Unifying approach to the quantification of bipartite correlations by Bures distance
The notion of distance defined on the set of states of a composite quantum system can be used to quantify total, quantum and classical correlations in a unifying way. We provide new closed formulae for classical and total correlations of two-qubit Bell-diagonal states by considering the Bures distance. Complementing the known corresponding expressions for entanglement and more general quantum correlations, we thus complete the quantitative hierarchy of Bures correlations for Bell-diagonal states. We then explicitly calculate Bures correlations for two relevant families of states: Werner states and rank-2 Bell-diagonal states, highlighting the subadditivity which holds for total correlations with respect to the sum of classical and quantum ones when using Bures distance. Finally, we analyse a dynamical model of two independent qubits locally exposed to non-dissipative decoherence channels, where both quantum and classical correlations measured by Bures distance exhibit freezing phenomena, in analogy with other known quantifiers of correlations.
pacs:03.67.Mn, 03.65.Ud, 03.65.Yz
Characterising the different forms of correlations shared by the constituents of a composite quantum system is essential for the theoretical understanding and for the operational exploitation of the quantum system itself [1, 2]. Correlations which cannot be amenable to a classical description, in particular, exhibit a rich variety in mixed states of bipartite and multipartite quantum systems. Nowadays, the notion of quantum correlations refers not only to entanglement, but to more general forms of correlations which are conventionally identified with the quantum discord [3, 4], and capture for instance the necessary disturbance induced on quantum states by any local measurement (which is nonzero even in all separable states apart from so-called classical-quantum states). Correspondingly, the portion of correlations which are left in the state after a minimally disturbing local measurement can be identified with the classical correlations originally shared by the subsystems.
Unlike entanglement, for which a resource theory is well established , proposals to quantify quantum and classical correlations in this more general paradigm are still relatively scarce, and the mathematical requirements that any such proposal has to obey to be regarded as a valid measure are still to be completely formalised . Yet an interesting phenomenology associated to these correlations is being uncovered in different physical contexts. For example, from a foundational perspective the sudden transition from a decay to a plateau regime for classical correlations between a quantum system and its measurement apparatus has been interpreted as characterising the finite-time emergence of the pointer basis in the apparatus [5, 6]. Moreover, quantum correlations between noninteracting qubits have been shown to dynamically revive despite of decoherence thanks to memory effects of the local environment, independently of the quantum or classical nature of the environment [7, 8, 9, 10]. Quantum correlations of ground states also appear to play an important role in the characterisation of exotic phases of quantum many-body systems [11, 12, 13]. More generally, from an operational viewpoint various forms of quantum correlations, including and beyond entanglement, can and typically do provide fundamental resources for quantum technologies [1, 2]. Consequently, rigorously addressing the quantification of correlations is of paramount importance.
The notion of distance defined on the convex set of states of a quantum system paves the way for several geometric approaches to the quantification of correlations [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. We refer in particular to Refs. [20, 23] for a comparison among these approaches with respect to the quantification of quantum correlations. In this paper, we focus on bipartite systems and we follow the approach of Ref. , according to which the minimum distance between a state and the set of states that do not possess a particular kind of correlations is a quantifier of that kind of correlations. Hence, the minimum distances between the state of a bipartite system and the sets of product, classical-quantum and separable states represent, respectively, the amount of total correlations, quantum correlations and entanglement of . Furthermore, the minimum distance between the set of closest classical-quantum states to and the set of product states represents the classical correlations of . This geometric approach to the quantification of correlations manifests several appealing features. First, it is unifying, thus allowing for a direct comparison among all the above mentioned notions of correlations . Second, it readily suggests generalisations to the multipartite setting .
In this paper we use specifically the Bures distance on the set of states to define geometric quantifiers of correlations. The Bures distance is defined as
where and are two arbitrary states while is the Uhlmann fidelity 
The reason for choosing this distance instead of others stems from its quite peculiar, but desirable, properties. Bures distance is at the same time contractive under completely positive trace-preserving maps, locally Riemannian and its metric coincides with the quantum Fisher information [26, 27, 28, 29], thus playing a crucial role in high precision interferometry and quantum metrology. Moreover, the minimum Bures distance between a state and the set of classical-quantum states is simply related to the maximal success probability in the ambiguous quantum state discrimination of a family of states and prior probabilities depending on [30, 31]. The task of minimal error quantum state discrimination plays a fundamental role both in quantum communication and cryptography and has been realised experimentally using polarised light [32, 33]. On the contrary, e.g., the Hilbert-Schmidt distance is locally Riemannian but not contractive [34, 35], the trace distance is contractive but not locally Riemannian  and the relative entropy, although widely used in information theory, is technically not even a proper distance as it is not symmetric .
