On the Brodutch and Modi method of constructing geometric measures of classical and quantum correlations
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Abstract
Recently, Brodutch and Modi proposed a general method of constructing meaningful measures of classical and quantum correlations. We systematically apply this method to obtain geometric classical and quantum correlations based on the Bures and the trace distances for twoqubit Bell diagonal states. Moreover, we argue that in general the Brodutch and Modi method may provide nonunique results, and we show how to modify this method to avoid this issue.
Keywords
Quantum correlations Classical correlations Geometric measures of correlations Bell diagonal states1 Introduction
In quantum information science, the problem of classification and quantification of correlations present in quantum states has been widely studied in the last two decades [1, 2]. In this regard, the most significant progress has been made in the case of bipartite quantum systems which have been studied initially in the entanglementseparability paradigm that was first formalized by Werner [3]. Within this paradigm, the correlations present in a quantum state can be classified as either classical or quantum, where the latter ones are identified with entanglement that can be quantified by a variety of entanglement measures [1].
However, it has gradually become clear that entanglement cannot be regarded as the only kind of quantum correlations, because separable quantum states can also have quantum correlations, other than entanglement, that are responsible for the improvements of some quantum tasks that cannot be simulated by classical methods [4, 5, 6, 7, 8, 9, 10]. Therefore, it has become evident that the entanglementseparability paradigm should be replaced by a new one.
The first step in this direction was taken independently by Ollivier and Zurek who introduced quantum discord as an informationtheoretic measure of quantum correlations beyond entanglement [11] and by Henderson and Vedral who studied the problem of separation of classical and quantum correlations from an informationtheoretic perspective [12]. Due to the discovery [13] that quantum discord may be the key resource in the deterministic quantum computation with one qubit [4], the problem of classification and quantification of correlations present in quantum states has been extensively studied within the informationtheoretic paradigm [2].
Because quantum discord cannot be computed analytically even for arbitrary twoqubit states [2], an alternative approach to classification and quantification of correlations within the informationtheoretic paradigm has been proposed in which different types of correlations are quantified by a distance from a given quantum state to the closest state which does not have the desired property [14]. Of course, within this approach the amount of quantum correlations present in a given quantum state is determined by the choice of distance measure for quantum states.
The first geometric measure of quantum correlations was geometric quantum discord in which the Schatten 2norm has been applied as the distance measure between a given quantum state and the closest zero discord state to obtain the analytical expression for geometric quantum discord for a general twoqubit state [15]. Of course, geometric quantum discord has attracted considerable interest due to its analytic computability for general twoqubit states [2].
However, recently it has been shown that geometric quantum discord cannot be regarded as a bona fide measure of quantum correlations [16], because of the lack of contractivity of the Schatten 2norm under tracepreserving quantum channels [17]. Moreover, it has turned out that among all geometric quantum discords based on the Schatten pnorms [18] only geometric quantum discord based on the Schatten 1norm is a meaningful measure of quantum correlations [17].
The problems with geometric quantum discord have highlighted the need for a general method of constructing meaningful measures of correlations within the informationtheoretic paradigm.
Recently, Brodutch and Modi [19] proposed a method in which quantum correlations are quantified by a distance between a given multipartite quantum state and the classicalquantum state emerging from a measurement performed on the considered state, where the measurement is chosen according to some strategy. Moreover, within this method classical correlations are quantified by a distance between the classicalquantum state and the completely separable state resulting from the same measurement performed on the tensor product of the states of the individual subsystems. Furthermore, Brodutch and Modi [19] identified the strategies that provide meaningful measures of classical and quantum correlations that satisfy the following necessary conditions: product states have no correlations, all correlations are invariant under local unitary operations, all correlations are nonnegative, and classical states have no quantum correlations.
The purpose of this paper is twofold. First, we systematically apply the Brodutch and Modi method to obtain for the first time geometric classical and quantum correlations based on the Bures distance for twoqubit Bell diagonal states using two natural strategies for constructing meaningful measures of correlations. Second, we consider geometric classical and quantum correlations based on the trace distance for twoqubit Bell diagonal states to show that the Brodutch and Modi method should be modified as in this case one of two possible strategies results in the nonuniqueness of geometric classical correlations. Moreover, we show how to modify the Brodutch and Modi method to avoid the problem of nonunique results in the general case.
2 Geometric classical and quantum correlations based on the Bures distance
In this section, we systematically apply the Brodutch and Modi method to obtain geometric classical and quantum correlations based on the Bures distance for twoqubit Bell diagonal states using two natural strategies for constructing meaningful measures of correlations.

