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arXiv:1409.2237 · International Journal of Quantum Information

Universal optimal quantum correlator

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

Published Oct 28, 2014• 18 citations
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Abstract

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.

DOI
10.1142/s0219749915600023

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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.
The expectation value of an arbitrary operator O can be obtained via a universal measuring apparatus that is independent of O, by changing only the data-processing of the outcomes. Such a ``universal detector'' performs a joint measurement on the system and on a suitable ancilla prepared in a fixed state, and is equivalent to a positive operator valued measure (POVM) for the system that is ``informationally complete''. The data processing functions generally are not unique, and we pose the problem of their optimization, providing some examples for covariant POVM's, in particular for SU(d) covariance group.
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.
Quantum algorithms are conventionally formulated for implementation on a single system of qubits amenable to projective measurements. However, in expectation value quantum computation, such as nuclear magnetic resonance realizations, the computer consists of an ensemble of identical qubit-systems amenable only to expectation value measurements. The prevalent strategy in such expectation value implementations of quantum algorithms has been to retain the conventional formulation's unitary operations but modify its initialization and measurement steps appropriately. This naive approach is not optimal for Grover's algorithm and a shortened version for expectation value quantum computers is presented.
The principle of complementarity is quantified in two ways: by a universal uncertainty relation valid for arbitrary joint estimates of any two observables from a given measurement setup and by a general uncertainty relation valid for the optimal estimates of the same two observables when the state of the system prior to measurement is known. A formula is given for the optimal estimate of any given observable, based on arbitrary measurement data and prior information about the state of the system, which generalizes and provides a more robust interpretation of previous formulas for ``local expectations'' and ``weak values'' of quantum observables. As an example, the canonical joint measurement of position XXX and momentum PPP corresponds to measuring the commuting operators ${X}_{J}=X+{X}^{\ensuremath{'}}$ and ${P}_{J}=P\ensuremath{-}{P}^{\ensuremath{'}}$, where the primed variables refer to an auxiliary system in a minimum-uncertainty state. It is well known that $\ensuremath{\Delta}{X}_{J}\ensuremath{\Delta}{P}_{J}\ensuremath{\geqslant}\ensuremath{\hbar}$. Here it is shown that given the same physical experimental setup and knowledge of the system density operator prior to measurement, one can make improved joint estimates Xopt{X}_{opt}Xopt​ and Popt{P}_{opt}Popt​ of XXX and PPP. These improved estimates are not only statistically closer to XXX and PPP but further satisfy $\ensuremath{\Delta}{X}_{opt}\ensuremath{\Delta}{P}_{opt}\ensuremath{\geqslant}\ensuremath{\hbar}∕4$, where equality can be achieved in certain cases. Thus one can do up to four times better than the standard lower bound (where the latter corresponds to the limit of no prior information). Other applications include the heterodyne detection of orthogonal quadratures of a single-mode optical field and joint measurements based on Einstein-Podolsky-Rosen correlations.
Abstract Quantum tomography is currently the mainly employed method to characterize a quantum system and therefore plays a fundamental role when trying to characterize the action of a particular channel. Nonetheless, quantum tomography works on the premise of a full characterization and description of the devices preparing the quantum state and realizing the measurements. Such an assumption was recently relaxed in Dall’Arno et al (arXiv: 1805.01159 ) and Dall’Arno et al (2017 Proc. R. Soc. A 473 20160721), where a theoretical framework for the device-independent inference of quantum channels was developed and experimentally implemented with superconducting qubits. Here, based on such a framework, we present a complete experimental test on a photonic setup of two semi-device-independent protocols that can be employed for the validation of the tomographic reconstruction or the characterization of a given quantum channel, not relying on many assumptions on the adopted device. Our implementation paves the way to the development of new experimental methods not relying on the assumptions typically taken for granted in all the previous protocols.
A new measure of non-classical correlations is introduced and characterized. It tests the ability of using a state ρ of a composite system AB as a probe for a quantum illumination task (e.g. see Lloyd 2008 Science 321 1463 ), in which one is asked to remotely discriminate between the two following scenarios: (i) either nothing happens to the probe, or (ii) the subsystem A is transformed via a local unitary whose properties are partially unspecified when producing ρ . This new measure can be seen as the discrete version of the recently introduced interferometric power measure (Girolami et al 2013 e-print arXiv: 1309.1472 ) and, at least for the case in which A is a qubit, it is shown to coincide (up to an irrelevant scaling factor) with the local quantum uncertainty measure of Girolami, Tufarelli and Adesso (2013 Phys. Rev. Lett. 110 240402 ). Analytical expressions are derived which allow us to formally prove that, within the set of separable configurations, the maximum value of our non-classicality measure is achieved over the set of quantum-classical states (i.e. states ρ which admit a statistical unravelling where each element of the associated ensemble is distinguishable via local measures on B ).