Diagonal quantum discord

Quantum discord measures quantum correlation by comparing the quantum mutual information with the maximal amount of mutual information accessible to a quantum measurement. This paper analyzes the properties of diagonal discord, a simplified version of discord that compares quantum mutual information with the mutual information revealed by a measurement that correspond to the eigenstates of the local density matrices. In contrast to the optimized discord, diagonal discord is easily computable; it also finds connections to thermodynamics and resource theory. Here we further show that, for the generic case of non-degenerate local density matrices, diagonal discord exhibits desirable properties as a preferable discord measure. We employ the theory of resource destroying maps [Liu/Hu/Lloyd, PRL 118, 060502 (2017)] to prove that diagonal discord is monotonically nonincreasing under the operation of local discord nongenerating qudit channels, $d>2$, and provide numerical evidence that such monotonicity holds for qubit channels as well. We also show that it is continuous, and derive a Fannes-like continuity bound. Our results hold for a variety of simple discord measures generalized from diagonal discord.

Quantum discord measures a very general form of nonclassical correlation, which can be present in quantum systems even in the absence of entanglement. Since the first expositions of this concept more than a decade ago [1,2], a substantial amount of research effort has been devoted to understanding the mathematical properties and physical meanings of discord and similar quantities. Comprehensive surveys of the properties of discord can be found in [3,4], and references [5][6][7][8] provide recent perspectives on the field.
The study of discord presents many challenges and open questions. One major difficulty with discord-like quantities is that they are typically hard to compute. The canonical version of quantum discord is defined to be the difference between quantum mutual information (total correlation) and the maximum possible amount of correlation that is locally accessible (classical correlation), which involves optimization over all possible local measurements. Such optimization renders the problem of estimating discord (along with other optimized quantities such as deficit, geometric discord) computationally intractable (NP-complete) [9]. Moreover, the analytic formulas for these optimized quantities are only known for very limited cases, such as two-qubit X-states [10] and highly symmetric states [11]. So it is generally very difficult to evaluate discord or to analyze the properties of discord. Another problem is that there appear to be both conceptual and technical obstacles in studying discord under operational frameworks, in particular, resource theory. The connection of discord-type quantities to resource theory remains unclear.
Diagonal discord is a natural simplification of discord, in which, rather than maximizing mutual information over all measurements, one looks at the mutual information revealed by a measurement with respect to the eigenbasis of the reduced density matrix of the subsys-tem under study [12]. That is, we allow the local density matrix to 'choose' to define mutual information by the locally minimally disturbing measurement. Because such measurement does not disturb the local states, diagonal discord truly represents the property of 'correlation'. By definition, diagonal discord is an upper bound for discord as originally defined, and is zero for states with with zero discord. Different entropic measures of discord (the original discord and deficit [13,14]) coincide with diagonal discord when the optimization procedure leads to measurements with respect to the local eigenbasis. We note that quantities defined by a similar local measurement strategy have been considered earlier: for instance, the so-called measurement induced disturbance [15] and nonlocality [16] are close variants of diagonal discord defined by a local eigenbasis as well. Diagonal discord has been shown to play key roles in thermodynamic scenarios, such as energy transport [12] and work extraction [17].
In contrast to optimized discord-type quantities, diagonal discord is generically efficiently computable. We were inspired by the example of the introduction of negativity as a measure of entanglement [18]: because it is efficiently computable, negativity greatly simplifies the study of entanglement in a wide variety of scenarios. Furthermore, diagonal discord naturally emerges from the theory of resource destroying maps [19], a recent framework for analyzing resource theories that can also be applied to nonconvex theories. Indeed, local measurement in an eigenbasis is a canonical discord destroying map, and the diagonal discord of a state is just the relative entropy of the state to its discord-destroyed counterpart. We believe that the study of diagonal discord may forge new links between discord and resource theory.