M is a nonselective rank1 projective measurement performed on one subsystem of the multipartite system in a state \(\rho \), and M minimizes the quantum correlations \(Q(\rho )\),

M is a nonselective rank1 projective measurement performed on one subsystem of the multipartite system in a state \(\rho \), and M maximizes the classical correlations \(C(\rho )\).
2.1 Strategy 1
In the framework of the first strategy, for a given twoqubit Bell diagonal state (5) we first identify the measurements M that minimize the geometric quantum correlations (3) and then we use these optimal measurements to compute the geometric classical correlations (4). In other words, for a given point \((c_{1}, c_{2}, c_{3})\) of tetrahedron (6) we first identify unit vectors \((n_{1}, n_{2}, n_{3})\) that maximize the fidelity (9) and then we use these optimal vectors to compute the fidelity (12).
 if \(c_{i}^{2} = c_{j}^{2} = c_{k}^{2}\) where \(i \ne j \ne k\), then all measurements M are optimal, and geometric quantum and classical correlations are given by$$\begin{aligned} Q_{B}(\rho )&= \sqrt{2  2\sqrt{w_{k}}}, \end{aligned}$$(13a)$$\begin{aligned} C_{B}(\rho )&= \sqrt{2  \sqrt{2 + 2\sqrt{1  c_{k}^{2}}}}, \end{aligned}$$(13b)

if \(c_{i}^{2} = c_{j}^{2} < c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(n_{i}^{2} = n_{j}^{2} = 0\) and \(n_{k}^{2} = 1\) are optimal, and geometric quantum and classical correlations are given by Eqs. (13),

if \(c_{i}^{2} < c_{j}^{2} = c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(n_{i}^{2} = 0\) and \(n_{j}^{2} + n_{k}^{2} = 1\) are optimal, and geometric quantum and classical correlations are given by Eqs. (13),

if \(c_{i}^{2}< c_{j}^{2} < c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(n_{i}^{2} = n_{j}^{2} = 0\) and \(n_{k}^{2} = 1\) are optimal, and geometric quantum and classical correlations are given by Eqs. (13),
Let us note here that the problem of identification of unit vectors \((n_{1}, n_{2}, n_{3})\) maximizing the fidelity (9) for a given point \((c_{1}, c_{2}, c_{3})\) of tetrahedron (6) is closely related to the problem of finding classicalquantum states \(\chi _{\rho }\) that maximize the fidelity \(F(\rho , \chi _{\rho })\). This problem has been studied in the literature [21, 22] in the context of the Bures geometric quantum discord in which the Bures distance was applied as the distance measure between a given quantum state and the closest classicalquantum state. Interestingly, in general the Bures geometric quantum discord \(D_{B}(\rho )\) [23] is less than or equal to geometric quantum correlations based on the Bures distance under the first strategy \(Q_{B}(\rho )\), i.e., \(D_{B}(\rho ) \le Q_{B}(\rho )\), since \(M(\rho )\) is always a classicalquantum state for a bipartite state \(\rho \) [24]. Of course, the relation \(D_{B}(\rho ) \le Q_{B}(\rho )\) holds in the case of twoqubit Bell diagonal states as one can verify taking into account our results regarding \(Q_{B}(\rho )\) and those found in the literature regarding \(D_{B}(\rho )\) [23]. Moreover, one can show that this inequality becomes an equality if and only if \(\rho = M(\rho )\) for optimal measurement M or \(\rho \) is a mixture of two Bell states. It is also worth noting that considering the Bures geometric quantum discord for twoqubit Bell diagonal states one cannot uniquely determine the closest classicalquantum state for a wide class of twoqubit Bell diagonal states [22]. More precisely, it can be done uniquely if and only if a twoqubit Bell diagonal state is represented by the point \((c_{1}, c_{2}, c_{3})\) being interior point of tetrahedron (6) and the index k such that \(c_{k}^2 = \max (c_{1}^{2}, c_{2}^2, c_{3}^{2})\) is uniquely given. For comparison, in the case of geometric quantum correlations based on the Bures distance under the first strategy the classicalquantum state \(M(\rho )\) can be uniquely determined if and only if a twoqubit Bell diagonal state is represented by the point \((c_{1}, c_{2}, c_{3})\) of tetrahedron (6) and the index k such that \(c_{k}^2 = \max (c_{1}^{2}, c_{2}^2, c_{3}^{2})\) is uniquely given, as it was shown above. Moreover, it is worth noting that an alternative approach to geometric classical correlations based on the Bures distance under the first strategy has been considered in the literature [23]. The classical correlations measure based on the Bures distance introduced in [23] can be computed analytically for twoqubit Bell diagonal states, like geometric classical correlations based on the Bures distance under the first strategy. However, it can be shown that they are not directly comparable measures of classical correlations, because unlike for \(D_{B}(\rho )\) and \(Q_{B}(\rho )\) for which the relation \(D_{B}(\rho ) \le Q_{B}(\rho )\) holds, a similar relation, valid for all twoqubit Bell diagonal states, cannot be established between these two measures of classical correlations.
2.2 Strategy 2
In the framework of the second strategy, for a given twoqubit Bell diagonal state (5) we first identify the measurements M that maximize the geometric classical correlations (4) and then we use these optimal measurements to compute the geometric quantum correlations (3). In other words, for a given point \((c_{1}, c_{2}, c_{3})\) of tetrahedron (6) we first identify unit vectors \((n_{1}, n_{2}, n_{3})\) that minimize the fidelity (12) and then we use these optimal vectors to compute the fidelity (9).
 if \(c_{i}^{2} = c_{j}^{2} = c_{k}^{2}\) where \(i \ne j \ne k\), then all measurements M are optimal, and geometric quantum and classical correlations are given by$$\begin{aligned} Q_{B}(\rho )&= \sqrt{2  2\sqrt{w_{k}}}, \end{aligned}$$(15a)$$\begin{aligned} C_{B}(\rho )&= \sqrt{2  \sqrt{2 + 2\sqrt{1  c_{k}^{2}}}}, \end{aligned}$$(15b)

