Abstract
As the precious resource for quantum information processing, quantum coherence can be created remotely if the involved two sites are quantum correlated. It can be expected that the amount of coherence created should depend on the quantity of the shared quantum correlation, which is also a resource. Here, we establish an operational connection between coherence induced by steering and the quantum correlation. We find that the steeringinduced coherence quantified by such as relative entropy of coherence and tracenorm of coherence is bounded from above by a known quantum correlation measure defined as the oneside measurementinduced disturbance. The condition that the upper bound saturated by the induced coherence varies for different measures of coherence. The tripartite scenario is also studied and similar conclusion can be obtained. Our results provide the operational connections between local and nonlocal resources in quantum information processing.
Introduction
Quantum coherence, being at the heart of quantum mechanics, plays a key role in quantum information processing such as quantum algorithms^{1} and quantum key distribution^{2}. Inspired by the recently proposed resource theory of quantum coherence^{3,4}, researches are focused on the quantification^{5,6} and evolution^{7,8} of quantum coherence, as well as its operational meaning^{5,9} and role in quantum information tasks^{10,11,12}. When multipartite systems are considered, coherence is closely related to the wellestablished quantum information resources, such as entanglement^{13} and discordtype quantum correlations^{14}. It is shown that the coherence of an open system is frozen under the identical dynamical condition where discordtype quantum correlation is shown to freeze^{15}. Further, discordtype quantum correlation can be interpreted as the minimum coherence of a multipartite system on tensorproduct basis^{16}. An operational connection between local coherence and nonlocal quantum resources (including entanglement^{17} and discord^{18}) is presented. It is shown that entanglement or discord between a coherent system and an incoherent ancilla can be built by using incoherent operations, and the generated entanglement or discord is bounded from above by the initial coherence. The converse procedure is of equal importance: to extract coherence locally from a spatially separated but quantum correlated bipartite state. The extraction of coherence with the assistance of a remote party has been studied in the asymptotical limit^{19}. In this paper, we ask how we extract coherence locally from a single copy of a bipartite state.
The quantum steering has long been noted as a distinct nonlocal quantum effect^{20} and has attracted recent research interest both theoretically and experimentally^{21,22,23,24,25,26,27,28,29,30,31}. It demonstrates that Alice can remotely change Bob’s state by her local selective measurement if they are correlated, and is hence a natural candidate to accomplish the task of remote coherence extraction.
In this paper, we present the study of coherence extraction induced by quantum steering and the involved quantum correlation. Precisely, we introduce the quantity of steeringinduced coherence (SIC) for bipartite quantum states. Here Bob is initially in an incoherent state but quantum correlated to Alice. Alice’s local projective measurement can thus steer Bob to a new state which might be coherent. The SIC is then defined as the maximal average coherent of Bob’s steered states that can be created by Alice’s selective projective measurement. When there is no obvious incoherent basis for Bob, (for example, Bob’s system is a polarized photon), the definition can be generalized to arbitrary bipartite system where Bob’s incoherent basis is chosen as the eigenbasis of his reduced state. In this case, the SIC can be considered as a basisfree measure of Bob’s coherence. The main result of this paper is building an operational connection between the SIC and the shared quantum correlation between Alice and Bob. We prove that the SIC can not surpass the initially shared Bside quantum correlation, which is a known quantum correlation measure named as measurementinduced disturbance (MID) ^{32}. States whose relative entropy SIC can reach its upper bound are identified as maximally correlated states. For twoqubit states, while the tracenorm SIC can always reach the corresponding , we find an example of twoqubit state whose is strictly less than . This indicates that the condition for to reach the upper bound strongly depends on the measure of coherence. We further generalize the results to a tripartite scenario, where Alice can induce entanglement between Bob and Charlie in a controlled way. Since coherence of a single party is generally robust than quantum correlations involving two parties, our work provides a way to “store” quantum correlation as coherence. Besides, the coherent state induced by steering can be widely used for quantum information processing. Our results establish the intrinsic connection between coherence and quantum correlation by steering.