Because diagonal is defined without optimization over local measurements, however, several of its important mathematical properties must be verified. The purpose arXiv:1708.09076v1 [quant-ph] 30 Aug 2017 of this paper is to provide such verification. First, monotonicity under operations that are considered free is a defining feature of resource measures; identifying such monotones is a central theme of resource theory. A curious property of discord is that it can even be created by some local operations [20,21]. One can easily show that the minimum (contractive) distances to classical states (sometimes known as geometric measures of discord) are monotonically nonincreasing under all local discord nongenerating operations, but it is not clear whether such a property holds for diagonal discord. Note that the monotonicity under all nongenerating operations is arguably an overly strong requirement [4], which automatically implies monotonicity under all stricter theories (with less free operations). Second, continuity is also a desirable feature [4,22], which indicates that the measure does not see a sudden jump under arbitrarily small perturbations. Again, continuity holds for optimized discord [22,23], but from examples given in [22,24], where the local states are both maximally mixed qubits, we know that diagonal discord can generally be discontinuous at degeneracies. However, the continuity properties otherwise remains unexplored. These two unclear features represent the most important concerns of restricting to local eigenbases.
In this paper, we address the above concerns by providing more complete analysis on the monotonicity and continuity of diagonal discord. We first find that, rather surprisingly, diagonal discord exhibits good monotonicity properties. We do so by showing that local isotropic channels commute with the canonical discord destroying map, which implies that diagonal discord is monotone under them by [19]. By the classification of commutativity-preserving operations [20,21,25], we conclude that monotonicity holds for all local commutativity-preserving operations except for unital qubit channels that are not isotropic. However, numerical studies imply that monotonicity holds for these channels as well. Then, we prove that, when the local density operator is nondegenerate, diagonal discord is continuous. We derive a Fannes-type continuity bound, which diverges as the minimum gap between eigenvalues tends to zero as expected. Proofs and some detailed discussions are left to the appendix.
Diagonal discord.-Here, we define the notion of diagonal discord more formally. Without loss of generality, we mainly study the one-sided discord of a bipartite state ρ AB , where the local measurements are made on subsystem A. It is straightforward to generalize the results to two-sided measurements or multipartite cases.
Let {Π A i ≡ |i A i|} be a local eigenbasis of A, i.e., suppose ρ A = tr B ρ AB admits spectral decomposition ρ A = i p i Π A i . Note that the eigenbasis is not uniquely defined in the presence of degeneracy. Define π A (ρ AB ) = which describes the local measurement in eigenbasis {Π A i } and the canonical discord destroying map [19]. Diagonal discord of ρ AB as measured by A, denoted as D A (ρ AB ), quantifies the reduction in mutual information induced by π A . Since π A does not perturb ρ A ,D A equals the increase in the global entropy. So diagonal discord represents a unified simplification of discord and deficit. Formally,D The optimized versions of the first and second line are respectively discord and deficit, which are inequivalent in general. It is crucial that diagonal discord can also take the form of relative entropy (a straightforward derivation in appendix): From Eq. (3), it can be seen that diagonal discord indeed only vanishes for classical-quantum states, the fixed points of π A , which is a necessary condition for a discord measure. Note that, in general, optimization is still needed within degenerate subspaces. It can be seen that, as long as the degenerate subspace is small, diagonal discord can be efficiently computed. In this paper, we are mostly concerned with the nondegenerate case, where π A is unique. Structure of π theory and monotonicity.-Local discord non-generating channels (X A (π A )), i.e. commutativity-preserving channels [21], consist of unital (equivalent to mixed-unitary [26]) channels for qubits (d A = 2) and isotropic channels for qudits (d A > 2) [20,21,25], in addition to all semiclassical channels, which always destroy discord. In general, identifying operations under which some measure behaves as a monotone is a highly nontrivial task. Due to Eq. (3), the general monotonicity condition emerged from the theory of resource destroying maps [19] can be applied to diagonal discord:D is monotonically nonincreasing under X A (π A ), the set of local operations that commute with π A . We use MU, ISO, and SC to denote the sets of mixed-unitary, isotropic and semiclassical channels (without the completely depolarizing channel, so that SC ∩ ISO = ∅) respectively. For simplicity, we assume that ρ A is nondegenerate. Now we identify operations that belong to X A (π A ). First, it is known that SC ⊂ X A (π A ) [19]. Also recall that isotropic channels take the form E ISO (ρ) = (1 − γ)W (ρ)+γI/d, where W is either unitary or antiunitary. Ref. [19] showed that unitary-isotropic channels are in X A (π A ). We analyze the remaining cases for qubits and qudits separately, since they exhibit different structures in the theory of π.