if \(c_{i}^{2} = c_{j}^{2} < c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(n_{i}^{2} = n_{j}^{2} = 0\) and \(n_{k}^{2} = 1\) are optimal, and geometric quantum and classical correlations are given by Eqs. (15),

if \(c_{i}^{2} < c_{j}^{2} = c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(n_{i}^{2} = 0\) and \(n_{j}^{2} + n_{k}^{2} = 1\) are optimal, and geometric quantum and classical correlations are given by Eqs. (15),

if \(c_{i}^{2}< c_{j}^{2} < c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(n_{i}^{2} = n_{j}^{2} = 0\) and \(n_{k}^{2} = 1\) are optimal, and geometric quantum and classical correlations are given by Eqs. (15),
3 Geometric classical and quantum correlations based on the trace distance
In this section, we consider geometric classical and quantum correlations based on the trace distance, induced by the Schatten 1norm, for twoqubit Bell diagonal states to show that the Brodutch and Modi method should be modified as in this case one of two possible strategies results in the nonuniqueness of geometric classical correlations. Moreover, we show how to modify the Brodutch and Modi method to avoid the nonunique results in the general case.
It is worth noting here that if \(\rho \) is a bipartite state and the measurement M is chosen according to the first strategy, then the geometric quantum correlations (16) coincide with the trace distance geometric quantum discord introduced in [17] if and only if the measured subsystem is a qubit [25]. Although the trace distance geometric quantum discord was evaluated explicitly for twoqubit Bell diagonal states [17, 25, 26], for these states all optimal measurements M and in consequence all classicalquantum states \(M(\rho )\) have not yet been identified.
3.1 Strategy 1
In the framework of the first strategy, for a given twoqubit Bell diagonal state (5) we first identify the measurements M that minimize the geometric quantum correlations (16) and then we use these optimal measurements to compute the geometric classical correlations (17). In other words, for a given point \((c_{1}, c_{2}, c_{3})\) of tetrahedron (6) we first identify unit vectors \((n_{1}, n_{2}, n_{3})\) that minimize \(\rho  M(\rho )_{1}\) given by Eq. (18) and then we use these optimal vectors to compute \(M(\rho )  M(\pi _{\rho })_{1}\) given by Eq. (19).
 if \(c_{i}^{2} = c_{j}^{2} = c_{k}^{2}\) where \(i \ne j \ne k\), then all measurements M are optimal, and geometric quantum and classical correlations are given by$$\begin{aligned} Q_{T}(\rho )&= c_{j}, \end{aligned}$$(20a)$$\begin{aligned} C_{T}(\rho )&= c_{k}, \end{aligned}$$(20b)

