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arXiv:1312.4240

Direct observation of any two-point quantum correlation function

Francesco Buscemi, Michele Dall’Arno, Masanao Ozawa, Vlatko Vedral

Published Dec 16, 2013• 9 citations
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Abstract

The existence of noncompatible observables in quantum theory makes a direct operational interpretation of two-point correlation functions problematic. Here we challenge such a view by explicitly constructing a measuring scheme that, independently of the input state $ρ$ and observables $A$ and $B$, performs an unbiased optimal estimation of the two-point correlation function $\operatorname{Tr}[A \ ρ\ B]$. This shows that, also in quantum theory, two-point correlation functions are as operational as any other expectation value. A very simple probabilistic implementation of our proposal is presented.

DOI
10.48550/arxiv.1312.4240

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The existence of noncompatible observables in quantum theory makes a direct operational interpretation of two-point correlation functions problematic. Here we challenge such a view by explicitly constructing a measuring scheme that, independently of the input state ρρρ and observables AAA and BBB, performs an unbiased optimal estimation of the two-point correlation function Tr⁡[A ρ B]\operatorname{Tr}[A \ ρ\ B]Tr[A ρ B]. This shows that, also in quantum theory, two-point correlation functions are as operational as any other expectation value. A very simple probabilistic implementation of our proposal is presented.
Recently, a novel operational strategy to access quantum correlation functions of the form Tr[AρB] was provided in [F. Buscemi, M. Dall'Arno, M. Ozawa and V. Vedral, arXiv:1312.4240]. Here we propose a realization scheme, that we call partial expectation values, implementing such strategy in terms of a unitary interaction with an ancillary system followed by the measurement of an observable on the ancilla. Our scheme is universal, being independent of ρ, A, and B, and it is optimal in a statistical sense. Our scheme is suitable for implementation with present quantum optical technology, and provides a new way to test uncertainty relations.
We introduce a measure Q of bipartite quantum correlations for arbitrary two-qubit states, expressed as a state-independent function of the density matrix elements. The amount of quantum correlations can be quantified experimentally by measuring the expectation value of a small set of observables on up to four copies of the state, without the need for a full tomography. We extend the measure to 2×d systems, providing its explicit form in terms of observables and applying it to the relevant class of multiqubit states employed in the deterministic quantum computation with one quantum bit model. The number of required measurements to determine Q in our scheme does not increase with d. Our results provide an experimentally friendly framework to estimate quantitatively the degree of general quantum correlations in composite systems.
Experimental characterizations of a quantum system involve the measurement of expectation values of observables for a preparable state $\ensuremath{\mid}\ensuremath{\psi}⟩$ of the quantum system. Such expectation values can be measured by repeatedly preparing $\ensuremath{\mid}\ensuremath{\psi}⟩$ and coupling the system to an apparatus. For this method, the precision of the measured value scales as 1N\frac{1}{\sqrt{N}}N​1​ for NNN repetitions of the experiment. For the problem of estimating the parameter $\ensuremath{\phi}$ in an evolution ${e}^{\ensuremath{-}i\ensuremath{\phi}H}$, it is possible to achieve precision 1N\frac{1}{N}N1​ [the quantum metrology limit; see Giovannetti et al., Phys. Rev. Lett. 96, 010401 (2006)] provided that sufficient information about HHH and its spectrum is available. We consider the more general problem of estimating expectations of operators AAA with minimal prior knowledge of AAA. We give explicit algorithms that approach precision 1N\frac{1}{N}N1​ given a bound on the eigenvalues of AAA or on their tail distribution. These algorithms are particularly useful for simulating quantum systems on quantum computers because they enable efficient measurement of observables and correlation functions. Our algorithms are based on a method for efficiently measuring the complex overlap of $\ensuremath{\mid}\ensuremath{\psi}⟩$ and $U\ensuremath{\mid}\ensuremath{\psi}⟩$, where UUU is an implementable unitary operator. We explicitly consider the issue of confidence levels in measuring observables and overlaps and show that, as expected, confidence levels can be improved exponentially with linear overhead. We further show that the algorithms given here can typically be parallelized with minimal increase in resource usage.
We introduce a procedure based on quantum expectation values of measurement observables to characterize quantum coherence. In a bipartite system, the two-point correlation function is used in combination with the Schatten-1 norm to estimate the quantum coherence. This definition of coherence allows the decomposition into contributions corresponding to the non-classical correlations between the subsystems and localized on each subsystem. We illustrate the method for several bipartite systems, showing that an incomplete basis still gives a qualitative measure of the coherence.
Accurately estimating expectation values of quantum observables with as few measurements as possible is crucial to many quantum computing applications. We introduce a framework that covers many of existing measurement strategies and introduce heuristics that can be used to enhance randomized schemes, including those based on Pauli grouping with inverse probability weighting and variants of the classical shadow algorithm. We show how to maximize information gain from such schemes, while carefully optimizing the distribution of possible measurements, and show that simple grouping algorithms can get close to, and in some cases exceed, state-of-the-art accuracy for unbiased estimation of expectation values on a standard quantum chemistry benchmark. We show how these randomized methods may be compared to more recent measurement schemes, such as shadow grouping, derandomized shadow, and overlapped grouping measurement, we show how the same strategies can be used to augment these schemes, and we demonstrate that we can reduce measurement costs by up to a factor of two by allowing Clifford measurement circuits for otherwise Clifford-less methods.
In Quantum Physics it is not always possible to directly perform the measurement of an obsevable; in some of these cases, however, its value can be {\sl detected}, i.e. it can be inferred by measuring {\sl another} observable characterized by perfect correlation with the observable of interest. Though a {\sl detection} is often interpreted as a {\sl measurement} of the detected observable, we prove that the two concepts cannot be identified in Quantum Physics. Furthermore, we establish what meaning and role can be ascribed to detections coherently with Quantum Theory.