By explicitly using Lemma 1, we can show that (details in the appendix) all isotropic channels are in X A (π A ): Therefore, we obtain the following result for qubits: However, the condition in Lemma 1 does not hold in general, which implies that where N is the normalization factor. Then So this probabilistic Hadamard is not in X A (π A ). We conjecture (not important for our current purpose) that ISO = X A (π A ). That is, qubit mixed-unitary channels that are not isotropic all fail to satisfy the condition. For qubit channels that live in MU \ X(π A ), the current idea for proving monotonicity do not apply. However, we provide numerical results which strongly indicate that diagonal discord is monotone under such channels as well. Fig. 1 displays the comparison between diagonal discord before and after the action of several typical non-isotropic mixed-unitary channels, for a large number of randomly generated input states. It can be seen that all data points reside on the nonincreasing side. All other channels that we have analyzed exhibit similar behaviors. We put this as a conjecture at the moment: Conjecture 1. For d A = 2, diagonal discord is monotonically nonincreasing under any local discordnongenerating channel.
The analysis for d A > 2 turns out to be simpler. In fact, it can be shown in general dimensions that ISO ⊂ X A (π A ). The main step of the proof is to explicitly write out the eigenbasis after an antiunitary transformation.
The complete result for qudits then follows: diagonal discord is monotonically nonincreasing under any local discordnongenerating channel. Continuity.-As mentioned, Refs. [22,24] brought up examples of states with maximally mixed marginals, where diagonal discord can be discontinuous. The discontinuity essentially comes from the maximization within the degenerate subspace: one can perturb the state in the direction that is far away from the optimal eigenbasis. However, in the absence of degeneracies, the eigenbasis is unique, so the above phenomenon cannot occur. We first formally prove that diagonal discord is indeed continuous when the local density operator being measured is nondegenerate, by deriving a Fannes-like continuity bound: Theorem 6. Diagonal discord is continuous at states such that the local density operator being measured is nondegenerate. More explicitly, let ρ AB be a bipartite state in finite dimensions such that ρ A = tr B ρ AB has distinct eigenvalues, and the smallest gap is ∆. Suppose ρ AB is a perturbed state such that ρ AB − ρ AB 1 ≤ . For sufficiently small , it holds that The main idea is that π changes continuously, which is also known as "weak continuity" [22]. Details of the proof are also given in the appendix.
Locally nondegenerate states such that the local eigenbasis minimizes discord (and deficit), which we call π states, represent an important class of states such that the restriction to eigenbasis is indeed optimal. Note that all locally nondegenerate zero discord states are π states. The above continuity result indicates some special properties of π states. For example, it directly follows from Theorem 6 that diagonal discord remains close to optimized discords in the vicinity of π states. Also, conti- Rn(π/2)ρRn(π/2) † where Rn(π/2) is the π/2 rotation with respect to the axis n ∝ (1, 1, 1), and (c) E MU (ρ) = 1 6 ρ + 1 3 RX (π/10)ρRX (π/10) † + 1 2 RZ (π/5)ρRZ (π/5) † where RX and RZ are rotations with respect to X axis and Z axis respectively. The choice of these channels is arbitrary. The number of samples is set to 1000 for each channel.

FIG. 2.
Structure of commutativity-preserving channels (X(π)). Note that we define SC to exclude the completely depolarizing channel. (a) Qubits: note that ISO MU, SC ∩ MU includes e.g. completely dephasing channels; (b) Qudits with d > 2. Grey area (ISO): in X(π); Dotted area (MU\(ISO ∪ SC)): numerical evidence of being in X(π); White area (SC): not in X(π). nuity of the optimal basis (termed "strong continuity" [22]) is known to fail for discord and deficit. However, we conjecture that strong continuity holds at π states.
Generalizations.-The above results can be generalized to a wide variety of simple discord-type measures defined by π, such as different distances and multi-sided measures, which can be seen as close variants of diagonal discord.
First, consider general contractive distance measures besides relative entropy. Let δ be a nonnegative real function such that i) δ(ρ, σ) = 0 iff ρ = σ and ii) δ(E(ρ), E(σ)) ≤ δ(ρ, σ). Consider as a discord measure defined by δ and the resource destroying map π A . AllD(ρ AB ) δ,π A are monotonically nonincreasing under X A (π A ) [19]: Furthermore, the continuity holds when δ is given by the Schatten-p norm: Theorem 8. Let ρ AB be a bipartite state in finite dimensions such that ρ B = tr A ρ AB has distinct eigenvalues, and the smallest gap is ∆. Suppose ρ AB − ρ AB 1 ≤ where is sufficiently small, it holds that In the above, we focused on the one-sided discord measures. The results can be easily extended to multi-sided measures where we also make a measurement on system {A k } n k=1 in such a way that it will not disturb the marginal state. Here, we assume that n is finite. Let ρ {A k } be a composite state over the systems A 1 , . . . , A n and ρ Aj be nondegenerate for all j = 1 . . . n. Denote where {|i k } is the local eigenbasis of system A k . Then we obtain the following.
where δ is Schatten-p norm or relative entropy, is continuous at states such that the local density operators being measured are nondegenerate.
