Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Communications in Mathematical Physics

Commuting Quantum Operations Factorise

Renato Renner, Ramona Wolf

Published Jun 6, 2026• 0 citations
Publisher

Abstract

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
10.1007/s00220-026-05655-9

Related papers

Commuting operations factorise

Renato Renner, Ramona Wolf

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.

arXiv:2308.05792•Aug 10, 2023•0 citations

Tsirelson's Problem

Volkher B. Scholz, Reinhard F. Werner

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.

arXiv:0812.4305•Dec 22, 2008•43 citations

Generalized no-flipping theorem as a consequence of commutation of spacelike operations

Anandamay Das Bhowmik, Tathagata Gupta, Preeti Parashar +2 more

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 $d$-dimensional Hilbert space $\mathcal{H}$, 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 ($d>2$) 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.

Physical Review A•Mar 25, 2025•0 citations

A Generalized Quantum Slepian–Wolf

Anurag Anshu, Rahul Jain, Naqueeb Ahmad Warsi

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.

arXiv:1703.09961•IEEE Transactions on Information Theory•Dec 22, 2017•24 citations

Quantum entanglement, interaction, and the classical limit

Thomas Durt

PACS number: O3.65.Bz Abstract: Two or more quantum systems are said to be in an entangled or nonfactorisable state if their joint (supposedly pure) wave-function is not expressible as a product of individual wave functions but is instead a superposition of product states. It is only when the systems are in a factorisable state that they can be considered to be separated (in the sense of Bell). We show that whenever two quantum systems interact with each other, it is impossible that all factorisable states remain factorisable during the interaction unless the full Hamiltonian does not couple these systems so to say unless they do not really interact. We also present certain conditions under which particular factorisable states remain factorisable although they represent a bipartite system whose components mutually interact. We identify certain quasi-classical regimes that satisfy these conditions and show that they correspond to classical, pre-quantum, paradigms associated to the concept of particle

arXiv:quant-ph/0401121•Jan 1, 2004•15 citations

Compressing quantum mixed-state sources by sending classical information

Emina Soljanin

We consider visible compression for discrete memoryless sources of mixed quantum states when only classical information can be sent from Alice to Bob. We assume that Bob knows the source statistics, and that Alice and Bob have access to the same source of random numbers. We put in an information-theoretic framework some previous results on visible compression for sources of states with commuting density operators, and remove the commutativity requirement. We derive a general achievable compression rate, which is for the noncommutative case still higher than the known lower bound. We also present several related problems of classical information theory, and show how they can be used to answer some questions of the mixed-state compression problem.

IEEE Transactions on Information Theory•Aug 1, 2002•19 citations
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.
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.
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.
PACS number: O3.65.Bz Abstract: Two or more quantum systems are said to be in an entangled or nonfactorisable state if their joint (supposedly pure) wave-function is not expressible as a product of individual wave functions but is instead a superposition of product states. It is only when the systems are in a factorisable state that they can be considered to be separated (in the sense of Bell). We show that whenever two quantum systems interact with each other, it is impossible that all factorisable states remain factorisable during the interaction unless the full Hamiltonian does not couple these systems so to say unless they do not really interact. We also present certain conditions under which particular factorisable states remain factorisable although they represent a bipartite system whose components mutually interact. We identify certain quasi-classical regimes that satisfy these conditions and show that they correspond to classical, pre-quantum, paradigms associated to the concept of particle
We consider visible compression for discrete memoryless sources of mixed quantum states when only classical information can be sent from Alice to Bob. We assume that Bob knows the source statistics, and that Alice and Bob have access to the same source of random numbers. We put in an information-theoretic framework some previous results on visible compression for sources of states with commuting density operators, and remove the commutativity requirement. We derive a general achievable compression rate, which is for the noncommutative case still higher than the known lower bound. We also present several related problems of classical information theory, and show how they can be used to answer some questions of the mixed-state compression problem.