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arXiv:1311.6842 · Physical Review A

Characteristics of universal embezzling families

Debbie Leung, Bingjie Wang

Published Oct 24, 2014• 18 citations
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Abstract

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.

DOI
10.1103/physreva.90.042331

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Quantum state embezzlement is the transformation ∣\ensuremathμ\ensuremath⟩\ensuremath↦∣\ensuremathμ\ensuremath⟩∣\ensuremathφ\ensuremath⟩|\ensuremath{\mu}\ensuremath{\rangle}\ensuremath{\mapsto}|\ensuremath{\mu}\ensuremath{\rangle}|\ensuremath{\varphi}\ensuremath{\rangle}∣\ensuremathμ\ensuremath⟩\ensuremath↦∣\ensuremathμ\ensuremath⟩∣\ensuremathφ\ensuremath⟩ using only local operations, where ∣\ensuremathφ\ensuremath⟩|\ensuremath{\varphi}\ensuremath{\rangle}∣\ensuremathφ\ensuremath⟩ and ∣\ensuremathμ\ensuremath⟩|\ensuremath{\mu}\ensuremath{\rangle}∣\ensuremathμ\ensuremath⟩ 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 ∣\ensuremathμ\ensuremath⟩|\ensuremath{\mu}\ensuremath{\rangle}∣\ensuremathμ\ensuremath⟩ that enables embezzlement in the bipartite setting with arbitrary precision as the dimension of ∣\ensuremathμ\ensuremath⟩|\ensuremath{\mu}\ensuremath{\rangle}∣\ensuremathμ\ensuremath⟩ increases. Furthermore, the family is independent of the state ∣\ensuremathφ\ensuremath⟩|\ensuremath{\varphi}\ensuremath{\rangle}∣\ensuremathφ\ensuremath⟩ 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.
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.
Using local operations and classical communication (LOCC), entanglement can be manipulated but not created. However, entanglement can be embezzled . In this work, we completely characterize universal embezzling families and demonstrate how this singles out the original family introduced by van Dam and Hayden. To achieve this, we first give a full characterization of pure to mixed state LOCC-conversions. Then, we introduce a new conversion distance and derive a closed-form expression for it. These results might be of independent interest.
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 prove that the family of embezzlement states defined by van Dam and Hayden [vanDamHayden2002] is universal for both quantum and classical entangled two-prover non-local games with an arbitrary number of rounds. More precisely, we show that for each ε>0ε>0ε>0 and each strategy for a k-round two-prover non-local game which uses a bipartite shared state on 2m qubits and makes the provers win with probability ωωω, there exists a strategy for the same game which uses an embezzlement state on 2m+2m/ε2m + 2m/ε2m+2m/ε qubits and makes the provers win with probability ω−2εω-\sqrt{2ε}ω−2ε​. Since the value of a game can be defined as the limit of the value of a maximal 2m-qubit strategy as m goes to infinity, our result implies that the classes QMIP*_{c,s}[2,k] and MIP*_{c,s}[2,k] remain invariant if we allow the provers to share only embezzlement states, for any completeness value c in [0,1] and any soundness value s < c. Finally we notice that the circuits applied by each prover may be put into a very simple universal form.
Embezzlement of entanglement is the counterintuitive process in which entanglement is extracted from a resource system using local unitary operations, with almost no detectable change in the resource's state. It has recently been argued that any state of a relativistic quantum field theory can serve as a resource for perfect embezzlement. We study the circuit complexity of embezzlement, using sequences of states that enable arbitrary precision for the process, commonly called universal embezzling families. In addition, we argue that this approach provides a well-defined model for the complexity of embezzlement from quantum field theories. Under fairly general assumptions, we establish a generic lower bound on the complexity, which increases with the precision of the process or embezzled entanglement, and diverges as these become infinite. As an example, we consider a 1d1d1d critical system as the resource and derive an exponentially growing lower bound on the complexity. Consequently, the findings imply that circuit complexity acts as a physical obstruction to perfect embezzlement. Supplementary to the main results, we derive lower bounds for common models of circuit complexity for state preparation, based on the difference between the Schatten norms of the initial and final states.
We show that, starting from a product state, local unitary quantum evolutions generate intermediate states which exhibit a multiscale entanglement-spectrum structure required for universal entanglement embezzlement. This facilitates entanglement extraction from a catalyst many-body state while leaving it asymptotically unchanged. Remarkably, these atypical structures emerge generically at intermediate stages, well before reaching maximum entropy where thermalization has flattened out the spectral hierarchy. The resulting state is accompanied by nonlocal nonstabilizerness that diverges with the system size, consistent with a recently established equivalence between universal embezzlement and divergent nonlocal magic. Thus, without any fine tuning, a chaotic quantum evolution generates intermediate states which form a universal family of catalytic reservoirs.