if \(c_{i}^{2} = c_{j}^{2} < c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(n_{i}^{2} = n_{j}^{2} = 0\) and \(n_{k}^{2} = 1\) are optimal, and geometric quantum and classical correlations are given by Eqs. (20),
 if \(c_{i}^{2} < c_{j}^{2} = c_{k}^{2}\) where \(i \ne j \ne k\), then all measurements M are optimal, and geometric quantum and classical correlations are given by (see Fig. 1)$$\begin{aligned} Q_{T}(\rho )&= c_{j}, \end{aligned}$$(21a)$$\begin{aligned} C_{T}(\rho )&= \sqrt{c_{k}^{2} + (c_{i}^{2}  c_{k}^{2}) n_{i}^{2}}, \end{aligned}$$(21b)
 if \(c_{i}^{2}< c_{j}^{2} < c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(0 \le n_{i}^{2} \le (c_{j}^{2}  c_{i}^{2})/(c_{k}^{2}  c_{i}^{2})\), \(n_{j}^{2} = 0\) and \(n_{k}^{2} = 1  n_{i}^{2}\) are optimal, and geometric quantum and classical correlations are given by (see Fig. 2)$$\begin{aligned} Q_{T}(\rho )&= c_{j}, \end{aligned}$$(22a)$$\begin{aligned} C_{T}(\rho )&= \sqrt{c_{k}^{2} + (c_{i}^{2}  c_{k}^{2}) n_{i}^{2}}. \end{aligned}$$(22b)
Let us note that the Broduch and Modi method can always result in uniquely determined measures of classical correlations under the first strategy of choosing measurement M, provided that we modify this strategy in the following way. If the classical correlations \(C(\rho )\) are not uniquely determined by the minimization procedure, then the classical correlations \(C(\rho )\) are additionally maximized over the all measurements M that minimizes the quantum correlations \(Q(\rho )\). Interestingly, a similar way of determining classical correlations was proposed in [23] where an alternative approach to geometric classical and quantum correlations based on the Bures distance was considered in the case of twoqubit Bell diagonal states.
 if \(c_{i}^{2} = c_{j}^{2} = c_{k}^{2}\) where \(i \ne j \ne k\), then all measurements M are optimal, and geometric quantum and classical correlations are given by$$\begin{aligned} Q_{T}(\rho )&= c_{j}, \end{aligned}$$(23a)$$\begin{aligned} C_{T}(\rho )&= c_{k}, \end{aligned}$$(23b)

if \(c_{i}^{2} = c_{j}^{2} < c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(n_{i}^{2} = n_{j}^{2} = 0\) and \(n_{k}^{2} = 1\) are optimal, and geometric quantum and classical correlations are given by Eqs. (23),

if \(c_{i}^{2} < c_{j}^{2} = c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(n_{i}^{2} = 0\) and \(n_{j}^{2} + n_{k}^{2} = 1\) are optimal, and geometric quantum and classical correlations are given by Eqs. (23),

if \(c_{i}^{2}< c_{j}^{2} < c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(n_{i}^{2} = n_{j}^{2} = 0\) and \(n_{k}^{2} = 1\) are optimal, and geometric quantum and classical correlations are given by Eqs. (23).
3.2 Strategy 2
In the framework of the second strategy, for a given twoqubit Bell diagonal state (5) we first identify the measurements M that maximize the geometric classical correlations (17) and then we use these optimal measurements to compute the geometric quantum correlations (16). In other words, for a given point \((c_{1}, c_{2}, c_{3})\) of tetrahedron (6) we first identify unit vectors \((n_{1}, n_{2}, n_{3})\) that maximize \(M(\rho )  M(\pi _{\rho })_{1}\) given by Eq. (19) and then we use these optimal vectors to compute \(\rho  M(\rho )_{1}\) given by Eq. (18).
 if \(c_{i}^{2} = c_{j}^{2} = c_{k}^{2}\) where \(i \ne j \ne k\), then all measurements M are optimal, and geometric quantum and classical correlations are given by$$\begin{aligned} Q_{T}(\rho )&= c_{j}, \end{aligned}$$(24a)$$\begin{aligned} C_{T}(\rho )&= c_{k}, \end{aligned}$$(24b)

