Hallmarking quantum states: unified framework for coherence and correlations

Quantum coherence and distributed correlations among subparties are often considered as separate, although operationally linked to each other, properties of a quantum state. Here, we propose a measure able to quantify the contributions derived by both the tensor structure of the multipartite Hilbert space and the presence of coherence inside each of the subparties. Our results hold for any number of partitions of the Hilbert space. Within this unified framework, global coherence of the state is identified as the ingredient responsible for the presence of distributed quantum correlations, while local coherence also contributes to the quantumness of the state. A new quantifier, the"hookup", is introduced within such a framework. We also provide a simple physical interpretation, in terms of coherence, of the difference between total correlations and the sum of classical and quantum correlations obtained using relative-entropy--based quantifiers.

The superposition principle is one of the axioms and most distinctive features of quantum mechanics and is responsible for the presence of coherence in quantum states [1]. When it comes to multipartite scenarios, the superposition principle is still the ingredient that makes quantum states intimately different from classical states and allows them to show inner correlations beyond any classical probabilistic model [2][3][4]. One of the most famous applications where the complete operational equivalence between local coherence and bipartite entanglement was shown is quantum cryptography [5]. Indeed, the original BB84 key distribution scheme, which makes use of ordinary single-particle states in noncommuting basis [6], was proved to be completely equivalent to Ekert's scheme, the security of which relies on the use of maximally entangled states [7].
Despite the previous considerations and the evident common roots between the phenomena of coherence and distributed correlations, the quantification of the quantum character of a state normally follows separate paths. While different kinds of correlations (entanglement, discord, etc.) are introduced in distributed scenarios, coherence is thought to measure the quantumness of a quantum system as a whole. In spite of being one of the most distinctive traits of quantum mechanics, the characterization and quantification of quantum coherence have only very recently become an intense field of investigation [13]. In analogy to what done in different contexts, such as entanglement theory ( [2,[8][9][10]) or quantum thermodynamics ( [11,12]), a resource theory for quantum coherence was proposed in Refs. [14][15][16][17][18][19]. Coherence has also been linked to asymmetry [20] and purity [21]. The distribution of coherence among subparties and its monogamy properties were studied in Refs. [22][23][24]. The interplay between entanglement and coherence was analyzed in Refs. [25][26][27]. A more quantititive relationship between correlations and coherence was established in [28].
In this Letter, we go beyond these approaches and propose a unified framework where the full character of a quantum state belonging to a multipartite Hilbert space can be determined by both local and collective properties. The founding observation is that both coherence and correlations can be measured using the same kind of quantifier. Within such a new approach, we are able to give a physical meaning to the excess of classical plus quantum correlations with respect to the total ones.
Our starting point for quantifying correlations is the geometric scheme presented in Ref. [29]. All the distances between pairs of states ρ and σ are measured by the quantum relative entropy S(ρ||σ) = − tr {ρ log σ} − S(ρ), where S(ρ) = − tr {ρ log ρ} is the von Neumann entropy of ρ. The relative entropy is commonly used and accepted as a distance measure in different contexts, as it fulfills a series of key requirements [30]. Different kinds of metrics (also symmetric), such as the l 1 -norm, could be be introduced as well [16].
Within this approach, the various kinds of correlations present in a quantum state are given by the distance between the state itself and the closest state without the desired property. Thus, the total correlations of a multipartite state ≡ A1,A2,...,An are given by its distance from the the closest product state. They are quantified by the total mutual information of that state: T ( ) = S( ||π π π[ ]) = S(π π π[ ]) − S( ). Here, π π π[ ] is the reduced state in its product form: π π π[ ] = π 1 ⊗ · · · ⊗ π n where π i is obtained from by calculating the complementary partial trace (i.e., over all the partitions with the exception of the ith). The quantum part of these correlations is measured by the (two-sided) quantum discord D( ) = S( ||χ ) = S(χ ) − S( ), where χ is the classically correlated state closest to and where classical states have the form χ = k p k | k k|, with | k = |k 1 ⊗ · · · ⊗ |k n . Henceforth, unless specifically indicated, states will always be represented in such local bases. This assumption is motivated by the fact that we are going to consider quantifiers for coherence, which is a basis-dependent quantity. Classical correlations are given by J ( ) = S(χ ||π π π[χ ]) = S(π π π[χ ]) − S(χ ), where π π π[χ ] is the product state closest to χ . arXiv:1706.03625v1 [quant-ph] 12 Jun 2017 Following this treatment, an apparent incongruity comes out, as in general the sum of classical and quantum correlations exceeds the total correlations: T ( ) ≤ D( ) + J ( ), with the equality holding only in some special cases. In fact, we have L( ) ≡ D( ) + J ( ) − T ( ) = S(π π π[ ]||π π π[χ ]). (1) As we shall see later, we are able to give a physical interpretation for L( ) in terms of the quantum coherence of the local sub-parties. Here, it is worth remarking that, in this context, χ is commonly referred as "the classical state closest to ". Within the framework we are going to introduce, χ is classical in terms of correlations, while there is only one special basis where its coherence vanishes and where it can be seen as a fully classical entity. A state exhibiting nonvanishing nondiagonal density-matrix elements can hardly be identified as a classical one. Analogously to correlations, also coherence can be quantified using the relative entropy [16,31]. Coherence is a basis-dependent quantity, and the relative entropy of coherence of a state is defined as where I is the set of totally incoherent (diagonal) states in that basis. It turns out that |i i| |i i| is the state obtained by keeping only the diagonal elements of and by setting all the other entries to 0.