Here we derive closed formulae for classical and total correlations of Bell-diagonal states of two qubits according to the Bures distance. Together with the known corresponding formulae for entanglement  and discord-type quantum correlations [38, 30, 31], these allow us to gain a complete and unifying view of Bures correlations for Bell-diagonal states. We then provide two applications of these results. We first report the explicit expressions of the Bures correlations for two special subclasses of Bell-diagonal states, namely Werner states and rank-2 Bell-diagonal states. Finally, we consider a dynamical system made of two independent qubits locally interacting with a bosonic non-dissipative channel and show that both quantum and classical correlations measured by Bures distance can alternatively freeze during the evolution, joining the ranks of other faithful correlation quantifiers [39, 38, 6]. It is worthwhile to note that the freezing analysis was addressed in Ref.  with similar methods for the trace distance, but with a different definition of classical correlations.
The paper is organised as follows. In Section 2 we review some known results concerning Bures quantum correlations and entanglement of Bell-diagonal states. In Sections 3 and 4 we provide, respectively, the closed formulae for Bures classical and total correlations of Bell-diagonal states. In Section 5 we compute the correlations of two particular classes of Bell-diagonal states, i.e., Werner states and rank-2 Bell-diagonal states. In Section 6 we analyse the dynamics of correlations between two noninteracting qubits initially prepared in a Bell-diagonal state and subject to identical local pure dephasing channels. We conclude in Section 7 with a summary and outlook.
2 Quantum Correlations
Quantum correlations stem from two peculiar ingredients of quantum mechanics, the superposition principle and the tensor product structure of the Hilbert space associated to a composite quantum system. They are completely characterised by entanglement in the case of pure states, whereas in the case of mixed states entanglement constitutes only a part of the quantumness of correlations [3, 4]. As a result, for any pair of comparable quantifiers of general quantum correlations and entanglement, the quantum correlations of a state should intuitively be always greater or equal to the corresponding entanglement, being equal if the state is pure [41, 42]. This is nicely captured by the aforementioned geometric approach. Specifically, in this paper, the quantum correlations of a state are quantified by the minimum Bures distance of to the set of classical-quantum states, namely
where is the set of classical-quantum states, i.e. states of the form with being a probability vector, an orthonormal basis of qubit and any state of qubit and is any of the closest classical-quantum states to . The entanglement of is measured by the minimum Bures distance of to the set of separable states, namely
where is the set of separable states, i.e. states of the form with being a probability vector, and any state of qubit and , respectively, while is any of the nearest separable states to . As the set of classical-quantum states is contained in the set of separable states, we immediately have that for every .
We shall restrict ourselves to the relevant but structurally simple class of Bell-diagonal (BD) states of two qubits, which are diagonal in the “magic basis” of the four maximally entangled Bell states. As a result, BD states are represented in the standard computational basis by the following matrix
where and , , are the identity and the Pauli matrices, respectively. The coefficients are the only correlation matrix elements of a BD state that can be different from zero, in terms of which the eigenvalues of are expressed as follows,
BD states are also called states with maximally mixed marginals, due to the fact that their reduced density matrices and are both equal to the maximally mixed state of a qubit, i.e., and . The class of BD states is particularly interesting: for instance, they include the well-known Bell states and Werner states  and constitute a resource for entanglement activation and distribution [42, 43, 44, 45].
where the index is such that , with
As a result, the closed expression for the quantum correlations of an arbitrary BD state, as quantified by the Bures distance, is given by
We now make some important remarks. The index characterising the state in (7) is such that . This can be easily proven by considering the expressions of the correlation matrix coefficients in terms of the eigenvalues of the BD state and noting that the condition is equivalent to the condition .
The closest classical-quantum state to a BD state is unique if, and only if, is within the interior of the tetrahedron of BD states () and the index such that is unique. Otherwise, there are infinitely many closest classical-quantum states to a BD state .
We finally note that the Bures quantum correlations of as captured by Eq. (10) are different (conceptually and quantitatively) from the “discord of response” of , where quantumness of correlations is alternatively defined in terms of the minimum (Bures) distance between and the set of states obtained by rotating via local root-of-unity unitary operations on one subsystem only .
In the case of a BD state , the concurrence specialises to
with being the eigenvalues in non-increasing order.
3 Classical Correlations
The classical correlations of a state can be quantified as follows. Given the set of all the closest classical-quantum states to , called , we define the classical correlations of to be
where is the set of product states, i.e. states of the form with () being an arbitrary state of qubit () while is any of the closest product states to . Notice that the definition in Eq. (15) represents an important improvement over previous attempts to quantify classical correlations geometrically [15, 40, 46]. In fact, without the inclusion of the additional minimisation over all the closest classical-quantum states to , a measure of classical correlations might be ill-defined, as the distances between each closest classical-quantum state and their respective closest product states can generally differ. This issue has been very recently highlighted in an independent work .
