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arXiv:2208.08141 · Physical Review A

Sequential generalized measurements: Asymptotics, typicality, and emergent projective measurements

Wen-Long Ma, Shu‐Shen Li, Ren‐Bao Liu

Published Jan 25, 2023• 9 citations
arXivHTMLPDF

Abstract

The relation between projective measurements and generalized quantum measurements is a fundamental problem in quantum physics, and clarifying this issue is also important to quantum technologies. While it has been intuitively known that projective measurements can be constructed from sequential generalized or weak measurements, there is still lack of a proof of this hypothesis in general cases. Here we prove it from the perspective of quantum channels. We show that projective measurements naturally arise from sequential generalized measurements in the asymptotic limit, when the measurement operators are normal and commuting with each other. Specifically, a selective projective measurement arises from a set of typical sequences of selective generalized measurements. We also provide an explicit scheme to construct projective measurements of a quantum system with sequential generalized measurements. Remarkably, a single ancilla qubit is sufficient to mediate sequential generalized measurements for constructing arbitrary projective measurements of a generic system, which can have applications in readout, initialization, and feedback control of a quantum system. As an example, we present a protocol to measure the modular excitation numbers of a bosonic mode with an ancilla qubit.

DOI
10.1103/physreva.107.012217

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The relation between projective measurements and generalized quantum measurements is a fundamental problem in quantum physics, and clarifying this issue is also important to quantum technologies. While it has been intuitively known that projective measurements can be constructed from sequential generalized or weak measurements, there is still lack of a proof of this hypothesis in general cases. Here we prove it from the perspective of quantum channels. We show that projective measurements naturally arise from sequential generalized measurements in the asymptotic limit, when the measurement operators are normal and commuting with each other. Specifically, a selective projective measurement arises from a set of typical sequences of selective generalized measurements. We also provide an explicit scheme to construct projective measurements of a quantum system with sequential generalized measurements. Remarkably, a single ancilla qubit is sufficient to mediate sequential generalized measurements for constructing arbitrary projective measurements of a generic system, which can have applications in readout, initialization, and feedback control of a quantum system. As an example, we present a protocol to measure the modular excitation numbers of a bosonic mode with an ancilla qubit.
It is well known that any projective measurement can be decomposed into a sequence of weak measurements, which cause only small changes to the state. Similar constructions for generalized measurements, however, have relied on the use of an ancilla system. We show that any generalized measurement can be decomposed into a sequence of weak measurements without the use of an ancilla, and give an explicit construction for these weak measurements. The measurement procedure has the structure of a random walk along a curve in state space, with the measurement ending when one of the end points is reached. This shows that any measurement can be generated by weak measurements, and hence that weak measurements are universal. This may have important applications to the theory of entanglement.
Quantum measurement can be strong or weak, the former being important in readout and initialization of quantum objects and the latter being useful for monitoring and maneuvering quantum evolution. However, the boundary between weak and strong measurement is unclear. Here we show that a phase transition occurs in sequential quantum measurement, which unambiguously separates the weak and strong measurement by a critical value of measurement strength or duration. We formulate the probability distribution of the output of a sequence of quantum measurements as the Boltzmann distribution of an interacting spin model. The measurement results present phase transitions similar to those in the spin model. In particular the sequential commuting positive operator-valued measurement is mapped to a long-range Ising model, and a projective measurement emerges from sequential weak measurements when the strength or the number of measurements becomes above certain critical values, corresponding to the ferromagnetic phase transition of the spin model. This finding sheds insights on sequential quantum measurement, and also provides the theoretical foundation for constructing projective measurements from sequential weak measurements, which have applications in steering the quantum evolution and initializing quantum systems where strong measurement in a single shot is often not possible.
The prevailing consensus is that the sequential sharing of nonlocality in a Bell experiment requires generalized unsharp measurements, given that a sharp measurement inevitably destroys the entanglement of the shared state. In contrast, a recent work [A. Steffinlongo and A. Tavakoli, Projective measurements are sufficient for recycling nonlocality, Phys. Rev. Lett. 129, 230402 (2022)PRLTAO0031-900710.1103/PhysRevLett.129.230402] demonstrated the sharing of nonlocality up to two sequential observers by employing projective measurement aided with local randomness. Here, we introduce a form of local randomness-assisted qubit projective measurement protocol that enables the sharing of nonlocality by an arbitrary number of sequential observers (Bobs) with a single spatially separated observer (Alice). We achieve this by inspecting the diversity involved in implementing generalized measurements that harness nonlocality by preserving a sufficient amount of entanglement of the shared two-qubit entangled state. Furthermore, we highlight the crucial interplay between the degrees of measurement incompatibility of Alice and Bob in demonstrating the unbounded sharing of nonlocality.
The projection postulate and spectral representation of observables remain central in quantum theory but describe only an idealized measurement limit. Realistic measurements may be inefficient, unsharp, weak, indirect, adaptive, destructive, or informationally incomplete. Positive-operator-valued measures (POVMs) provide the probability rule for processes, whereas quantum instruments specify state transformation. This review traces the hierarchy from von Neumann projective measurement to generalized measurement, emphasizing distinctions among effects, instruments, channels, and implementations. Naimark dilation realizes arbitrary POVMs through unitary coupling to an ancillary meter and projective readout. Stinespring dilation represents completely positive trace-preserving dynamics by unitary evolution on an enlarged system followed by partial trace. A qubit–ancilla interaction yields a partial-measurement instrument, its POVM effects, and its information–disturbance trade-off. It compares projective, generalized, and ancilla-mediated schemes; examines compatibility, instrument incompatibility, and disturbance; and surveys applications to state discrimination, tomography, control, and error correction. Recent developments in dynamic-circuit measurements and noisy-POVM compatibility are integrated with questions concerning measurement back-action and outcome definiteness. The review stresses that a POVM does not uniquely determine state update, dilation theorems establish realizability rather than unique mechanisms, and operational measurement theory does not settle definite outcomes, arguing quantum instruments provide a complete description of measurement.
The problem of estimating a parameter of a quantum system through a series of measurements performed sequentially on a quantum probe is analyzed in the general setting where the underlying statistics is explicitly non-i.i.d. We present a generalization of the central limit theorem in the present context, which under fairly general assumptions shows that as the number N of measurement data increases the probability distribution of functionals of the data (e.g., the average of the data) through which the target parameter is estimated becomes asymptotically normal and independent of the initial state of the probe. At variance with the previous studies (Guţă M 2011 Phys. Rev. A 83 062324 ; van Horssen M and Guţă M 2015 J. Math. Phys. 56 022109 ) we take a diagrammatic approach, which allows one to compute not only the leading orders in N of the moments of the average of the data but also those of the correlations among subsequent measurement outcomes. In particular our analysis points out that the latter, which are not available in usual i.i.d. data, can be exploited in order to improve the accuracy of the parameter estimation. An explicit application of our scheme is discussed by studying how the temperature of a thermal reservoir can be estimated via sequential measurements on a quantum probe in contact with the reservoir.
For systems of controllable qubits, we provide a method for experimentally obtaining a useful class of multitime correlators using sequential generalized measurements of arbitrary strength. Specifically, if a correlator can be expressed as an average of nested (anti)commutators of operators that square to the identity, then that correlator can be determined exactly from the average of a measurement sequence. As a relevant example, we provide quantum circuits for measuring multiqubit out-of-time-order correlators using optimized control-ZZZ or ZX\mathit{ZX}ZX-90 two-qubit gates common in superconducting transmon implementations.