Results
Coherence and measurementinduced disturbance
A state is said to be incoherent on the reference basis , if it can be written as^{3}
Let be the set of incoherent state on basis Ξ. The incoherent completely positive tracepreserving (ICPTP) channel is defined as
where the Kraus operators K_{n} satisfy . According to ref. 3, a proper coherence measure of a quantum state ρ on a fixed reference basis Ξ should satisfy the following three conditions. (C1) C(ρ, Ξ) = 0 iff . (C2) Monotonicity under selective measurements on average: satisfying and , where , occurring with probability , is the state corresponding to outcome n. (C3) Convexity: .
A candidate of coherence measure is the minimum distance between ρ and the set of incoherent states
where is a distance measure on quantum states and satisfies the following five conditions. (D1) D(ρ, σ) = 0 iff ρ = σ. (D2) Monotonicity under selective measurements on average: . (D3) Convexity: . (D4) , , where U is a unitary operation, and denotes the projective measurement on basis Ξ: . (D5) . Conditions (D1D3) make sure that (C1C3) is satisfied by the coherence measure defined in Eq. (3). When (D4) is satisfied, the coherence of ρ on the reference basis Ξ can be written as
As proved in ref. 3, the relative entropy and the l_{1} matrix norm satisfies all the conditions (D1D4), which makes the corresponding coherence measures and satisfy the conditions (C1C3). As discovered recently^{33}, the tracenorm distance does not satisfy (D2).
Introduced in ref. 32, MID characterizes the quantumness of correlations. MID of a bipartite system ρ is defined as the minimum disturbance caused by local projective measurements that do not change the reduced states and
where the infimum is taken over projective measurements which satisfy and , and is a distance on quantum states, which satisfies conditions (D1D5) and further (D6) . It can be checked that (D6) can be satisfied by relative entropy but not satisfied by l_{1}norm. Comparing Eq. (5) with Eq. (4), we find MID is just the coherence of the bipartite state ρ on the local eigenbasis .
For later convenience, we introduce Bside MID as
goes to zero for Bside classical states, which can be written as , while is strictly positive for if . Notice that for one do not have a coherence interpretation.
Definition of steeringinduced coherence
As shown in Fig. 1, Alice and Bob initially share a quantum correlated state ρ, and Bob’s reduced state ρ_{B} is incoherent on his own basis. Now Alice implements a local projective measurement on basis Ξ_{A}. When she obtains the result i (which happens with probability ), Bob is “steered” to a coherent state . We introduce the concept of SIC for characterizing Alice’s ability to create Bob’s coherence on average using her local selective measurement.
Definition (Steeringinduced coherence, SIC)
For a bipartite quantum state ρ, Alice implements projective measurement on basis (). With probability , she obtains the result , which steers Bob’s state to . Let () be the eigenbasis of reduced states ρ_{B}. The steeringinduced coherence is defined as the maximum average coherence of Bob’s steered states on the reference basis
where the maximization is taken over all of Alice’s projective measurement basis Ξ_{A}, and the infimum over is taken when ρ_{B} is degenerate and hence is not unique.
Since Bob’s initial state ρ_{B} is incoherent on its own basis , the SIC describes the maximum ability of Alice’s local selective measurement to create Bob’s coherence on average. We verify the following properties for .
(E1) , and iff ρ is a Bside classical state.
(E2) Nonincreasing under Alice’s local completelypositive tracepreserving channel: .
(E3) Monotonicity under Bob’s local selective measurements on average: satisfying , where and .
(E4) Convexity: .
Proof. Condition (E1) can be proved using the method in ref. 31, where it is proved that vanishes iff ρ is a Bside classical state. (E2) is verified by noticing that the local channel can not increase the set of Bob’s steered states, and hence the optimal steered states may not be steered to after the action of channel . The conditions (E3) and (E4) are directly derived from conditions (C2) and (C3) for coherence.
Relation between SIC and MID
Intuitively, Alice’s ability to extract coherence on Bob’s side should depend on the quantum correlation between them. The following theorem gives a quantitative relation between the SIC and quantum correlation measured by Bside MID .
