Abstract
Simultaneous existence of correlation in complementary bases is a fundamental feature of quantum correlation and we show that this characteristic is present in any nonproduct bipartite state. We propose a measure via mutually unbiased bases to study this feature of quantum correlation and compare it with other measures of quantum correlation for several families of bipartite states.
Introduction
Quantum systems can be correlated in ways inaccessible to classical objects. This quantum feature of correlations not only is the key to our understanding of quantum world, but also is essential for the powerful applications of quantum information and quantum computation^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}. In order to characterize the correlation in quantum state, many approaches have been proposed to reveal different aspects of quantum correlation, such as the various measures of entanglement^{6,7,8,9,10} and the various measures of discord and related measures^{17,18,19,20,21}, etc. It is believed that some aspects of quantum correlation could still exist without the presence of entanglement and these aspects could be revealed via local measurements with respect to some basis of a local system.
The simultaneous existence of complementary correlations in different bases is revealed very early by the Bell's inequalities^{22}. Bell's inequalities quantify quantum correlation via expectation values of local complementary observables. In Ref. 23, the feature of genuine quantum correlation is revealed by defining measures based on invariance under a basis change: for a bipartite quantum state, the classical correlation is the maximal correlation present in a certain optimum basis, while the quantum correlation is characterized as a series of residual correlations in the bases mutually unbiased (MU) to the optimum basis. In this paper, we use the fact that the essential feature of the quantum correlation is that it can be present in any two mutually unbiased bases (MUBs) simultaneously. Thus, one of the two bases is not necessarily the optimum basis to reveal the maximal classical correlation in this paper. With respect to the measure proposed here, we shall show that only the product states do not contain quantum correlation. A product state contains neither any quantum correlation nor any classical correlation; while any nonproduct bipartite state contains correlation that is fundamentally quantum! We shall also reveal interesting properties of this measure by comparing this measure to other measures of quantum correlation for several families of bipartite states.
The MUBs constitute now a basic ingredient in many applications of quantum information processing: quantum state tomography^{24}, quantum cryptography^{25}, discrete Wigner function^{26}, quantum teleportation^{27}, quantum error correction codes^{28} and the mean king's problem^{29}. Two orthonormal bases {ψ_{i}〉} and {ϕ_{j}〉} of a ddimensional Hilbert space H are said to be mutually unbiased if and only if
In a ddimensional Hilbert space, there exist at least 3 MUBs (when d is a power of a prime number, a full set of d + 1 MUBs exists, more details can be found in Ref. 30).
We recall the quantity defined in Ref. 23. Let H_{ab} = H_{a}H_{b} with dim H_{a} = d_{a} and dim H_{b} = d_{b} be the state space of the bipartite system A+B shared by Alice and Bob. Let {i〉} and j′〉 be the orthonormal bases of H_{a} and H_{b} respectively. Alice selects a basis {i〉} of H_{a} and performs a measurement projecting her system onto the basis states. The Holevo quantity χ{ρ_{ab}{i〉}} of ρ_{ab} with respect to Alice's local projective measurement onto the basis {i〉〈i}, is defined as . A basis {i〉} that achieves the maximum (denoted as C_{1}(ρ_{ab})) of the Holevo quantity is called a C_{1}basis of ρ_{ab}. There could exist many C_{1}bases for a state ρ_{ab} and the set of these bases is denoted as . Let be the set of all bases that are mutually unbiased to Π^{a}, . The quantity of quantum correlation in Ref. 23, denoted by Q_{2}(ρ_{ab}), is defined as
In other words, Q_{2} is defined as the Holevo quantity of Bob's accessible information about Alice's results, maximized over Alice's projective measurements in the bases that are mutually unbiased to a C_{1}basis and further maximized over all possible C_{1}bases (if not unique). Thus, Q_{2} is actually the maximum correlation present simultaneously in any set of two MUBs of which one is a C_{1}basis.
Results
Our approach  Correlation measure based on MUBs
We now present our approach in a more general way.
Definition
Let Δ denote the set of all twoMUB sets, i.e.,
We define
The quantity represents the maximal amount of correlation that is present simultaneously in any two MUBs. Similar to the other usual measures of quantum correlation, is local unitary invariant, that is, for any unitary operators U_{a} and U_{b} acting on H_{a} and H_{b} respectively.
versus
Although both and Q_{2} represent quantum correlation (here the symbol is associated with Correlation in two MUBs) instead of classical correlation (represented by C_{1}), they are actually quite different. As is defined as the maximum correlation present simultaneously in any two MUBs while Q_{2} is the maximum correlation present simultaneously in two MUBs of which one is a C_{1}basis, it is obvious that
for any ρ_{ab}. In a sense, is more essential than Q_{2} since the maximum in the former one is taken over arbitrarily two MUBs. Thus, may reveal more quantum correlation than Q_{2}.
