Tight bounds for the quantum discord
Quantum discord quantifies quantum correlations beyond entanglement and assumes nonzero values, which are notoriously hard to compute, for almost all quantum states. Here we provide computable tight bounds for the quantum discord for qubit-qudit states. In the case of two qubits our lower and upper bounds coincide for a 7-parameter family of filtered -states, whose quantum discords can therefore be evaluated analytically. An application to the accessible information of the binary qubit channel is also presented. For the qubit-qudit state output by the circuit of deterministic computation with one qubit, nontrivial lower and upper bounds that respect the zero-discord conditions are obtained for its quantum discord.
pacs:03.67.-a, 03.65.Ta, 03.67.Lx
The quantum discord discord1 (); discord2 () has gradually become another important resource for quantum informational processing tasks besides the entanglement. Firstly, in certain quantum mechanical tasks such as the deterministic quantum computation with one qubit (DQC1) DQC1 (), the quantum advantages can be gained Datta () with the presence of quantum discord while the entanglement is absent. Secondly the quantum discord is shown to be a better indicator of the quantum phase transition in certain physical systems than the entanglement pt (). Thirdly, in addition to its original interpretation via Maxwell demon demon (), the operational interpretations of quantum discord via state merging merg () establish firmly the status of the quantum discord as an essential resource amidst other concepts of quantum information.
where denotes the von Neumann entropy and the minimum is taken over all the positive operator valued measures (POVMs) on the subsystem with being the probability of the -th outcome and being the conditional state of subsystem . The minimum can also be taken over all the von Neumann measurements discord1 (). These two definitions are in general inequivalent even for qubits and our proposed bounds in Eq.(5) below apply to both of them. When the measurements are made on subsystem the quantum discord can be defined similarly and is in general different from .
Quantum discord assumes nonnegative values and zero-discord states are relatively well understood: if and only if there exists a complete orthonormal basis for the subsystem together with some density matrices for the subsystem such that . Recently various methods to detect zero discord cond1 (); condition () have been proposed for a given state as well as for an unknown state witness (). Besides its initial motivation in pointer states in measurement problem discord1 (), vanishing quantum discord is also found to be related to the complete positivity of a map CP () and the local broadcasting of quantum correlations broadcast ().
Unfortunately zero-discord states are of zero measure almost () and the nonzero values of the quantum discord are notoriously difficult to compute because of the minimization over all the possible measurements. There are only a few analytical results including the Bell-diagonal states bells () and rank-2 states r2 (), in addition to a thorough numerical calculation max () in the case of von Neumann measurements. For two-qubit -states, since there are counter examples chen (); lu () for the algorithm proposed in X2 (), the evaluation of their quantum discords remains a nontrivial task. It is therefore desirable to have some computable bounds for the quantum discord so that the question of how large or small the quantum discord can possibly be, e.g., in the DQC1 circuit, can be answered more reasonably.
In this Letter we shall provide computable tight bounds for the quantum discord of qubit-qudit states. For a family of two-qubit filtered -states with 7 parameters up to local unitary transformations (LUTs) our lower and upper bounds coincide so that their quantum discords can be evaluated analytically. Also we present an application to the accessible information of the binary qubit channel for which our lower and upper bounds can coincide. For the quantum discord of the qubit-qudit state output by the DQC1 circuit we derive nontrivial lower and upper bounds, which qualitatively agree with the zero discord conditions cond1 (); witness () comparing with the estimation in Datta ().
To present our main result we need some notations. First of all, we denote by a qubit-qudit state with the reduced density matrices and for the qubit and the qudit respectively. Let be the Bloch vector for with its length given by . Without loss of generality we assume that the reduced density matrix of qubit , on which the measurement is performed, is invertible since otherwise we have a product state, which has zero discord. Therefore the following filtered density matrix is well defined:
Secondly, let be the identity matrix and three standard Pauli matrices. We associate with every qubit-qudit state (or ) a positive semi-definite two-qubit operator
in which stands for the two-qudit swapping operator. Then the equation , where , has four real solutions hil (); ulm (). It should be noted that each is as readily computable as the concurrence of , which equals to filter (), in the case of two-qubit states for which we have with being the correlation matrix defined by . These values , showing explicitly the dependence on , are invariant under an arbitrary Lorentz transformation (LT) , satisfying , that brings to . The local filter acting on qubit brings to and induces an LT to , from which it follows that .