As we have already mentioned, if is an arbitrary BD state then, within , there always exists a BD classical-quantum state of the form of Eq. (7). In the following we shall prove that, for any BD state , the BD state achieves the infimum over in Eq. (15) and that one of the product states nearest to the BD classical-quantum state is the tensor product of the marginals of a BD state , i.e.
Thus, the Bures classical correlations of any BD state are quantified by
where () is defined in Eq. (2).
We now prove the announced result. In  the authors presented an explicit construction of all the closest classical-quantum states to any BD state . In order to express in a general mathematical form we need to introduce some notations. Let , , , be the eigenvalues of and be any index such that . Let us finally introduce the orthonormal product basis of defined as follows ,
if is unique, then are the eigenvectors of ;
if can take two distinct values, then , and are defined by:
where and are arbitrary;
if can take all three different values, i.e. with , then we define
(22) (23) (24)
where and are arbitrary.
Now we are ready to write down the general form of any closest classical-quantum states to an arbitrary BD state :
if and for , , , then
where is a parameter which can take any value in the interval ;
if and for , , , then
where is a parameter which can take any value in the interval ;
if , then
Now, for the sake of simplicity, let us focus on BD states such that is unique and equal to and such that and , which from now on will be referred to as reference BD states. Later, we will generalise the analysis valid for this particular reference BD states, to a general BD state. In the case of the reference BD states we have that the set of all closest classical-quantum states to is given by the following -parameter family of states:
Let us consider a general product state of two qubits, , where and are any two states of qubits and , respectively. Due to the Bloch representations of and , i.e. and with , we have that is represented in the standard computational basis by the following matrix
A general product state of two qubits is clearly characterised by the Bloch vectors of each qubit, i.e. and . We now take into account the product states and , where , with , is characterised by the Bloch vectors and , whereas is characterised by the Bloch vectors and . Then, the following holds
where is any closest classical-quantum state of the form (28). To prove the above equality it suffices to consider the invariance of the fidelity under general unitaries and the invariance of any under the action of the particular local unitary .
It is known that the fidelity is a concave function on the convex set of states, i.e.
for any states , and . As a result, by substituting , , and into (31), one obtains
By symmetry, a similar result holds also by flipping the first two components of the Bloch vector . As a result, in order to maximise the fidelity between any closest classical-quantum state to a reference BD state and any product state , the Bloch vectors that characterise must be necessarily such that and . The square root of the fidelity between any of the form (28) and any product state with and is
where we have used the fact that any of the form (28) commutes with any product state with and , so that the square root of their fidelity is nothing but the square root of their classical fidelity which, in turn, is given by the sum of the square roots of the products of the corresponding eigenvalues of and . By maximising Eq. (3) with respect to , and , one obtains that reaches the global maximum for any , so in particular for any . As a consequence, for any reference BD state , the infimum over in Eq. (15) is achieved by and one of the nearest product states to is .
Before generalising the above analysis from the reference BD states to any BD state we need to make the following two remarks. First, for any orthonormal product basis , any classical-quantum state in Eq. (25) can be transformed through a local unitary into the reference classical-quantum state in Eq. (28) with the same value of and , any classical-quantum state in Eq. (26) can be transformed through a local unitary into the reference classical-quantum state (28) with the same value of and , and finally any classical-quantum state in Eq. (27) can be transformed through a local unitary into the reference classical-quantum state (28) with and . Second, we note that the minimal Bures distance from the set of product states is invariant under local unitaries, indeed
where in the first equality we use the invariance of the Bures distance under unitaries and in the second equality we use the locality and bijectivity of local unitaries.
As promised, we are now ready to generalise the above analysis from the reference BD states to any BD state. Let us start from the BD states satisfying Condition (1), i.e. the ones such that and for , , . As we mentioned, we have necessarily a reference BD state with such that for any pair of and with the same value of , there always exists a local unitary such that . Also,
where in the first equality, we use the fact that all the states with the same value of , that may depend on or or both, are local unitarily equivalent and that the minimal Bures distance from the set of product states is invariant under local unitaries. In the second equality we use the fact that all the states are local unitarily equivalent to the states with the same value of and again that the minimal Bures distance from the set of product states is invariant under local unitaries. In the third equality we use the fact that achieves the infimum over is and one of its nearest product states is . In the last equality we use the fact that is local unitarily equivalent to , both corresponding to , and the fact that and the Bures distance are invariant under general unitaries. Equation (35) means that, for any BD state satisfying Condition (1), i.e. such that and for , , , the infimum over in Eq. (15) is achieved by the BD closest classical-quantum states to , , and one of the nearest product states to is . An identical reasoning holds for BD states satisfying Condition (2), i.e. such that and for , , , with the only exception of using a reference BD state such that . Finally, for BD states satisfying Condition (3), i.e. such that , one simply needs to use the fact that each of the closest classical-quantum states , all obtained by setting in the previous two cases, is local unitarily equivalent to directly.
