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Physical Review A

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

Anandamay Das Bhowmik, Tathagata Gupta, Preeti Parashar, Guruprasad Kar, Manik Banik

Published Mar 25, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.111.032431

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•Jan 23, 2018•3 citations
A quantum flipping device is a theoretical apparatus designed to transform an input quantum state, represented by the density operator \ensuremathρ\ensuremath{\rho}\ensuremathρ acting on a ddd-dimensional Hilbert space H\mathcal{H}H, into an output state \ensuremathσ\ensuremath{\sigma}\ensuremathσ residing within the subspace orthogonal to \ensuremathρ\ensuremath{\rho}\ensuremathρ. 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.
The no-signalling principle lies at the heart of Bell-type experiments involving spacelike-separated parties. Although not all no-signalling correlations can be realized within quantum theory, the no-sinalling principle itself remains fundamental even for post-quantum theories, since its violation would imply superluminal signalling. In the spirit of device independent nature of the Bell inequalities, there has recently been growing interest in causal inequalities, which identify correlations admitting no causal explanation. Such correlations have been rigorously studied within the process-matrix formalism, which is built upon a constraint analogous to spacelike separation in Bell scenarios. We refer to this constraint as the single-interaction constraint, whereby each party interacts with the environment only once. Despite considerable effort, no fundamental guiding principle is currently known that characterizes the restrictions this constraint imposes on the correlations generated by Process Matrices. In this work, we address this gap by proposing a generalized causality principle for the bipartite case. We show that for certain cases, the generalised causality principle defines a strict subset of the set of all signalling correlations. This principle thus provides a fundamental limit on quantum and even post-quantum theories subject to the single-interaction constraint. By analogy with no-signalling, whose violation excludes certain causal structures among the parties, a violation of the generalised causality principle rules out a corresponding class of causal structures(causal structures with the single interaction constraint). More broadly, our techniques provide a general framework for deriving generalised causal constraints, applicable to a wide class of indefinite-causal-order scenarios.
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.
We have studied the possibility of post-quantum theories more nonlocal than the (standard) quantum theory using the modification of the quantum probability rule under the no-signaling condition. For this purpose we have considered the situation that two spacelike separate parties Alice and Bob share an entangled two qubit system. We have modified the quantum probability rule as small as possible such that the first local measurements are governed by the usual Born rule and the second measurement by the modified quantum probability rule. We have shown that only the maximally entangled states can have higher nonlocality than the quantum upper bound while satisfying the no-signaling condition. This fact could be a partial explanation for why the nonlocality of the quantum theory is limited. As a by-product we have found the systematic way to obtain a variety of nonlocal boxes.
An astonishing feature of higher-order quantum theory is that it can accommodate indefinite causal order. In the simplest bipartite setting, there exist signaling correlations for which it is fundamentally impossible to ascribe a definite causal order for the parties' actions. Moreover, the assumptions required to arrive at such a statement (local quantum transformations and well-behaved probabilities) result in a nontrivial set of correlations, whose boundary is, to date, uncharacterized. In this work, we investigate indefinite causal order in boxworld theories. We construct a higher-order theory whose descriptor is the generalized bit (gbit) - the natural successor of the classical bit and quantum qubit. By fixing the local transformations in boxworld and asking about the global causal structure, we find that we trivially recover the full set of two-way signaling correlations. In light of this, we motivate and propose two physical principles in order to limit the set of achievable correlations: nonsignaling preservation and no signaling without system exchange. We find that a higher-order boxworld theory that respects these physical principles leads to (i) a nontrivial set of achievable correlations and (ii) a violation of some causal inequalities that is higher than what can be achieved in higher-order quantum theory. These results lead us to conjecture that the set of correlations of our higher-order boxworld theory is an outer approximation to the set of correlations produced by higher-order quantum theory.
It was recently found that it is theoretically possible for there to exist quantum processes in which the order of the operations performed by separate parties cannot be ascribed a definite causal order. Some of these processes have been claimed to have a physical realization in standard quantum mechanics via coherent control of the times of the operations. A prominent example is the quantum SWITCH, which was recently demonstrated experimentally. However, up until now, there has been no rigorous justification for the interpretation of such an experiment as a genuine realization of a process with indefinite causal order as opposed to a simulation of such a process. Where exactly are the local operations of the parties in such an experiment? On what spaces do they act given that their times are indefinite? Can we probe them directly rather than assume what they ought to be based on heuristic considerations? How can we reconcile the claim that these operations really take place, each once as required, with the fact that the structure of the presumed process implies that they cannot be part of any acyclic circuit? Here, I offer a precise answer to these questions: the input and output systems of the operations in such a process are generally nontrivial subsystems of Hilbert spaces that are tensor products of Hilbert spaces associated with different times---a fact that is directly experimentally verifiable. With respect to these time-delocalized subsystems, the structure of the process is that of a circuit with a cycle, which cannot be reduced to a probabilistic mixture of acyclic circuits. I further show that all bipartite processes that admit a certain type of isometric extension have a physical realization on such time-delocalized subsystems. These results unveil a novel structure within quantum mechanics, which may have important implications for physics and information processing.