In recent years, exploring the possible use of separable states as resource for achieving quantum information processing(QIP) tasks has been gaining increasing significance. In this context, a particularly important demonstration has been that non-vanishing discord is the necessary condition for the separable states to be used as resource for remotely preparing any arbitrary pure target state [Nature Physics 8888, 666(2012)6662012666(2012)666 ( 2012 )]. The present work stems from our observation that not only resource states with same discord can imply different efficiencies (in terms of average fidelity) of the remote state preparation (RSP) protocol, but also states with higher discord can imply lower RSP efficiency. This, therefore, necessitates identification of the relevant feature of quantum correlations which can appropriately quantify effectiveness of the resource state for the RSP protocol. To this end, for the two-qubit Bell-diagonal states, we show that an appropriate measure of simultaneous correlations in three mutually unbiased bases can serve to quantify usefulness of the resource for the RSP task using entangled as well as separable states, including non-discordant states as resource. In particular, it is revealed that zero-discord states having such non-vanishing measure can be useful for remotely preparing a subset of pure target states. Thus, this work shows that, using separable states, an effective resource for QIP tasks such as RSP can be provided by simultaneous correlations in mutually unbiased bases.
pacs:
03.65.Ud, 03.67.Mn, 03.65.Ta
The study of far-reaching implications of quantum foundational aspects of entanglement and its various applications in quantum information constitutes one of the most vibrant areas of research in contemporary science. Neverthless, it has been recognized that the paradigm of entanglement captures only a particular segment of correlations inherent in the quantum regime, and the study of useful correlations inherent in separable states, along with their applicability in quantum information processing(QIP) tasks, is attracting increasing attention. This is essentially because, compared to entangled states, separable states are easier to produce, manipulate and protect against decohering effects. In this context, various measures of ”quantumness” of correlations beyond entanglement have been suggested Ollivier and Zurek (2001); Guo and Wu (2014); Baumgratz et al. (2014), chief amongst them being quantum discordHenderson and Vedral (2001); Ollivier and Zurek (2001). It is also important to note that while entanglement has been established as a resource for QIP protocols using pure states Bennett et al. (1993); Bennett and Wiesner (1992), there exist examples of QIP tasks where bipartite separable states can act as resource. For instance, consider the protocol of quantum state merging where, given an unknown quantum state distributed over two systems, the task is to determine how much quantum communication is needed to transfer the complete state to one system Horodecki et al. (2005/08/04/online). It has been shown Horodecki et al. (2007); Madhok and Datta (2011) that quantum discord quantifies the usefulness of separable states in the context of this protocol. Further, consider the task of deterministic quantum computational protocol of efficient estimation of the trace of a unitary matrix, namely, what is known as the ”power of one qubit” model; it has been shown that quantum discord can be used as a figure of merit to characterize whether such a protocol is successfully implemented Datta et al. (2008). Girolami et.al. have also shown that quantum discord determines the necessary condition for bipartite qubit state to be useful for showing the so-called interferometric power of quantum states Girolami et al. (2014).
Against the above backdrop, focusing on a particular QIP task, viz, remote state preparation (RSP) Pati (2000); Lo (2000), the motivation underlying the present paper stems from the fact that while non-vanishing discord has been shown to provide the necessary condition for successful implementation of the RSP task Dakić et al. (2012), we’ve found that higher discord states do not necessarily imply higher efficiency (in terms of optimal quadratic fidelity) of the RSP protocol with respect to a specific target state. Gaming news Thus, there is a need for characterizing the efficiency of RSP for a general class of resource states, including entangled as well as separable states, in terms of a suitable measure of quantum correlations. On the other hand, for the analogous QIP task of teleportation, entanglement is deemed to be the necessary condition Bennett et al. (1993); Cavalcanti et al. (2017), as well as it characterizes the efficiency of the protocol Horodecki et al. (1999). The central result of this paper is that a suitable measure of efficiency of the RSP protocol (say, in terms of average optimal quadratic fidelity) is found to be in direct correspondence with an appropriate measure of simultaneous correlations in three mutually unbiased bases Guo and Wu (2014), which is non-zero for any non-product state. This, in particular, enables to capture effectiveness of the zero discord states for remotely preparing a class of pure target states. Further, given any two non-zero discord states as resource for remotely preparing a target state, we show that it is the simultaneous correlation in mutually unbiased bases that determines which particular resource state can be more useful than the other. Explanation of the reason for discord to fail as the quantifier of RSP efficiency is a highlight of the present paper.
The recently introduced measures of simultaneous correlations in mutually unbiased bases(SCMUB) inherent in the quantum state seek to quantify ”quantumness” by the persistence of correlations in incompatible bases used to measure the stateGuo and Wu (2014); Wu et al. (2014). Correlations in a given basis are quantified using the Holevo quantity, and comparing amongst incompatible bases their respective Holevo quantities, one can obtain a series of quantumness measures by choosing sets of bases that maximize the amount of simultaneous correlations in the sense whose precise meaning will be explained later. Incompatible bases used to define these measures of correlations are chosen to be mutually unbiased, meaning that if a particular member of a basis is measured, it is projected with equal probability onto all members of the corresponding incompatible basis. A class of these measuresGuo and Wu (2014) has been shown to vanish if and only if the state in question is a product state, which in turn raises the question as to whether the correlations captured using such measures can provide an effective characterization of efficiency in QIP tasks using separable states. An affirmative answer to this question is provided in the present paper in the context of a specific QIP task, namely, Remote State Preparation.
The plan of the paper is as follows In Section II we give an overview of the general RSP protocol along with the nuances regarding the condition pertaining to successful implementation of the RSP protocol and the role of quantum discord in this regard. This is followed by examples of resource states for which quantum discord cannot successfully explain the higher efficiency of the RSP protocol. In Section III we discuss the effectiveness/strength of the resource state pertaining to RSP protocol in terms of average optimal quadratic fidelity. In Section IV, a suitable measure of simultaneous correlations in mutually unbiased bases is identified and related to the measure of RSP efficiency. In Section V, we take zero discord states as resource and show in what way it can be used as an effective resource for remotely preparing a particular set of pure target states. The ramifications of our results are discussed in Section VI.
II Remote State Preparation (RSP)-an overview
The RSP protocol Pati (2000) seeks to demonstrate that by using a shared bipartite state as resource, a specified quantum state (hereby referred to as target state) can be prepared at a distant location by classical communication of local quantum measurement outcomes. Operationally, we can outline the RSP protocol as follows: Alice and Bob share a general bipartite qubit state given by
ρ=14(ℐ⊗ℐ+∑iaiσi⊗ℐ+∑jbjℐ⊗σj+∑kEkσk⊗σk)