In summary, we have proven that for any BD state , the closest BD classical-quantum state and the product of the marginals of quite miraculously achieve the double minimisation in Eq. (15). This result is highly nontrivial (and not obvious a priori) and suggests that the definition of classical correlations proposed here is particularly natural for BD states .
4 Total Correlations
The Bures total correlations of a state are defined by the minimum Bures distance of to the set of product states, namely
where is the set of product states, while is any of the product states closest to . Therefore, in order to obtain the total correlations of a given BD state , we simply need to maximise the fidelity between and any product state .
However, the argument used in the previous section to maximise the fidelity between any closest classical-quantum state and any product state , does not apply anymore to the present problem. We note in fact that Eq. (30) does not hold for a general BD state, i.e.,
The concavity of the fidelity thus cannot be utilised to extend the previous analysis from classical to total correlations.
We then formulate an ansatz on the form of one of the closest product states to a general BD state , which is of the form
where the index is such that . This allows us to accomplish the optimisation of the Bures distance analytically. Interestingly, the ansatz form of one of the closest product states to a BD state using the trace distance, formulated in Ref. , is the same as Eq. (38), but with given by .
The ansatz in Eq. (38) is supported and verified by an extensive numerical investigation, which was implemented in the following way. We begin by generating a random set of four normalised probabilities and forming a BD state by setting these probabilities as the eigenvalues , , and . We then numerically maximise the fidelity between this random BD state and a general product state. The result of this numerical maximisation is compared with the analytical maximisation of the fidelity between the random BD state and our ansatz product states, Eq. (38). This process was repeated for randomly generated BD states, and in all cases the analytically maximised fidelity between the random BD state and the ansatz product states exceeded or equalled the numerical maximisation over all product states.
In order to get the explicit expression of the above coefficient characterising the Bloch vectors of , in terms of the BD state parameters, let us proceed as follows. We define the auxiliary set of coefficients as given by suitable reorderings of the BD state eigenvalues. Specifically, for we have , for we have , and for we have .
For any BD state with, respectively, and , the square root of the fidelity between and any product state having the Bloch vectors of Eq. (38) is
By maximising now the square root of the fidelity between and with respect to , one obtains after some algebra that:
if and the condition
is fulfilled, then with
if and the condition
is fulfilled, then with
if none of the two conditions above hold, we have .
As a result, the Bures total correlations of an arbitrary BD state are measured by
The contour plot in Figure 1 shows the fidelity between an example BD state obeying Condition (1) with , and product states obtained as perturbations of the ansatz from Eq. (38), by allowing and to vary over the interval . The values of and that maximise this fidelity are shown to coincide with the values given by Eq. (42). Similar plots may be created by perturbing the ansatz states in such a way that the index is no longer , i.e. , but rather or . These plots both show two maxima, but the fidelity in these two maxima never exceeds the maximal value corresponding to .
It is worth highlighting that the product of the marginals does not represent, in general, the closest product state to an arbitrary BD state ; indeed, from the above classification, only when Condition (3) holds the corresponding is then precisely the fidelity between and . This apparently counterintuitive feature has been observed as well when the trace distance is used to measure total correlations . Differently, when Hilbert-Schmidt  and relative entropy  distances are taken into account, the closest product state to any BD state is always the product of the marginals, i.e. the maximally mixed state.
Another aspect to be remarked in this context is that, according to a naive intuition, total correlations may be expected to be equal to the sum of the classical and quantum correlations. Indeed, this is true for BD states when considering the relative entropy and Hilbert-Schmidt distance measures of correlations . However, the triangle inequality which is a requirement for any metric imposes that, for geometric quantifiers of correlations, only a subadditivity property needs to be satisfied, of the form . This turns out to be in general a sharp inequality when correlations are measured either by the trace distance [40, 46] or by the Bures distance, as shall become apparent in the following.
In this Section the Bures distance correlations are analysed for two families of one-parameter BD states: Werner states  and rank-2 BD states.
Werner states are conventionally defined, for two qubits, as the mixture of a maximally entangled Bell state with the maximally mixed state, namely , with . Werner states are therefore a subclass of BD states, with eigenvalues given simply by and . Combining the results of [37, 38, 30, 31] with the above analysis, the Bures distance based correlations of , as shown in Figure 2 as functions of , are given overall by