As mentioned in the introduction, recent studies can be found about the distribution of coherence in multipartite settings [22][23][24]28]. In this context, it is useful to introduce the concept of local coherence [22,28]. In the framework of relative entropy, the local coherence of a state A1,A2,...,An is the sum of the coherences of the reduced states π i s: with C defined in Eq. (2). Using the additivity of the relative entropy, one can show that In the same framework, a measure for the purely multipartite contribution to coherence can be introduced by subtracting the contribution of local terms: It can be shown that C M ( ) is a nonnegative quantity, and, for this purpose we anticipate the following [see also the illustration in Fig. 1(a)]: Lemma 1. The total dephasing operation ∆[·] commutes with π π π[·], that is, Proof. The proof can be given by calculating explicitly the matrix elements of the two operators. Given a generic bipartite state (the proof is identical irrespective of the number of parties) = i,j,k,l c i,j,k,l |i, j k, l|, Then, the hierarchical relationship C( ) ≥ C L ( ) can be proven using the fact that C( ) is the relative entropy between two states, while C L ( ) is relative to two new states obtained from the previous ones by applying the quantum operation π π π: by using Lemma 1 in Eq. (4), we have As relative entropy is known not to increase under any completely positive, trace preserving quantum operation, C M cannot be negative. C M ( ) also amounts to the difference between the total mutual information of and the total mutual information of . The connection to mutual information hints at a broader context for correlations and coherence. Interestingly, the use of the quantum mutual information as a measure of total correlations has a deep operational interpretation, as it is equal to the work required to erase such correlations, that is, to convert a correlated state into a product one [32]. This result was the generalization to quantum information of Landauer's theory of thermodynamics and information erasure [33]. On the other hand, coherence has also been shown to play a relevant role in thermodynamical processes [13]. Apart from being related to the work extraction problem [34], quantum coherence is at the root of fundamental issues, as, for example, irreversibility [35]. Given these two different physical origins of resources, a legitimate question arises about how much information is actually contained in a generic quantum state. This question can be answered by introducing a unified framework within which both the correlations and the local quantum character of a state are taken into account.
Such a unified framework can be built up by defining the class of states that are operationally useless as the set of incoherent product states I Π . A meaningful measure, the hookup M, for the amount of noise necessary to erase both the coherence and the correlations is represented by the distance between a state and its closest incoherent product state: Theorem 1. The closest incoherent product state is obtained by applying the dephasing operation to π π π[ ]: Being a relative entropy, the hookup can be interpreted as a distance [30]. It is a non-negative quantity and it is non-increasing under all the local, strictly incoherent operations. A clear physical interpretation for M can be obtained observing that, as promised, it can be decomposed as the sum of two terms, one of them associated to multipartite correlations and the other one being, according to Eq. (3), the local coherence of the state [see Using Eq. (1) we can also write To summarize, we have identified a distance that is able to capture both the nonlocal and the local effects of coherence. Exploiting Lemma 1, a different decomposition of M can be given where the total coherence appears explicitly. In fact, we have [see Fig. 1

(a)]
where also the totally classical correlations K( ) ≡ T (∆[ ]) appear. The quantity K( ), naturally emerged in our unified framework, has already been proven to play a relevant role in the context of many-body localization and quantum ergodicity [36]. It measures the amount of information that survives to total dephasing and is given by the classical mutual information of . Thus, it can be termed irreducible classical information. It can also be written as K = T − C M . Notice that while Eq. (11) implies that all the quantum content of is contained in C, the classical content of correlations is not necessarily all included in K( ). The information is split into contributions from diagonal (K) and nondiagonal (C) components of the state.
The previous observation has an important consequence. Indeed, it is possible to find a special basis where M( ) is the sum of discord and classical correlations. As already shown in Ref. [23], the minimum value of coherence, calculated over all the possible local unitaries, is the quantum discord D( ) [see the upper distances in Fig. 1(b)]. The following theorem demonstrates that the basis where this minimum is attained is the one where χ is completely incoherent.
Theorem 2. The minimum value of coherence is reached in the basis where χ is completely incoherent.
Proof. Any incoherent state is also a purely classically correlated one. Then, in the basis where χ is diagonal, In such a special basis, the classical correlations are given by K( ), and then M( ) = D( ) + J ( ). For any other basis choice, we need to take into account that, as the closest classically correlated state χ exhibits coherence, discord fails to be a good indicator of quantumness.