Thirdly, we denote by the matrix obtained from the matrix by deleting its first row and column. Let be the largest eigenvalue of corresponding to the eigenvector . We define to be the unit vector along the direction , where and . In the case of we define . By measuring the observable on the qubit we obtain a suboptimal value for the quantum discord as in Eq.(1) without minimization. For a two-qubit state , since , we have where is the correlation matrix for defined by .
Lastly, we need to introduce an increasing convex function of where
and . For any density matrix it holds since we always have and if we have which can be read off from the information diagram between the entropy and index of coincidence ci ().
Theorem For a qubit-qudit state , with subsystem being a qubit, the following bounds hold
where . In the case of two qubits, the lower and upper bounds coincide if .
A detailed proof can be found in online () and an outline is given in what follows. The upper bound holds true by definition. The key to prove the lower bound is the Koashi-Winter relation rel (): , where is the entanglement of formation eof () of the state with being a purification of in a system . The lower bound follows from , where is the concurrence of , and an evaluation of the concurrence by applying Theorem 4 in ulm (). Furthermore the ensemble of induced by measuring on qubit is optimal for the tangle of whose value can be readily obtained by applying Theorem 1 in osb (). Since is a concave function of we have bound , which coincides with the lower bound if , i.e., , since in the case of two qubits.
For two-qubit states we have carried out some numerical calculations and a comparison with our bounds is shown in Fig.1, where for convenience the upper bound is taken to be , which is slightly weaker than Eq.(5). For about 70% of randomly chosen states of rank 4 the differences between our bounds and the values of quantum discord obtained by minimization over von Neumann measurements lie in the range of with a maximal difference about . We note that for some states our lower bounds are negative and therefore are trivial. On the other hand there exist several nontrivial families of states, for which the upper and lower bounds coincide so that analytical expressions of quantum discord for these states can be obtained. In these cases the two definitions of quantum discord become identical.
Example 1: Bell-diagonal states bells (). A Bell-diagonal state has a density matrix with being real and . It is clear that and . As a result we have so that upper and lower bounds as in Eq.(5) coincide.
Example 2: Rank-2 states r2 (). Every two-qubit state of rank 2 admits a 3-qubit purification with the reduced density matrix being a two-qubit state of rank 2. As a result so that the upper and lower bounds as in Eq.(5) coincide.
Example 3: -states. In many scenarios -state arises as the two-particle reduced density matrix as long as there is a certain symmetry of the physical system. In general a 2-qubit -state
has 7 independent real parameters. Since the quantum discord is invariant under LUTs, we can assume without loss of generality that and are real and therefore we have in fact only 5 real parameters, which can be conveniently taken as those nonzero entries of the correlation matrix , i.e., , , and (). Since holds for qubits it is easy to see that four solutions to the equation are and . Furthermore the correlation matrix is diagonal with entries and . Thus as long as
we have so that the upper and lower bounds of the quantum discord coincide. We denote by the -state that satisfies the condition Eq.(7) and the optimal observable to measure is if and otherwise. Explicitly,
Example 4: Filtered -states. Take an arbitrary 2-qubit state satisfying i) , i.e., the upper and lower bounds for coincide, and ii) is proportional to identity, for example the state with as in example 3, and take an arbitrary invertible Hermitian operator acting only on qubit . Then for the filtered state we have from the definition Eq.(2) so that has also a coincided lower and upper bound. Because and represent the same filter for any nonzero real , there are independent real parameters for . Thus we have a family of filtered -states with 7 parameters for which the quantum discords can be evaluated analytically.
Example 5: Accessible information. A quantum communication channel is defined by the action of sending states with probabilities to a receiver who can perform any possible POVM to gain the information of . The accessible information is the maximum of the mutual information of the joint probability distribution over all POVMs . If we introduce a bipartite state where is a complete orthonormal basis then we have and where is the Holevo bound. For qubit channels the theorem provides an upper bound as well as a lower bound given by the measurement along for the accessible information.
Consider a binary qubit channel and denote by and the Bloch vectors of qubit states and , respectively, and . The lower bound for provided by coincides with the lower bound () in fuchs (). The correlation matrix of the two-qubit state reads
where . We note that is of rank 2 and therefore four solutions to the equation are , where
from which we obtain . The correlation matrix of is of rank 1 and the largest eigenvalue of is . By solving we obtain in which case the upper and lower bounds for coincide, meaning that the optimal POVM for is a von Neumann measurement (along the direction ). This proves explicitly, for the first time as far as we know, that the optimal von Neumann measurement given in fuchs () is also optimal among POVMs.
