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arXiv:2112.03233

Fundamental trade-off relation in probabilistic entanglement generation

Yuanbo Chen, Yoshihiko Hasegawa

Published Dec 6, 2021• 0 citations
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Abstract

We investigate the generation of entanglement between two non-interacting systems by synthesizing a new quantum process from the superposition of distinct processes characterized by local-only operations. Our analysis leads to the derivation of a universal trade-off relation, $P_{\text{succ}}(1+\mathcal{C})\le1$, that fundamentally bounds the success probability ($P_{\text{succ}}$) and the generated entanglement (concurrence $\mathcal{C}$). The derivation of this trade-off relation is inspired by indefinite causal order, but applies for a broader class of quantum processes. Next, we show that the mathematical structure of this bound predicts the existence of a "quasi-deterministic" mode of operation, a surprising phenomenon which we then confirm with concrete entanglement generation protocols, where a maximally entangled state is guaranteed to be produced. In this mode of operation, both outcomes of the post-selection measurement on the auxiliary control system result in a maximally entangled state of the target system. Furthermore, we demonstrate how this general principle can be realized using a quantum switch, which leverages an indefinite causal order as a physical resource, and explore the rich variety of dynamical behaviors governed by the universal trade-off. Our results establish a general principle for entanglement generation with superposition of quantum processes and introduce a novel way of controlling entanglement generation.

