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Register a new userLogical pre- and post-selection paradoxes are proofs of contextuality
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Matthew F. Pusey
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If a quantum system is prepared and later post-selected in certain states, paradoxical predictions for intermediate measurements can be obtained. This is the case both when the intermediate measurement is strong, i.e. a projective measurement with Luders-von Neumann update rule, or with weak measurements where they show up in anomalous weak values. Leifer and Spekkens [quant-ph/0412178] identified a striking class of such paradoxes, known as logical pre- and post-selection paradoxes, and showed that they are indirectly connected with contextuality. By analysing the measurement-disturbance required in models of these phenomena, we find that the strong measurement version of logical pre- and post-selection paradoxes actually constitute a direct manifestation of quantum contextuality. The proof hinges on under-appreciated features of the paradoxes. In particular, we show by example that it is not possible to prove contextuality without Luders-von Neumann updates for the intermediate measurements, nonorthogonal pre- and post-selection, and 0/1 probabilities for the intermediate measurements. Since one of us has recently shown that anomalous weak values are also a direct manifestation of contextuality [arXiv:1409.1535], we now know that this is true for both realizations of logical pre- and post-selection paradoxes.
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While quantum computers are expected to yield considerable advantages over classical devices, the precise features of quantum theory enabling these advantages remain unclear. Contextuality--the denial of a notion of classical physical reality--has emerged as a promising hypothesis. Magic states are quantum resources critical for practically achieving universal quantum computation. They exhibit the standard form of contextuality that is known to enable probabilistic advantages in a variety of computational and communicational tasks. Strong contextuality is an extremal form of contextuality describing systems that exhibit logically paradoxical behaviour. Here, we consider special magic states that deterministically enable quantum computation. After introducing number-theoretic techniques for constructing exotic quantum paradoxes, we present large classes of strongly contextual magic states that enable deterministic implementation of gates from the Clifford hierarchy. These surprising discoveries bolster a refinement of the resource theory of contextuality that emphasises the computational power of logical paradoxes.
We provide a cohomological framework for contextuality of quantum mechanics that is suited to describing contextuality as a resource in measurement-based quantum computation. This framework applies to the parity proofs first discussed by Mermin, as well as a different type of contextuality proofs based on symmetry transformations. The topological arguments presented can be used in the state-dependent and the state-independent case.
A recent quantum protocol for counterfactual communication [Y. Aharonov and L. Vaidman, Phys. Rev. A 99, 010103(R), 2019] relies on post-selection to eliminate the weak trace in the transmission channel. We show that the post-selection in this protocol also eliminates the flow of Fisher information from transmitter to receiver. However, we also show that a classical communication protocol with post-selection can be counterfactual. Hence, we argue that post-selection should not be allowed in genuine counterfactual communication. In the quantum counterfactual communication protocol, the probability of discarding an event by post-selection tends to zero with an increasing number of ideal optical components. But the counterfactual violation strength tends to infinity at a faster rate. Consequently, the quantum protocol is not counterfactual proper.
Many quantum state preparation methods rely on a combination of dissipative quantum state initialization, followed by unitary evolution to a desired target state. Here we demonstrate the usefulness of quantum measurement as an additional tool for quantum state preparation. Starting from a pure separable multipartite state, a control sequence, which includes rotation, spin squeezing via one-axis twisting, quantum measurement and post-selection, generates a highly entangled multipartite state, which we refer to as Projected Squeezed states (or PS states). Through an optimization method, we then identify parameters required to maximize the overlap fidelity of the PS states with the maximally entangled Greenberger-Horne-Zeilinger states (or GHZ states). The method leads to an appreciable decrease in state preparation time of GHZ states when compared to preparation through unitary evolution with one-axis twisting only.
A weak measurement performed on a pre- and post-selected quantum system can result in an average value that lies outside of the observables spectrum. This effect, usually referred to as an anomalous weak value, is generally believed to be possible only when a non-trivial post-selection is performed, i.e., when only a particular subset of the data is considered. Here we show, however, that this is not the case in general: in scenarios in which several weak measurements are sequentially performed, an anomalous weak value can be obtained without post-selection, i.e., without discarding any data. We discuss several questions that this raises about the subtle relation between weak values and pointer positions for sequential weak measurements. Finally, we consider some implications of our results for the problem of distinguishing different causal structures.
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