if \(c_{i}^{2} = c_{j}^{2} < c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(n_{i}^{2} = n_{j}^{2} = 0\) and \(n_{k}^{2} = 1\) are optimal, and geometric quantum and classical correlations are given by Eqs. (24),

if \(c_{i}^{2} < c_{j}^{2} = c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(n_{i}^{2} = 0\) and \(n_{j}^{2} + n_{k}^{2} = 1\) are optimal, and geometric quantum and classical correlations are given by Eqs. (24),

if \(c_{i}^{2}< c_{j}^{2} < c_{k}^{2}\) where \(i \ne j \ne k\), then only measurements M with \(n_{i}^{2} = n_{j}^{2} = 0\) and \(n_{k}^{2} = 1\) are optimal, and geometric quantum and classical correlations are given by Eqs. (24).
The question whether the Brodutch and Modi method always results in uniquely determined measures of quantum correlations under the second strategy of choosing measurement M remains open for the future research. However, if the answer will be negative, then the second strategy of choosing measurement M should be modified in the following way. If the quantum correlations \(Q(\rho )\) are not uniquely determined by the maximization procedure, then the quantum correlations \(Q(\rho )\) are additionally minimized over the all measurements M that maximizes the classical correlations \(C(\rho )\).
4 Conclusions
In the framework of the Brodutch and Modi method, quantum correlations are quantified by a distance between a given multipartite state \(\rho \) and the classicalquantum state \(M(\rho )\) emerging from a measurement M performed on the considered state. However, classical correlations are quantified by a distance between the postmeasurement classicalquantum state \(M(\rho )\) and the completely separable state \(M(\pi _{\rho })\) resulting from the same measurement performed on the tensor product of the states of the individual subsystems. Within this method, there are two natural strategies of choosing a measurement M that provide meaningful measures of classical and quantum correlations present in a multipartite state \(\rho \). In both strategies, M is a nonselective projective measurement performed on one subsystem of the multipartite system in a state \(\rho \). However, the measurement M minimizes quantum correlations according to the first strategy, and it maximizes classical correlations according to the second one.
In this work, we have applied the Brodutch and Modi method to obtain for the first time geometric classical and quantum correlations based on the Bures distance under the two natural strategies of choosing the measurement M performed on the first qubit of the twoqubit system in a Bell diagonal state. Under the first strategy, we first identified the measurements M that minimize the geometric quantum correlations and then we used these optimal measurements to compute the geometric classical correlations, while under the second one, we first identified the measurements M that maximize the geometric classical correlations and then we used these optimal measurements to compute the geometric quantum correlations. For the both strategies, we have identified not only all optimal measurements M but also all twoqubit Bell diagonal states for which there exists more than one optimal measurement M. However, the nonuniqueness of \(M(\rho )\) does not affect the geometric classical and quantum correlations for the first and the second strategy, respectively. Remarkably, it turned out that the both strategies provide the same results with regard to the geometric classical and quantum correlations and the optimal measurements.
Moreover, we have shown that the Brodutch and Modi method should be modified as in general it may provide nonunique results. As an explicit example, we have applied the Brodutch and Modi method to obtain geometric classical and quantum correlations based on the trace distance under the two natural strategies of choosing the measurement M performed on the first qubit of the twoqubit system in a Bell diagonal state. For the both strategies, we have computed the geometric classical and quantum correlations, although the geometric quantum correlations under the first strategy had been studied in the literature. Moreover, for the both strategies, we have identified not only all optimal measurements M but also all twoqubit Bell diagonal states for which there exists more than one optimal measurement M. Remarkably, it turned out that for the first strategy the nonuniqueness of \(M(\rho )\) results in the nonuniqueness of geometric classical correlations, and therefore, the Brodutch and Modi method should be modified. Finally, we have shown how to modify the Brodutch and Modi method to avoid the problem of nonunique results with regard to classical and quantum correlations in the general case.
Notes
Acknowledgements
We thank Paweł Caban for helpful comments. This work was supported by the University of Lodz. Iwona Wintrowicz acknowledges the support from the European Union under the Human Capital Operational Programme, Measure 8.2.1.
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