Discriminating strength: a bona fide measure of non-classical correlations

Abstract

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 R A 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).

Measures and applications of quantum correlations

“…Closed analytical formulae are available for both measures on all bipartite states ρ AB when subsystem A is a qubit [54,57], demonstrating that these measures reconcile the typically contrasting requirements of computability and reliability [96]. In this case (dropping the spectrum superscript without loss of generality as discussed in Section 3.2.5), the local quantum uncertainty Q A (ρ AB ) [58], as discussed in Section 3.7. Surprisingly, a similar equivalence does not hold between the interferometric power Q C QF A (ρ AB ) and the unitary response measure based on the Bures distance, and it is still unknown whether the former may admit any alternative interpretation in terms of a geometric or a unitary response type measure.…”

“…Notice the resemblance with the case of unitary response measures in Eq. (58). Similarly to such a case, in fact, it is not immediate to extend Eq.…”

“…The discriminating strength, whose defining operational interpretation in the context of channel discrimination will be discussed in Section 4.2.2, can be thus regarded as a unitary response measure of QCs based on the Chernoff distance D C (ρ, σ) := 1 − C(ρ, σ). The choice of the spectrum Γ in order to maximise the measure Q U Γ C A was analysed as well in [58]. When subsystem A is a qubit (in which case any spectrum leads to an equivalent measure up to a normalisation factor, hence the harmonic spectrum can be assumed without any loss of generality [54]), it was further shown in [58] that the discriminating strength coincides up to a constant multiplicative factor with the local quantum uncertainty [54], and thus with the Hellinger unitary response measure of QCs, via Eq.…”

Gaussian interferometric power

“…[24] it shows that their role in metrology transcends specific schemes and Hilbert space dimensions. The formalism applied here can be immediately useful to calculate other discord-type quantities for Gaussian states, which capture geometrically their sensitivity under local unitary transformations, e.g., the local quantum uncertainty [31], the discriminating strength [42], and the discord of response [43] (see also [44]). …”

Building versatile bipartite probes for quantum metrology

“…As another example, we could be asked to prepare a passe-partout probe state that must be good whenever the interaction with the measured system is described by a Hamiltonian picked at random from a given ensemble, so that we have no interest in optimizing the probe for a particular element of the ensemble. It turns out that in such and similar situations, that we may describe as instances of 'black-box' quantum metrology, the presence of correlations gives another fundamental advantage [6][7][8][9][10]. While with a single probe system we always run the risk of preparing the probe in a state which is left unmodified by some unlucky interaction mechanism with the target system, by exploiting correlations between the probe and an ancillary system kept as a reference we can instead guarantee a minimum detection efficiency.…”