Theorem 1. When the distance measure in the definition of MID and coherence satisfies conditions (D1D6), the SIC is bounded from above by the Bside MID, i.e.,
Proof. We start with the situation that ρ_{B} is nondegenerate and hence one do not need to take the infimum in Eqs (5) and (7). By definition, we have
where .
After Alice implements a selective measurement on basis Ξ_{A}, the average coherence of Bob’s state becomes
The second equality holds because (condition (D5)) and . Since selective measurement does not increase the state distance (condition (D2)), we have , and hence Eq. (8) holds.
The generalization to degenerate state is straightforward. We choose to reach the infimum of , which may not be the optimal eigenbasis for . Hence we have .
According to ref. 17, the coherence of a quantum system B can in turn be transferred to the entanglement between the system and an ancilla C by incoherent operations. The established entanglement, measured by the minimum distance between the state ρ^{BC} and a separable state as , is bounded from above by the initial coherence of B. Here is the set of separable states and the state distance D is required not to increase under tracepreserving channels , which is automatically satisfied when we combine conditions (D2) and (D3).
This leads to the threeparty protocol as shown in Fig. 2, where Alice’s local selective measurement can create entanglement between Bob and Charlie. In this protocol, Bob and Charlie try to build entanglement between them from a product state , but are limited to use incoherent operations. Since ρ_{B} is incoherent on his eigenbasis , Bob and Charlie can build only classically correlated state without Alice’s help. Now Alice implement projective measurement and on the outcome i, the state shared between Bob and Charlie is steered to which can be entangled. The following corollary of theorem 1 gives the upper bound of the steeringinduced entanglement.
Corollary 1 Alice, Bob and Charlie share a tripartite state ρ, which is prepared from the product state using an ICPTP channel on BC: . Here is the reference basis of coherence. Alice’s local selective measurement can establish entanglement between Bob and Charlie, and the established entanglement on average is bounded from above by the initial Bside MID between Alice and Bob
Proof. Before Alice implement the measurement, the state shared between Bob and Charlie is incoherent on basis and hence can be written as . Apparently, , so Bob and Charlie is classically correlated.
On the measurement outcome i, the entanglement between Bob and Charlie becomes which satisfies . Notice that and hence . Eq. (11) is arrived by noticing that from theorem 1.
Now we consider a general tripartite state ρ. If the reduced state is nondegenerate, one can follow the same steps and prove that
whenever ρ^{BC} is incoherent on basis . Here is the {BC}side MID between Alice and the combination of Bob and Charlie. However, when ρ^{BC} is degenerate, the condition that the tripartite state ρ is prepared from by an ICPTP channel on BC is stringent. For example, the state where , with ρ^{BC} incoherent on basis , violates Eq. (12), since but the lefthandside reaches unity for Alice’s measurement basis . It indicate that the state ρ^{X} can not be prepared from a product state in the form using only incoherent operations.
States to reach the upper bound
According to theorem 1, Bob’s maximal coherence that can be extracted by Alice’s local selective measurement is bounded from above by the initial quantum correlation between them. Since the relative entropy is the only distance measure found to date which satisfies all the conditions (D1D6), we employ relative entropy as the distance in the definition of coherence and MID, and discuss the states which can reach the upper bound of theorem 1.
Theorem 2. The SIC can reach Bside MID
for maximally correlated states .
Proof. Any maximally correlated state can be written in a pure state decomposition form with and . Here has eigenbasis . In order to calculate the Bside MID, we consider Bob’s projective measurement , which takes the bipartite state to . Apparently, . By definition, we have
In order to extract the maximum average coherence on Bob’s side, Alice measures her quantum system on basis , where , and d_{A} is the dimension of A. On the measurement result k, Bob’s state is steered to where , which happens with probability . Apparently, and hence . Meanwhile, we have . The coherence of steered state is then
for any outcome k. Therefore we arrive at Eq. (13).