From the following Theorem and Examples, one knows that there do exist states such that (Example 4) and (Examples 1,2,3). A clear illustration of the difference between and Q_{2} is also given in Fig. 3. We know that Q_{2} does not exceed quantum discord D for all the known examples^{23}. However, one can easily find states such that exceeds quantum discord D (See Fig. 3).
The nullity of
Now we show that any bipartite quantum state contains nonzero correlation simultaneously in two mutually unbiased bases unless it is a product state, this result is stated as the following theorem, while the proof is left to the Method.
Theorem
if and only if ρ_{ab} is a product state.
In a sense, this theorem implies that, any nonproduct bipartite state contains genuine quantum correlation and reveals the amount of quantum correlation in the state. In addition, we know that is different from the quantity Q_{2} in Ref. 23 since Q_{2}(ρ_{cq}) = 0 for any classicalquantum state ρ_{cq} while only for product states. The difference between the measure and other measures of quantum correlation shall be discussed below for several families of bipartite states in more details.
Now, we shall calculate the quantity for several families of bipartite states and see how our measure in terms of MUBs is well justified as a measure of quantum correlation.
Example 1  Pure states
For a bipartite pure state with the Schmidt decomposition , . It can be easily checked that coincides with the entropy of either reduced state for any pure state, which is also the usual measure of entanglement in a pure state.
Example 2  Werner states
Next, we consider the Werner states of a dd dimensional system^{5},
where −1 ≤ α ≤ 1, I is the identity operator in the d^{2}dimensional Hilbert space and is the operator that exchanges A and B. For a local measurement with respect to basis states {e_{i}〉} of H_{a}, with probability , Alice will obtain the kth basis state e_{k}〉 and Bob will be left with the state , where with α_{kj} = 〈e_{k}j〉. It is straightforward to show that
The entanglement of formation E_{f} for the Werner states is given as , with h(x) ≡ −x log_{2}x − (1 − x) log_{2}(1 − x)^{31}. The three different measures of quantum correlation, i.e., , the quantum discord D and the entanglement of formation E_{f}, are illustrated in Fig. 1 for comparison. From this figure, we see that the curve for entanglement of formation intersects the other two curves; thus, E_{f} can be larger or smaller than .
Example 3  Isotropic states
For the dd isotropic states
where P^{+} = Φ^{+}〉 〈Φ^{+}, is the maximally entangled pure state in . Let {e_{k}〉 〈e_{k}} be an arbitrarily given projective measurement on Alice's part. Bob's state after after Alice gets the kth measurement result is
where with α_{kj} = 〈e_{k}j〉. As the eigenvalues of does not depend on the basis for Alice's measurement, one can easily show that
The entanglement of formation E_{f} for the isotropic states is given as^{32,33}
where . The quantum discord of the isotropic state is^{34}
The three different measures of quantum correlation, i.e., , the quantum discord D and the entanglement of formation E_{f}, are illustrated in Fig. 2 for comparison. From this figure, we see that the curve for entanglement of formation intersects the other two curves; thus, E_{f} can be larger or smaller than .
Example 4  A family of twoqubit states
As the last example, we consider a family of twoqubit states that are Belldiagonal states. This family of states admit the form
We rearrange the three numbers {r_{1}, r_{2}, r_{3}} according to their absolute values and denote the rearranged set as {, , } such that . Next we show that
Without loss of generality, we prove (14) only for the case . A projective measurement performed on qubit A can be written as , parameterized by the unit vector . When Alice obtains p_{±}, Bob will be in the corresponding states , each occurring with probability . The entropy reaches its minimum value when . Let and with ax + by = 0, then is mutually unbiased to , where , . It is immediate that and . Thus as desired since h(c) is a monotonic decreasing function when . Our quantity is compared with the quantum discord D and the entanglement of formation E_{f} for ρ_{1} and ρ_{2} in Fig. 3.
