III_1 factor via the GNS construction, consistent with recent classification results. We further extend the construction to allow exact embezzlement of all states, at the cost of working with a non-separable C*-algebra. Despite the increase in algebraic size, the operational structure remains simple and localized. This offers a conceptually intuitive model for universal entanglement embezzlement in infinite-dimensional settings.
We show that the optimal commuting operator correlation for a variant of Coladangelo's two-player generalized nonlocal game, based on the embezzlement of entanglement, is a commuting operator self-test. Our result gives the first correlation self-test for an exact entanglement embezzlement protocol in the commuting operator framework. Because the unique optimal correlation necessitates the exact embezzlement of entanglement, the correlation is unrealizable using quantum tensor-product models, even if the local spaces are allowed to be infinite dimensional. As such, our result provides an example of a commuting operator self-test where the Hilbert space shared by the two parties cannot be factored into a tensor-product of local spaces that is compatible with the local measurement operators. Our main proof technique relies on Harris' unique abstract state characterization of exact embezzlement of entanglement involving Cuntz algebras.
Embezzlement of entanglement refers to the task of extracting entanglement from an entanglement resource via local operations and without communication while perturbing the resource arbitrarily little. Recently, the existence of embezzling states of bipartite systems of type III von Neumann algebras was shown. However, both the multipartite case and the precise relation between embezzling states and the notion of embezzling families, as originally defined by van Dam and Hayden, was left open. Here, we show that finite-dimensional approximations of multipartite embezzling states form multipartite embezzling families. In contrast, not every embezzling family converges to an embezzling state. We identify an additional consistency condition that ensures that an embezzling family converges to an embezzling state. This criterion distinguishes the embezzling family of van Dam and Hayden from the one by Leung, Toner, and Watrous. The latter generalizes to the multipartite setting. By taking a limit, we obtain a multipartite system of commuting type III 1 factors on which every state is an embezzling state. We discuss our results in the context of quantum field theory and quantum many-body physics. As open problems, we ask whether vacua of relativistic quantum fields in more than two spacetime dimensions are multipartite embezzling states and whether multipartite embezzlement allows for an operator-algebraic characterization.
Quantum state embezzlement is the transformation $|\ensuremath{\mu}\ensuremath{\rangle}\ensuremath{\mapsto}|\ensuremath{\mu}\ensuremath{\rangle}|\ensuremath{\varphi}\ensuremath{\rangle}$ using only local operations, where $|\ensuremath{\varphi}\ensuremath{\rangle}$ and $|\ensuremath{\mu}\ensuremath{\rangle}$ are multipartite quantum states. Exact embezzlement is an impossible task since it implies the increase of entanglement without communication. Surprisingly, van Dam and Hayden [Phys. Rev. A 67, 060302 (2003)] find a universal embezzling family of states $|\ensuremath{\mu}\ensuremath{\rangle}$ that enables embezzlement in the bipartite setting with arbitrary precision as the dimension of $|\ensuremath{\mu}\ensuremath{\rangle}$ increases. Furthermore, the family is independent of the state $|\ensuremath{\varphi}\ensuremath{\rangle}$ to be embezzled. We study embezzlement in the bipartite setting. We present various requirements and consequences, and infinitely many universal embezzling families inequivalent to that proposed by van Dam and Hayden. We include numerical studies of up to 33-qubit large local systems.
We investigate the operator-algebraic structure underlying entanglement embezzlement, a phenomenon where a fixed entangled state (the catalyst) can be used to generate arbitrary target entangled states without being consumed. We show that the ability to embezzle a target state ggg imposes strong internal constraints on the catalyst state fff: specifically, fff must contain infinitely many mutually commuting, locally structured copies of ggg. This property is formalized using C*-algebraic tools and is analogous to a form of self-testing, certifying the presence of infinite copies of ggg within fff. Using this infinite-copy certification. Our results clarify the structural requirements for embezzlement and provide new conceptual tools for analyzing state certification in infinite-dimensional quantum systems.
We consider bipartite exact entanglement embezzlement with a catalyst state vector ψψψ in a Hilbert space H\mathcal{H}H using unitaries (or more generally, contractions). If M⊆B(H)\mathcal{M} \subseteq \mathcal{B}(\mathcal{H})M⊆B(H) is a von Neumann algebra and U∈Md⊗MU \in M_d \otimes \mathcal{M}U∈Md​⊗M and V∈M′⊗MdV \in \mathcal{M}' \otimes M_dV∈M′⊗Md​ are unitaries (or more generally contractions), then such a protocol is of the form (U⊗Id)(Id⊗V)(e0⊗ψ⊗e0)=∑i=0d−1αiei⊗ψ⊗ei(U \otimes I_d)(I_d \otimes V)(e_0 \otimes ψ\otimes e_0)=\sum_{i=0}^{d-1} α_i e_i \otimes ψ\otimes e_i(U⊗Id​)(Id​⊗V)(e0​⊗ψ⊗e0​)=∑i=0d−1​αi​ei​⊗ψ⊗ei​, where each αi>0α_i>0αi​>0 and ∑i=0d−1αi2=1\sum_{i=0}^{d-1} α_i^2=1∑i=0d−1​αi2​=1. We show that any such protocol must arise from a unique state on the tensor product Od⊗Od\mathcal{O}_d \otimes \mathcal{O}_dOd​⊗Od​ of the Cuntz algebra with itself. As a result, we prove that exact entanglement embezzlement is a self-test for a collection of ddd Cuntz isometries for each party and a unique quasi-free state on the Cuntz algebra Od\mathcal{O}_dOd​ in the sense of \cite{Iz93}. Moreover, we use modular theory to show that the von Neumann algebra generated by the copy of Od\mathcal{O}_dOd​ is the unique separable approximately finite-dimensional Type IIIλ\text{III}_λIIIλ​ factor for some 0<λ≤10<λ\leq 10<λ≤1, where λλλ can be determined by an algebraic condition on the Schmidt coefficients of the state φ=∑i=0d−1αiei⊗ei\varphi=\sum_{i=0}^{d-1} α_i e_i \otimes e_iφ=∑i=0d−1​αi​ei​⊗ei​.