DOI
10.48550/arxiv.2112.03233

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We investigate the generation of entanglement between two non-interacting systems by synthesizing a new quantum process from the superposition of distinct processes characterized by local-only operations. Our analysis leads to the derivation of a universal trade-off relation, Psucc(1+C)≤1P_{\text{succ}}(1+\mathcal{C})\le1Psucc​(1+C)≤1, that fundamentally bounds the success probability (PsuccP_{\text{succ}}Psucc​) and the generated entanglement (concurrence C\mathcal{C}C). The derivation of this trade-off relation is inspired by indefinite causal order, but applies for a broader class of quantum processes. Next, we show that the mathematical structure of this bound predicts the existence of a "quasi-deterministic" mode of operation, a surprising phenomenon which we then confirm with concrete entanglement generation protocols, where a maximally entangled state is guaranteed to be produced. In this mode of operation, both outcomes of the post-selection measurement on the auxiliary control system result in a maximally entangled state of the target system. Furthermore, we demonstrate how this general principle can be realized using a quantum switch, which leverages an indefinite causal order as a physical resource, and explore the rich variety of dynamical behaviors governed by the universal trade-off. Our results establish a general principle for entanglement generation with superposition of quantum processes and introduce a novel way of controlling entanglement generation.
The controlled generation of quantum entanglement from separable states remains a central challenge in quantum information science. Here, we investigate entanglement generation using two related control paradigms: coherent path superposition of local unitary operations and stochastic implementations of Pauli channels under coherent control. We show that entangled states belonging to the Bell, GHZ and W classes, can be deterministically generated from fully separable inputs by coherently superposing alternative sets of local unitary transformations. Conditions on the local operators for entanglement generation are derived, and the resulting states are shown to be locally unitary equivalent to standard multipartite entangled states. We further extend the analysis to noisy scenarios, where separable mixed states evolve through pairs of Pauli channels arranged in path-superposition and indefinite causal order configurations. Closed-form expressions for the output states are obtained, and entanglement is quantified using concurrence. By exploring representative channel families across their parameter space, we identify regimes where stochastic entanglement emerges, determine the associated success probabilities, and characterize trade-offs between entanglement and purity.
Quantum theory allows for the superposition of causal orders between operations, i.e., for an indefinite causal order; an implication of the principle of quantum superposition. Since a higher theory might also admit this feature, an understanding of superposition and indefinite causal order in a generalised probabilistic framework is needed. We present a possible notion of superposition for such a framework and show that in maximal theories, respecting non-signalling relations, single system state-spaces do not admit superposition; however, composite systems do. Additionally, we show that superposition does not imply entanglement. Next, we provide a concrete example of a maximally Bell-nonlocal theory, which not only admits the presented notion of superposition, but also allows for post-quantum violations of theory-independent inequalities that certify indefinite causal order; even up to an algebraic bound. These findings might point towards potential connections between a theory's ability to admit indefinite causal order, Bell-nonlocal correlations and the structure of its state spaces.
Entanglement represents "the" key resource for several applications of quantum information processing, ranging from quantum communications to distributed quantum computing. Despite its fundamental importance, deterministic generation of maximally entangled qubits represents an on-going open problem. Here, we design a novel generation scheme exhibiting two attractive features, namely, i) deterministically generating genuinely multipartite entangled states, ii) without requiring any direct interaction between the qubits. Indeed, the only necessary condition is the possibility of coherently controlling -- according to the indefinite causal order framework -- the causal order among some unitaries acting on the qubits. Through the paper, we analyze and derive the conditions on the unitaries for deterministic generation, and we provide examples for unitaries practical implementation. We conclude the paper by discussing the scalability of the proposed scheme to higher dimensional GME states and by introducing some possible applications of the proposal for quantum networks.
Quantum processes with indefinite causal order challenge the classical assumption that operations must occur in a single fixed temporal sequence. The quantum switch provides a concrete setting in which two operation orders, A≺B and B≺A, are coherently controlled by a quantum system. In the strict process matrix formulation of the lazy guess your neighbour’s input (LGYNI) game, however, quantum theory, including the quantum switch, does not violate the standard causal inequality when probabilities are computed solely from local instruments. In this work, we study an extended control-assisted operational protocol in which the control system of the quantum switch is measured and used to define the task output. We compare increasingly expressive strategy classes, including single-qubit SU(2) operations, product target-ancilla operations, and entangling Cartan-decomposed two-qubit operations with generalized POVMs. Restricted models saturate or remain below the 3/4 fixed-order benchmark, whereas the optimized Cartan + ancilla + POVM strategy reaches Psuccext≈0.83596, demonstrating enhanced task performance within the extended protocol. The optimized strategy remains operationally no-signaling to numerical precision and retains its extended protocol advantage under more than 25% white noise admixture. These results identify the operational resources required for control-assisted quantum switch enhancement and support the view that indefinite temporal order can be used as a quantum informational resource without implying a breakdown of operational causality.
Quantum theory is in principle compatible with scenarios where physical processes occur in an indefinite order, potentially yielding advantages in a broad range of information processing tasks. However, advantages in communication, the most basic form of information processing, have so far remained controversial and hard to prove. Here we provide a framework for assessing the role of causal order in communication, by comparing different causal structures under the constraint that the allowed operations must not generate signaling from signaling-incapable devices. Using this framework, we establish a clear-cut advantage of indefinite causal order, and, at the same time, we identify a series of fundamental limits to the communication power of causal structures in quantum mechanics. Notably, we find that a special form of indefinite causal order, obtained by coherently controlling the order of two processes, enhances the transmission of classical messages in a one-shot scenario, but no quantum operation with indefinite order can offer advantages over shared entanglement when asymptotically many uses of the same communication device are employed. Overall, our results unveil non-trivial relations between communication, causal order, entanglement, and no-signaling quantum processes.
Abstract In this proceeding, we revisit the discussion presented in Ref. [Commun Phys 7, 373 (2024)], which examines the behavior of a quantum switch involving two arbitrary quantum operations when the control is exposed to environmental effects. Our study extends this analysis by focusing on the evolution of entanglement in the target system within the quantum switch framework, taking into account the influence of environmental conditions and control post-selection. We find that entanglement evolution is highly sensitive to these factors. While entanglement sudden death occurs under definite causal order, indefinite causal order can reverse this loss. We observe entanglement revival in high-temperature regimes and a sudden reappearance of entanglement under weak monitoring conditions at low temperatures. These findings provide insights into the resilience of the quantum switch in the presence of environmental disturbances and highlight its potential for applications where preserving entanglement is essential.