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arXiv:1109.5154 · Physical Review A

Perfect discrimination of no-signalling channels via quantum superposition of causal structures

Giulio Chiribella

Published Oct 10, 2012• 253 citations
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Abstract

A no-signalling channel transforming quantum systems in Alice's and Bob's local laboratories is compatible with two different causal structures: ($A⪯B$) Alice's output causally precedes Bob's input and ($B⪯A$) Bob's output causally precedes Alice's input. Here I prove that two no-signalling channels that are not perfectly distinguishable in any ordinary quantum circuit can become perfectly distinguishable through the quantum superposition of circuits with different causal structures.

DOI
10.1103/physreva.86.040301

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A no-signalling channel transforming quantum systems in Alice's and Bob's local laboratories is compatible with two different causal structures: (A⪯BA⪯BA⪯B) Alice's output causally precedes Bob's input and (B⪯AB⪯AB⪯A) Bob's output causally precedes Alice's input. Here I prove that two no-signalling channels that are not perfectly distinguishable in any ordinary quantum circuit can become perfectly distinguishable through the quantum superposition of circuits with different causal structures.
A causal relation between quantum agents, say Alice and Bob, is necessarily mediated by an interaction. Modeling the latter as a reversible quantum channel, an intervention of Alice can have causal influence on Bob's system, modifying correlations between Alice and Bob's systems. Causal influence between quantum systems necessarily allows for signaling. Here we prove a mismatch between causal influence and signaling via direct computation of the two quantities for the Cnot gate. Finally we show a continuity theorem for causal effects of unitary channels: a channel has small causal influence if it allows for small signaling. Published by the American Physical Society 2024
Recent developments in the formalisation of quantum causal structures have made it possible to test and compare hypotheses about causal structure empirically, rather than being a-priori assumptions. Such differences in causal structure may be leveraged to distinguish between the processes they belong to, akin to distinguishing between quantum states known to belong to different eigenspaces of an observable. So how far this analogy can be pushed? Can causal order be interpreted as a kind of observable? Can it be measured? To this end we construct a completely operational definition of observables in terms of `discrimination tasks'. This begins with some base set of classes (analogous to a quantum observable's eigenspaces) before we impose three conditions on how these classes relate under discrimination tasks. These conditions recover the properties of a standard observable, letting us apply this definition to non-state entities such as quantum processes, which naturally encode causal structure. These conditions can be described in plain language as (1) `members of any class are perfectly distinguishable from members of any other' (i.e. our classes are `sharp'), (2) `entities perfectly distinguishable pairwise are all perfectly distinguishable by a joint intervention', and (3) `if all members of a class are perfectly distinguishable from an entity, then the whole class is perfectly distinguishable from it jointly'. By analysing classes of quantum processes having strict causal orders, we find causal classes which are sharp but violate condition 2, and classes which violate condition 3. This implies that causal order cannot be an observable. We note cases in the literature which claim to prove the opposite, and discuss them. We find their analyses presume special circumstances equivalent to conditions (2) and (3), and so do not contradict our own findings.
We consider devices with two inputs and two outputs, Alice and Bob each having access to one input and one output. To such a device we associate time-reverses by exchanging the roles of the inputs and the outputs. We find that there are devices which admit a local hidden variable representation, but for which time-reverses enable perfect signaling between Alice and Bob. That is, a `perfect channel in one time direction' becomes a `non-channel in the other direction'. Also, for PR boxes time-reverses enable signaling between Alice and Bob, but never as a perfect channel. These results undermine the representation of causal structures as partial orders or similar `time-symmetric structures', as is often assumed in search of a theory of quantum gravity. They also provide new insights into the structure of the polytope of generalized probabilistic correlations, and contribute to the growing area of research into quantum information processing in relativistic spacetimes.
We show that quantum theory allows for transformations of black boxes that cannot be realized by inserting the input black boxes within a circuit in a predefined causal order. The simplest example of such a transformation is the classical switch of black boxes, where two input black boxes are arranged in two different orders conditionally on the value of a classical bit. The quantum version of this transformation---the quantum switch---produces an output circuit where the order of the connections is controlled by a quantum bit, which becomes entangled with the circuit structure. Simulating these transformations in a circuit with fixed causal structure requires either postselection or an extra query to the input black boxes.
We show that quantum theory allows for transformations of black boxes that cannot be realized by inserting the input black boxes within a circuit in a pre-defined causal order. The simplest example of such a transformation is the classical switch of black boxes, where two input black boxes are arranged in two different orders conditionally on the value of a classical bit. The quantum version of this transformation-the quantum switch-produces an output circuit where the order of the connections is controlled by a quantum bit, which becomes entangled with the circuit structure. Simulating these transformations in a circuit with fixed causal structure requires either postselection, or an extra query to the input black boxes.
The causal effects activated by a quantum interaction are studied, modelling the last one as a bipartite unitary channel. The two parties, say Alice and Bob, can use the channel to exchange messages -- i.e. to signal. On the other hand, the most general form of causal influence includes also the possibility for Alice, via a local operation on her system, to modify Bob's correlations and viceversa. The presence or absence of these two effects are equivalent, but when they both occur, they can differ in their magnitude. We study the properties of two functions that quantify the amount of signalling and causal influence conveyed by an arbitrary unitary channel. The functions are proved to be continuous and monotonically increasing with respect to the tensor product of channels. Monotonicity is instead disproved in the case of sequential composition. Signalling and causal influence are analytically computed for the quantum SWAP and CNOT gates, in the single use scenario, in the nnn-parallel uses scenario, and in the asymptotic regime. A finite gap is found between signalling and causal influence for the quantum CNOT, thus proving the existence of extra causal effects that cannot be explained in terms of communication only. However, the gap disappears in the asymptotic limit of an infinite number of parallel uses, leaving room for asymptotic equivalence between signalling and causal influence.