At this point, it is interesting to notice that, within the unified framework, a physical interpretation for the term L( ), previously introduced as a mere mathematical entity [29], can be given. Let us consider the case where is written in the basis of the eigenstates of χ . In that basis, as just shown, χ = ∆[ ]. Consequently, by applying Lemma 1, L( ) = S(∆[π π π[ ]]) − S(π π π[ ]) ≡ C L ( ). Thus, L is nothing else than the local coherence of calculated in the basis where χ can be considered sensu stricto classical. A nonvanishing value of L implies that the (classical) basis that minimizes the total coherence is not the same one that minimizes its local contribution C L ( ), as it is always possible to find a basis where the latter is zero. The basis mismatch term L( ) represents the lack of completeness of the correlation framework, as it tries to quantify quantumness by omitting the conceptually fundamental component of coherence. A diagrammatic comparison between our unified framework and the correlation scheme of [29] is given in Fig. 1.
Let us remark that, while there exists a hierarchy between coherence and discord, it is not possible to find such an ordering relationship between K and J and between C L and L. As an instructive example, let us consider the maximally discordant mixed state M D = |Φ + Φ + | + (1 − )[m|01 01| + (1 − m)|10 10|] (the choice is suggested by the fact that this state is known to have low classical correlations compared to discord) [37], together with the family of states obtained by applying local unitaries˜ M D . For the sake of simplicity, we will take m = 0. Depending on the value of , the closest classically correlated state is obtained by dephasing M D either in the computational basis for < or in a rotated basis for > , where the threshold is given by = 2/3. After a second threshold, for > 0.76,

a) b)
FIG . 1: (a) The unified framework of coherence and correlations: from any initial state , the closest useless state can be reached either by applying first the dephasing operation ∆ (blue) and then the product operation π π π (red) or the other way around. Depending on the chosen path, the total hookup M can be decomposed into the sum of the total mutual information and the local coherence or into the sum of the total coherence and the irreducible classical information.
(b) Comparison between the correlation scheme of Ref. [29] (solid lines) and the unified framework of coherence and correlations (dashed lines). While the discord is a lower bound for the total coherence (that is, the distance between and χ is always smaller than the one between and ∆[ ]), K can be longer or shorter than J and CL can be longer or shorter than L.
the optimal basis is the x-basis. On the other hand, K reaches its maximum in the x-basis irrespective of . This means that, for < In the previous example, as L = 0, we obviously have C L ≥ L. A case where this ordering relationship is violated can be found by considering the state υ = 8 27 |000 000|+ 12 27 |W W |+ 6 27 |W W |+ 1 27 |111 111| [38], where |W = 1 √ 3 (|011 + |110 + |101 ). In fact, we have L(υ) = 0.24 [29]. It can immediately checked that the local coherence vanishes in the computational basis and it can also be shown that C L (υ) ≤ L(υ).
As discussed in Ref. [29], there exists a simple relationship between the two-sided discord we have considered so far and the one-sided discord introduced in Refs. [39,40]. The latter can be obtained by finding the quantum-classical state χ such that the quantity δ( ) ≡ D( ) − L( ) = T ( ) − J ( ) is minimized. Notice that the prime has been introduced here to stress the fact that, in general, the optimal state is not the same appearing in the two-sided discord minimization. Bipartite quantum-classical states admit the general form χ = k p k A k ⊗ |k k| B . Theorem 3. In the bipartite scenario, δ( ) is equal to the minimum purely bipartite coherence obtained by applying to part B all the possible unitaries: To summarize, when it comes to characterize a quantum state, the (total) mutual information is not necessarily an adequate indicator, for it fails to take into account the coherence properties of the state itself. This is the reason why the puzzling term L( ) comes out in the relative entropy framework. Such a term can be taken as a witness of the fact that the local coherence and the global one are minimized in different basis. This seemingly side observation actually reveals how much a unified framework is needed to build a consistent theory.
In the approach proposed here, quantum coherence and multipartite correlations cannot be thought as distinct labels, as they cooperate to hallmark the state, and allow for providing a full description of the quantumness of the state. We have introduced the hookup M and shown that it amounts to the sum of coherence and irreducible classical information. Such a comprehensive hallmark has different conceivable applications, as it fully determines the power of a quantum state. Earlier we have mentioned that the hookup can be used to measure the amount of work necessary to erase such correlations and has obvious thermodynamical implications. The interplay between quantum local and distributed contributions could also be employed in computing tasks, as the algorithmic performances are studied by analyzing either the single-qubit power or the presence of correlations as entanglement or discord. Another field where correlations and coherence are separately essential resources is quantum metrology, where our approach can be used to find the optimal quantum advantage and tighter bounds.
Valuable discussions with Fernando Galve and Francesco Plastina are kindly acknowledged. This work was supported by the EU through the H2020 Project QuProCS (Grant Agreement 641277) and by MINECO/AEI/FEDER through projects NoMaQ FIS2014-60343-P and EPheQuCS FIS2016-78010-P.