Any pure bipartite state can be written in a Schmidt decomposition form , and hence belongs to the set of maximally correlated states. As introduced in ref. 17, a maximally correlated states ρ^{mc} is prepared from an product states using an incoherent unitary operator, and its entanglement E(ρ^{mc}) can reach the initial coherence of ρ_{B}. Further, for maximally correlated states, one can check the equality, . Therefore, ρ^{mc} can be used in a scenario where coherence is precious and entanglement is not as robust as singleparty coherence. Precisely, consider the situation where Alice and Bob share a maximally correlated state but they are not use it in a hurry. To store the resource for latter use, she can transfer the entanglement between them into Bob’s coherence using her local selective measurement. Bob stores his coherent state as well as Alice’s measurement results. When required, Bob can perfectly retrieve the entanglement by preparing a maximally correlated state using only incoherent operations.
Twoqubit case, relation between l_{1}norm of SIC and tracenorm distance of Bside MID
One cannot define MID based on the l_{1}norm distance, since it does not satisfy (D6) in general. However, it can be checked that for singlequbit states ρ_{B} and σ_{B}, ^{34}, where r^{ρ} and r^{σ} are Bloch vectors of ρ_{B} and σ_{B} respectively. Hence the l_{1}norm of coherence for a singlequbit state ρ_{B} can be written as
Besides, D^{t}, which satisfies condition (D6), is proper to be used as a distance measure for MID. Therefore, when the Bob’s particle is a qubit, it is meaningful to study the relation between l_{1}norm of SIC and tracenorm distance of Bside MID. Now we consider a twoqubit state ρ, and employ in the definition of as in Eq. (7) and prove the following theorem.
Theorem 3. For a twoqubit state ρ, we have
Proof. The state of a twoqubit state can be written as , where the coefficient matrix can be written in the block form .
For nondegenerate case b ≠ 0, we choose the eigenbasis of ρ_{B} for the basis of density matrix and hence b = (0, 0, b_{3}). Further, a proper basis of qubit A is chosen such that the matrix T is in a triangle form with . We calculated the explicit form of and and obtain
For degenerate case with b = 0, we can always chose proper local basis such that T is diagonal. Here we impose T_{11} ≥ T_{22} ≥ T_{33} without loss of generality. Direct calculations lead to
We check that, for the state , we have , but according to theorem 3, . It means that relative entropy of coherence and norm of coherence are truly different measures of coherence.
Discussion
In this paper, we have introduced the notion of SIC which characterizes the power of Alice’s selective measurement to remotely create quantum coherence on Bob’s site. Quantitative connection has been built between SIC and the initially shared quantum correlation measured by side MID. We show that SIC is always less than or equal to side MID. Our results are also generalized to a tripartite scenario where Alice can build the entanglement between Bob and Charlie in a controlled way.
Next, we discuss a potential application of SIC in secrete sharing. Suppose Alice and Bob share a twoqubit state , whose SIC reaches unity. When Alice measures her state on different basis, Bob’s state is steered to, e.g., or with . The coherence of states in reach unity on basis and vise visa. Consequently, when we measure the states in the set on basis , the outcome is completely random. It is essential to quantum secret sharing using . In this sense, the SIC is potentially related to the ability for Alice to share secret with Bob.
Coherence and various quantum correlations, such as entanglement and discordlike correlations, are generally considered as resources in the framework of resource theories^{9,35}. By coining the concept of SIC, we present an operational interpretation between measures of those two resources, SIC and MID, and open the avenue to study their (ir)reversibility. The applications of various coherence quantities like SIC in manybody systems, as in the case of entanglement^{36,37,38}, can be expected.
Additional Information
How to cite this article: Hu, X. and Fan, H. Extracting quantum coherence via steering. Sci. Rep. 6, 34380; doi: 10.1038/srep34380 (2016).
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Acknowledgements
This work was supported by NSFC under Grant Nos 11447161, 11504205 and 91536108, the Fundamental Research Funds of Shandong University under Grant No. 2014TB018, the National Key Basic Research Program of China under Grant No. 2015CB921003, and Chinese Academy of Sciences Grant No. XDB01010000.
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X.H. and H.F. contributed the idea. X.H. performed the calculations. X.H. and H.F. wrote the paper. All authors reviewed the manuscript and agreed with the submission.
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Hu, X., Fan, H. Extracting quantum coherence via steering. Sci Rep 6, 34380 (2016). https://doi.org/10.1038/srep34380
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