We note that the known discord-type quantities given by local measurement in the eigenbasis belong to such generalizations.D(ρ AB ) S,π A π B on a bipartite state (where S denotes relative entropy) gives the measurement-induced disturbance [27]; When ρ A is nondegenerate,D(ρ AB ) · 2 ,π A gives the measurementinduced nonlocality [28] (the similar quantity given by geometric distance measure is investigated in [29]).
Concluding remarks.-Diagonal discord is an easily computable and meaningful measure of discord that has potentially wide application. Here we showed that diagonal discord and a variety of similar measures exhibit desirable mathematical properties of monotonicity and continuity in the generic case that the measured subsystem is nondegenerate: our analysis indicates the somewhat surprising result that diagonal discord is a monotonone under all local discord nongenerating qudit channels, d > 2, and is very likely a monotone for discord nongenerating qubit channels as well. References [12] and [17] show that diagonal discord has a direct thermodynamic interpretation in terms of heat flow and locally extractable work. Its direct connection with physical properties suggests that diagonal discord can play a useful role in resource theories in general.
We thank Xueyuan Hu for helpful comments on the draft. This work is supported by AFOSR, ARO, IARPA, and NSF under an INSPIRE grant. R.T. acknowledges the support of the Takenaka scholarship.

Diagonal discord as relative entropy
Here we provide a simple argument that diagonal discord can be expressed in the form of relative entropy. Notice that where the second line follows from the fact that each (Π A i ⊗ I B ) commutes with π A (ρ AB ), the third line follows from the cyclic property of trace, the fourth line follows from the idempotence of Π A i ⊗ I B , and the fifth line follows from the completeness relation i (Π A i ⊗ I B ) = I. Therefore, by Eq. (2), That is, diagonal discord of ρ AB equals the relative entropy to π A (ρ AB ), minimized over eigenbases π A in the presence of degeneracies. We first obtain On the other hand, where the first line follows from the spectral decomposition of E MU (ρ A ). Therefore, the two sides of the commuting condition Eqs. (11) and (15) coincide if and only if is a matrix defined on B. In other words, M is a skew-Hermitian matrix: it has zero or pure imaginary diagonal entries. Since the diagonals of 0|ρ AB |1 can be real or imaginary depending on ρ AB (for example, for can only be the zero matrix so that the skew-Hermitian condition holds for arbitrary ρ AB . Furthermore, notice that the eigenbasis can vary arbitrarily depending on ρ AB , so Eq. Proof. Here we show that ISO ⊂ X A (π A ) for qubits by directly employing the condition introduced in Lemma 1.
We now show that any antiunitary-isotropic channelū(ρ) = (1 − γ)U ρ T U † + γI/2 also satisfies the condition. Let {V |ψ , V |ψ } be the basis with respect to which the transpose is taken, where V is unitary. Notice that transpose operation can be written as We are now ready to examine whetherū(ρ) satisfies the condition. Due to Eq. (20), the first term gives zero. Notice that the new eigenbasis is |η +,− = {U |ψ , U |ψ }, where U = U Y V is unitary. So the second term also gives zero due to Eq. (19). Soū ∈ X A (π A ).
