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Register a new userTime-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics
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Ognyan Oreshkov
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It has been shown that it is theoretically possible for there to exist higher-order quantum processes in which the operations performed by separate parties cannot be ascribed a definite causal order. Some of these processes are believed 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, the interpretation of such experiments as realizations of a process with indefinite causal structure as opposed to some form of simulation of such a process has remained controversial. 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 systems at different times---a fact that is directly experimentally verifiable. With respect to these time-delocalized subsystems, the structure of the process is one of a circuit with a causal cycle. I also identify a whole class of isometric processes, of which the quantum SWITCH is a special case, that admit a physical realization on time-delocalized subsystems. These results unveil a novel structure within quantum mechanics, which may have important implications for physics and information processing.
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The fundamental dynamics of quantum particles is neutral with respect to the arrow of time. And yet, our experiments are not: we observe quantum systems evolving from the past to the future, but not the other way round. A fundamental question is whether it is in principle possible to probe a quantum dynamics in the backward direction, or in more general combinations of the forward and the backward direction. To answer this question, we characterise all possible time-reversals that satisfy four natural requirements and we identify the largest set of quantum processes that can in principle be probed in both time directions. Then, we show that quantum theory is compatible with the existence of a new kind of operations where the arrow of time is indefinite. We explicitly construct one such operation, called the quantum time flip, and show that it cannot be realised by any quantum circuit with a definite direction of time. The quantum time flip offers an advantage in a game where a referee challenges a player to identify a hidden relation between two gates, and can be experimentally simulated with photonic systems, shedding light on the information-processing capabilities of exotic scenarios in which the arrow of time is in a quantum superposition.
Recently, the possible existence of quantum processes with indefinite causal order has been extensively discussed, in particular using the formalism of process matrices. Here we give a new perspective on this question, by establishing a direct connection to the theory of multi-time quantum states. Specifically, we show that process matrices are equivalent to a particular class of pre- and post- selected quantum states. This offers a new conceptual point of view to the nature of process matrices. Our results also provide an explicit recipe to experimentally implement any process matrix in a probabilistic way, and allow us to generalize some of the previously known properties of process matrices. Furthermore we raise the issue of the difference between the notions of indefinite temporal order and indefinite causal order, and show that one can have indefinite causal order even with definite temporal order.
This paper investigates the relationship between subsystems and time in a closed nonrelativistic system of interacting bosons and fermions. It is possible to write any state vector in such a system as an unentangled tensor product of subsystem vectors, and to do so in infinitely many ways. This requires the superposition of different numbers of particles, but the theory can describe in full the equivalence relation that leads to a particle-number superselection rule in conventionally defined subsystems. Time is defined as a functional of subsystem changes, thus eliminating the need for any reference to an external time variable. The dynamics of the unentangled subsystem decomposition is derived from a variational principle of dynamical stability, which requires the decomposition to change as little as possible in any given infinitesimal time interval, subject to the constraint that the state of the total system satisfy the Schroedinger equation. The resulting subsystem dynamics is deterministic. This determinism is regarded as a conceptual tool that observers can use to make inferences about the outside world, not as a law of nature. The experiences of each observer define some properties of that observers subsystem during an infinitesimal interval of time (i.e., the present moment); everything else must be inferred from this information. The overall structure of the theory has some features in common with quantum Bayesianism, the Everett interpretation, and dynamical reduction models, but it differs significantly from all of these. The theory of information described here is largely qualitative, as the most important equations have not yet been solved. The quantitative level of agreement between theory and experiment thus remains an open question.
To study which are the most general causal structures which are compatible with local quantum mechanics, Oreshkov et al. introduced the notion of a process: a resource shared between some parties that allows for quantum communication between them without a predetermined causal order. These processes can be used to perform several tasks that are impossible in standard quantum mechanics: they allow for the violation of causal inequalities, and provide an advantage for computational and communication complexity. Nonetheless, no process that can be used to violate a causal inequality is known to be physically implementable. There is therefore considerable interest in determining which processes are physical and which are just mathematical artefacts of the framework. Here we make the first step in this direction, by proposing a purification postulate: processes are physical only if they are purifiable. We derive necessary conditions for a process to be purifiable, and show that several known processes do not satisfy them.
In quantum mechanics events can happen in no definite causal order: in practice this can be verified by measuring a causal witness, in the same way that an entanglement witness verifies entanglement. Indefinite causal order can be observed in a quantum switch, where two operations act in a quantum superposition of the two possible orders. Here we realise a photonic quantum switch, where polarisation coherently controls the order of two operations, $hat{A}$ and $hat{B}$, on the transverse spatial mode of the photons. Our setup avoids the limitations of earlier implementations: the operations cannot be distinguished by spatial or temporal position. We show that our quantum switch has no definite causal order, by constructing a causal witness and measuring its value to be 18 standard deviations beyond the definite-order bound.
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