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arXiv:2308.05792

Commuting operations factorise

Renato Renner, Ramona Wolf

Published Aug 10, 2023• 0 citations
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Abstract

Consider two agents, Alice and Bob, each of whom takes a quantum input, operates on a shared quantum system $K$, and produces a quantum output. Alice and Bob's operations may commute, in the sense that the joint input-output behaviour is independent of the order in which they access $K$. Here we ask whether this commutation property implies that $K$ can be split into two factors on which Alice and Bob act separately. The question can be regarded as a "fully quantum" generalisation of a problem posed by Tsirelson, who considered the case where Alice and Bob's inputs and outputs are classical. In this case, the answer is negative in general, but it is known that a factorisation exists in finite dimensions. Here we show the same holds in the fully quantum case, i.e., commuting operations factorise, provided that all input systems are finite-dimensional.

DOI
10.48550/arxiv.2308.05792

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Consider two agents, Alice and Bob, each of whom takes a quantum input, operates on a shared quantum system KKK, and produces a quantum output. Alice and Bob's operations may commute, in the sense that the joint input-output behaviour is independent of the order in which they access KKK. Here we ask whether this commutation property implies that KKK can be split into two factors on which Alice and Bob act separately. The question can be regarded as a "fully quantum" generalisation of a problem posed by Tsirelson, who considered the case where Alice and Bob's inputs and outputs are classical. In this case, the answer is negative in general, but it is known that a factorisation exists in finite dimensions. Here we show the same holds in the fully quantum case, i.e., commuting operations factorise, provided that all input systems are finite-dimensional.
Abstract Consider two agents, Alice and Bob, each of whom takes a quantum input, operates on a shared quantum system K , and produces a quantum output. Alice and Bob’s operations may commute, in the sense that the joint input-output behaviour is independent of the order in which they access K . Here we ask whether this commutation property implies that K can be split into two factors on which Alice and Bob act separately. The question can be regarded as a “fully quantum” generalisation of a problem posed by Tsirelson, who considered the case where Alice and Bob’s inputs and outputs are classical. In this case, the answer is negative in general, but it is known that a factorisation exists in finite dimensions. Here we show the same holds in the fully quantum case, i.e., commuting operations factorise, provided that all input systems are finite-dimensional.
The situation of two independent observers conducting measurements on a joint quantum system is usually modelled using a Hilbert space of tensor product form, each factor associated to one observer. Correspondingly, the operators describing the observables are then acting non-trivially only on one of the tensor factors. However, the same situation can also be modelled by just using one joint Hilbert space, and requiring that all operators associated to different observers commute, i.e. are jointly measurable without causing disturbance. The problem of Tsirelson is now to decide the question whether all quantum correlation functions between two independent observers derived from commuting observables can also be expressed using observables defined on a Hilbert space of tensor product form. Tsirelson showed already that the distinction is irrelevant in the case that the ambient Hilbert space is of finite dimension. We show here that the problem is equivalent to the question whether all quantum correlation functions can be approximated by correlation function derived from finite-dimensional systems. We also discuss some physical examples which fulfill this requirement.
A quantum flipping device is a theoretical apparatus designed to transform an input quantum state, represented by the density operator $\ensuremath{\rho}$ acting on a ddd-dimensional Hilbert space H\mathcal{H}H, into an output state $\ensuremath{\sigma}$ residing within the subspace orthogonal to $\ensuremath{\rho}$. While the no-flipping theorem establishes the impossibility of such a device for qubits (two-dimensional systems), its extension to higher-dimensional systems (d>2d>2d>2) allows for the possibility of such devices due to the increased space available for the output state. In this paper, we derive an impossibility theorem for generalized flipping devices, applicable to higher-dimensional systems, based on a causal principle, namely, commutation of spacelike operations. Given a bipartite quantum state shared between two parties Alice and Bob, this principle prohibits the resulting state from depending on the time order of Alice's and Bob's spacelike separated local operations applied on the respective parts of the bipartite systems. Importantly, unlike the argument applied to qubits, our approach does not rely on the unitary dynamics postulate of quantum theory. Furthermore, we investigate the implications of violating the principle of commutation of spacelike operations and demonstrate that such violations do not necessarily entail a violation of the no-signaling principle.
In this paper, we consider a quantum generalization of the task considered by Slepian and Wolf regarding distributed source compression. In our task, Alice, Bob, Charlie, and Reference share a joint pure state. Alice and Bob wish to send a part of their respective systems to Charlie without collaborating with each other. We give achievability bounds for this task in the one-shot setting and provide the asymptotic and independent identically distributed analysis in the case when there is no side information with Charlie. Our result implies the result of Abeyesinghe et al., who studied a special case of this problem. As another special case wherein Bob holds trivial registers, we recover the result of Devetak and Yard regarding quantum state redistribution.
Suppose Alice and Bob jointly possess a pure state, $|\ensuremath{\psi}〉$. Using local operations on their respective systems and classical communication it may be possible for Alice and Bob to transform $|\ensuremath{\psi}〉$ into another joint state $|\ensuremath{\varphi}〉$. This Letter gives necessary and sufficient conditions for this process of entanglement transformation to be possible. These conditions reveal a partial ordering on the entangled states and connect quantum entanglement to the algebraic theory of majorization. As a consequence, we find that there exist essentially different types of entanglement for bipartite quantum systems.