Example 6: DQC1 states. Let us consider the qubit-qudit state output by the DQC1 circuit, aiming at calculating efficiently the trace of a unitary operation acting on the qudit , with a density matrix DQC1 ()
where and . Let be the eigensystem of and obviously is separable where with . The nonvanishing quantum discord of , which has been estimated in Datta (), is argued to be responsible for the quantum speedup. Recently it is found that witness (); cond1 () if and only if or where . The estimation in Datta (), though respects the trivial condition , is insensitive to the condition , i.e., a nonzero value of the quantum discord is estimated in this case, in which and can be in the same time typical, i.e., . Our Theorem provides nontrivial bounds that respect both two zero-discord conditions.
For simplicity we shall assume first, i.e., . In this case we have and . It turns out that where has eigenvalues . As a result we have () which is also the largest eigenvalue of with eigenvector where . From Eq.(5) and for it follows
In deriving the upper bound we have used in which the first inequality is valid for arbitrary and becomes an equality in the case of . Thus the upper bound for given in Eq.(13) holds for all unitary and reaches its maximum at and . It is obvious that in the case of or in the case of pure qubit and the lower and upper bounds coincide so that the quantum discord vanishes.
To conclude, in view of the hardship of computing the nonzero values of quantum discord, the computable tight bounds provided here for qubit-qudit states should be useful in further quantitative studies of the relation between quantum discord and phase transitions, quantum speedups, and so on. Notably our bounds enable us to evaluate analytically the quantum discords of a family of filtered -states with 7 parameters up to LUTs (for comparison there are 9 parameters for a general 2-qubit state) and to estimate the quantum discord in the DQC1 circuit more reasonably. Also our bounds are applicable for the classical correlation and accessible information. Though we have restricted to qubit-qudit systems, for which the concurrence can be evaluated exactly, the bounds for the quantum discord of a general bipartite state, which might not be specially tight, can be obtained in a similar way via the bounds for the entanglement of formation .
We acknowledge the financial supports of A*STAR Grant No. R-144-000-189-305, CQT project WBS: R-710-000-008-271, and NNSF of China (Grant No. 11075227).
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Appendix A Supplementary material: Proof of the theorem
Let be the eigenstates of a given qubit-qudit state with eigenvalues . Then is a purification of in a system . Then is of rank 2 and is supported on the 2-dimensional subspace spanned by an orthornormal basis
Any state supported on the subspace spanned by can be expanded with the help of four generalized Pauli operators
where coefficients () are real . For example we have . From the fact it follows that and
in which has been defined in Eq.(3). The proof of the theorem is divided into the following four steps.
Lemma i) , ii) , iii) , and iv) For a two-qubit state the bound
holds true. The entanglement of formation, concurrence, and tangle of are defined to be, respectively,
where the minimization is taken over all possible ensembles with being pure and .
Proof. i) Let be the optimal ensemble for and we obtain
in which the first inequality is due to the second and third inequalities are due to the fact that is a convex and increasing function of , respectively.
ii) Since is of rank 2, its concurrence can be exactly evaluated according to Theorem 4 in ulm (), where is the second largest solution to the equation . Since the local filter acting on qubit brings to and induces an LT to we have and consequently .
iii) For an arbitrary state , with and denoting the reduced density matrices, we introduce a matrix whose elements are given by
According to Theorem 1 in osb () we have where is the smallest eigenvalue of the matrix obtained by deleting the first row and column of the matrix defined above. It turns out that where
Thus which proves Lemma iii.
iv) Let be an optimal decomposition of for the tangle and we have , the eigenvector of corresponding to its largest eigenvalue , and . If is the unit vector along the direction then we have and . Therefore if we measure the observable on the qubit we obtain outcome with probability leaving qubit in the state . Thus
in which the second equality is due to the fact that is a qubit state and the inequality is due to the fact that is a concave function of . It follows immediately that
Proof of the theorem. The upper bound is trivially true. Because of the Kaishi-Winter relation rel () , Lemmas i and ii lead to the lower bound in the first statement. In the case of two qubits, since leading to , the lower bound of in Eq.(5) becomes . From Lemma iv we see that is an upper bound of and therefore also an upper bound of . If then because of Lemma iii, which means that the upper and lower bound coincide.