Quantum circuits with classical versus quantum control of causal order

Abstract

Quantum supermaps are transformations that map quantum operations to quantum operations. It is known that quantum supermaps which respect a definite, predefined causal order between their input operations correspond to fixed-order quantum circuits, also called quantum combs. A systematic understanding of the physical interpretation of more general types of quantum supermaps-in particular, those incompatible with a definite causal structure-is however lacking. In this paper, we identify two new types of circuits that naturally generalise the fixed-order case and that likewise correspond to distinct classes of quantum supermaps, which we fully characterise. We first introduce "quantum circuits with classical control of causal order ", in which the order of operations is still well-defined, but not necessarily fixed in advance: it can in particular be established dynamically, in a classically-controlled manner, as the circuit is being used. We then consider "quantum circuits with quantum control of causal order ", in which the order of operations is controlled coherently. The supermaps described by these classes of circuits are physically realisable, and the latter encompasses all known examples of physically realisable processes with indefinite causal order, including the celebrated "quantum switch". Interestingly, it also contains new examples arising from the combination of dynamical and coherent control of causal order, and we detail explicitly one such process. Nevertheless, we show that quantum circuits with quantum control of causal order can only generate "causal" correlations, compatible with a well-defined causal order. We furthermore extend our considerations to probabilistic circuits that produce also classical outcomes, and we demonstrate by an example how the characterisations derived in this work allow us to identify new advantages for quantum information processing tasks that could be demonstrated in practice.

Measuring time with stationary quantum clocks

“…The substantival theory of time would say that there was a well-defined time between the two flashes of light while the relative theory would say that time ceased to exist between the two flashes of light, at least from the perspective of inside the region of space and time. 2 However, now suppose that there was a clock which was stationary to begin with and the 1st event applied the unitary U to it, while the 2nd event is responsible for applying U m and invoking an appropriate measurement on the clock. Our counterfactual clock protocol would have been realised, and if one of the interactionfree outcomes were obtained, the flow of time would have been detected, quantified and recorded, without any dynamics between the two events occurring, thus allowing for experimental verification of the substantival theory of time.…”

“…which holds for both p = 0, 1 since G σ is a symmetric function. Similarly, we have Dif (2) p (σ) :=…”

“…which, similarly to Dif (1) p (σ), is observed to be t 1 independent. For the special case of one tick described in the main text ( x 0 = 1, θ = −1, N T = 1, p = 0, 1), the expression reduces to Dif (2) p (σ) =…”

“…Quantum mechanics, on the other hand, while very strange and mysterious in many ways, did not bring any novel insights as far as time is concerned: time is just a parameter which increases in line with any mundane classical clock -just like in Newton's laws. Even more recently, it has been proven, in the context of quantum mechanics, that the time-analogue of Bell's inequality always has a perfectly sound classical explanation [1,2].…”

The quantum switch is uniquely defined by its action on unitary operations

“…In order to tackle more general processes, it would be useful to find a simple criterium to identify which processes are uniquely defined by its action on unitary operations. Here particular cases of interest would be to analyze the n-slot quantum switch [9], and processes which may be viewed as generalizations of the switch, such as the ones which can be described by coherent control of causal order [25,30,31].…”