Proof of Theorem 4 Proof. Here we provide a general proof of ISO ⊂ X A (π A ). Note that Theorem 2 for qubit systems is just a special case of this result. For d A > 2 we haveX A (π A ) = SC ∪ ISO [21,25], so ISO = X A (π A ). Again, recall that unitary-isotropic channels are shown to be in X A (π A ) [19]. Here we show that any antiunitaryisotropic channelū(ρ) = (1 − γ)U ρ T U † + γI/d is also in X A (π A ) for any d A . Let {|t i } be the basis with respect to which the transposition is taken. Suppose the input state ρ AB reads ρ AB = ijkl q ijkl |t i t j | ⊗ |r k r l |, where i, j and k, l are respectively indices of A and B, and {|r k,l } denotes some basis of the Hilbert space of B. Given that the spectral decomposition of A reads ρ A = α λ α |α α|, we have and so On the other hand, which involves a partial transpose. In order to express the action of π A , we need to find the eigenbasis of the reduced density operator We essentially need to find the eigenbasis of ρ T A . Rewrite ρ A as ρ A = ij α λ α t i |α α|t j |t i t j |, that is, k q ijkk = α λ α t i |α α|t j . So we obtain where we used the fact that eigenvalues λ α are real for the second line. Therefore, {|ᾱ } with |ᾱ ≡ i α|t i |t i forms the eigenbasis of ρ T A , and hence {U |ᾱ } is the eigenbasis of tr B ([ū A ⊗ I B ](ρ AB )). So starting from Eq. (26), we obtain where we used t i |ᾱ = t i |( j α|t j |t j ) = α|t i for the third line. By comparing to Eq. (25), we conclude that

Proof of Theorem 6
Note that we adopt matrix norms given by vectorization, i.e., for an operator M , M p := vec(M ) p . For density matrices · p is the equivalent to the Schatten p-norm. In particular, p = 1 yields the trace norm, and p = 2 yields the Frobenius norm, also known as Hilbert-Schmidt norm or Schur norm. Theorem 6. Diagonal discord is continuous at states such that the local density operator being measured is nondegenerate. More explicitly, let ρ AB be a bipartite state in finite dimensions such that ρ A = tr B ρ AB has distinct eigenvalues, and the smallest gap is ∆. Suppose ρ AB is a perturbed state such that ρ AB − ρ AB 1 ≤ . For sufficiently small , it holds that Proof. Notice that where the inequality follows from the triangle inequality. So by the continuity of von Neumann entropy, diagonal discord continuous as long as the discord-destroyed state π A (ρ AB ) is continuous, that is, π A (ρ AB ) and π A (ρ AB ) remain close. We show that it is so when ρ A is nondegenerate. (Indeed, discontinuity can occur in the vicinity of degeneracies, since the local eigenbases of perturbed states can be far from one another due to the nonuniqueness of eigenbases within the degenerate subspace, and hence π A (ρ AB ) cannot be continuous. This is the essence behind the examples of discontinuities given in [22,24].) Given ρ A = i p i Π i , the spectral decomposition of the perturbed marginal can take the form ρ A = i p i Π i with perturbed eigenvalues and eigenvectors, since they change continuously [32]. By triangle inequality, Since the trace distance is contractive [30], the second term directly satisfies i (Π i ⊗ I)ρ AB (Π i ⊗ I) − π A (ρ AB ) 1 ≤ ρ AB − ρ AB 1 ≤ . The first term is also well bounded due to the continuity of eigenprojection π A [32]. Now we derive an explicit bound for the first term. We assume that is sufficiently small so that any ρ A still remains nondegenerate. (This is always possible since the spectrum is bounded away from a degenerate one by assumption.) By triangle inequality, Notice that where the second line follows from M 1 ≤ √ rankM M 2 [33], and the third line follows from submultiplicativity of the Frobenius norm. Similarly for the second term. So we obtain Suppose ρ AB = (1 − ξ)ρ AB + ξτ AB where ξ ≤ and τ AB is a density operator. Then where the first inequality follows from [34], the second inequality follows from triangle inequality, and the third inequality follows from · 2 ≤ · 1 [33]. By nondegenerate perturbation theory, where O k is the operator corresponding to the k-th order, which is a bounded operator that grows at most linearly in k according to perturbation theory. So for any c > 0, there exists ζ > 0 such that for any ξ < ζ, ∞ k=2 ξ k O k 2 < c. Note that λ i = λ j when i = j and min i =j |λ i − λ j | = ∆. Let c = √ d B /∆. For sufficiently small ξ, Plugging this result into Eq. (43) and then Eq. (36), By Fannes' inequality [30], |S(π A (ρ AB )) − S(π A (ρ AB ))| ≤ 2 6 d 3 where H is the binary entropy function. By Eq. (35), The source of discontinuity in the presence of degeneracies is essentially the first term of the right hand side of Eq. (36): i (Π i ⊗ I)ρ AB (Π i ⊗ I) is not necessarily close to π A (ρ AB ).