From the left figure of Fig. 3, it is clear that is quite different from both D and Q_{2}. Unlike Q_{2} that does not exceed D for all known examples, can exceed D. We have when p is closed to , while when p is closed to 0 or 1; we also have when and increases monotonously while Q_{2}(ρ_{1}) decreases monotonously when p deviates from . In Fig. 3, the difference between our measure and the other measures is well illustrated by the extreme cases when p = 0 or 1 in the left figure and when p = 0 in the right figure. For example, for , our measure has a finite value while the other measures vanish.
Correlation revealed via more MUBs
In addition, we can define a quantity based on m MUBs (3 ≤ m ≤ dim H_{a} + 1), namely,
where
It is clear that . The following are obvious from the arguments in the previous examples: i) if and only if ρ is a product state, ii) for both the Werner states and the isotropic states and iii) for the family of twoqubit states in Eq. (13).
Discussion
We have provided a very different approach to quantify quantum correlation in a bipartite quantum state. Our approach captures the essential feature of quantum correlation: the simultaneous existence of correlations in complementary bases. We have proved that the only states that don't have this feature are the product states, which contains no correlation (classical or quantum) at all. Thus, any nonproduct state contains correlation that is fundamentally quantum. This feature of quantum correlation characterized here could be the key feature that enables quantum key distribution (QKD) with entangled states, since the quantum correlation that exists simultaneously in MUBs, which can be quantified by , is the resource for entanglementbased QKD via MUBs.
Method
Proof of the Theorem in the Main Text
The ‘if’ part is obvious and we only need to show the ‘only if’ part. In other words, we only need to prove that ρ_{ab} = ρ_{a} ρ_{b} if either or for any MUB pair . It is equivalent to show that both and for a certain MUB pair if ρ_{ab} is not a product state.
We assume that ρ_{ab} is not a product state, then the maximal classical correlation is nonzero, i.e., C_{1}(ρ_{ab}) ≠ 0. Let , we have χ(ρ_{ab}{e_{i}〉}) ≠ 0. Therefore, we only need to find a second basis (MU to {e_{i}〉}) such that the corresponding Holevo quantity is nonzero. We denote the projective measurement corresponding to {e_{i}〉} by . Then . As C_{1}(ρ_{ab}) ≠ 0, we know that and at least for some k_{0} and l_{0}. We arbitrarily choose a basis {f_{i}〉} that is MU to {e_{i}〉}. If χ{ρ_{ab}{f_{i}〉}} ≠ 0, then we already obtain the second basis and the theorem is true.
If χ{ρ_{ab}{f_{i}〉}} = 0, we can construct the MUB pair as follows. As in this case, the measurement corresponding to {f_{i}〉} yields the following output state
Thus, ρ_{ab} can be represented as
with respect to the local basis {f_{i}〉} and at least one of the offdiagonal blocks is not zero (otherwise, ρ_{ab} is a product state). Without loss of generality we assume that the (1,2)blockentry of the above matrix is nonzero. It follows that there exists a 2 by 2 unitary matrix U_{2}, such that, under the local basis {U_{2} ⊕ I_{d}_{−2}f_{i}〉}, the state admits the form
with and σ_{b} ≠ ρ_{b}. That is χ{ρ_{ab}{U_{2} ⊕ I_{d}_{−2}f_{i}〉}} ≠ 0. This unitary matrix U_{2} can be chosen as
with a very small positive number. Even though χ{ρ_{ab}{U_{2} ⊕ I_{d}_{−2}f_{i}〉}} could be very small, it is nonzero. As is a very small and χ{ρ_{ab}{e_{i}〉}} ≠ 0, we also have χ{ρ_{ab}{U_{2} ⊕ I_{d}_{−2}e_{i}〉}} ≠ 0. Thus, the Holevo quantity is nonzero at least for a certain MUB pair (i.e., {U_{2} ⊕ I_{d}_{−2}e_{i}〉} and {U_{2} ⊕ I_{d}_{−2}f_{i}〉}) and therefore .
Thus, for any ρ_{ab} that is not a product state. The proof is completed.
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Acknowledgements
Y. Guo acknowledges support from the Natural Science Foundation of China (No. 11301312, No. 11171249), the Natural Science Foundation of Shanxi (No. 20130210011, No. 20120110012) and the Research startup fund for Doctors of Shanxi Datong University (No. 2011B01). S. Wu acknowledges support from the Natural Science Foundation of China (No. 11275181) and the Fundamental Research Funds for the Central Universities (No. 20620140007).
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Y.G. and S.W. contributed equally to the paper.
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Guo, Y., Wu, S. Quantum correlation exists in any nonproduct state. Sci Rep 4, 7179 (2014). https://doi.org/10.1038/srep07179
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