Introduction

Quantum coherence, which arises from quantum superposition, is a fundamental feature of quantum mechanics, and it is also an essential ingredient in quantum information and computation1. Furthermore, in some emergent fields, such as quantum metrology2,3, nanoscale thermodynamics4,5,6,7,8 and quantum biology9,10,11,12, quantum coherence plays a central role.

The information-theoretic quantification of quantum coherence is a successful application of quantum resource theory13. Baumgratz et al. proposed the basic notions of incoherent states, incoherent operations and a series of necessary conditions any measures of coherence should satisfy. In this sense, coherence is defined as the resource relative to the set of incoherent operations. According to the postulates in the framework, relative entropy of coherence13, l1-norm of coherence13 and other coherence metrics14,15,16,17,18 have been put forward. Based on coherence measures, the relations between quantum coherence and other resources14,19,20, the complementarity relations of quantum coherence21 and other properties of quantum coherence22,23 have been investigated. Mainly due to the interest aroused by the resource theory of quantum coherence, there are several attempts at understanding the role of coherence as a resource for quantum protocols. For example, in the incoherent quantum state merging, which is the same as standard quantum state merging up to the fact that one of the parties has free access to local incoherent operations only and has to consume a coherent resource for more general operations, the entanglement-coherence sum is non-negative, and no merging procedure can gain entanglement and coherence at the same time24. Perfect incoherent teleportation of an unknown state of one qubit is possible with one singlet and two bits of classical communications25. Here, the incoherent teleportation is the same as standard teleportation up to the fact that local operations and classical communications are replaced by local incoherent operations and classical communications. Furthermore, the notion of coherence as a symmetry relative to a group of translations naturally shows up in the context of quantum speed limits because the speed of evolution is itself a measure of asymmetry relative to time translations26.

As we know, both quantum coherence and entanglement closely relate to quantum superposition. Moreover, many quantum information protocols, such as dense coding27 and teleportation28, would be impossible without the assistance of entanglement. Therefore, inspired by work on entanglement, we want to directly relate quantum coherence with the protocols of quantum information. Specifically, we want to give the quantitative relation between quantum coherence and the dense coding capacity or teleportation fidelity.

In a realistic scenario, the inevitable interactions between the system and the environment always lead to decoherence of the system and the rapid destruction of quantum properties. The dynamics of quantum coherence has been extensively investigated29,30,31,32. Dense coding in the presence of noise has also attracted much attention33,34,35,36,37,38,39, as well as teleportation40,41,42,43,44,45,46. In particular, dense coding for the case that the subsystems of the entangled resource state have to pass a noisy unital quantum channel between the sender and the receiver is considered in ref. 33. We try to apply the quantitative relation between quantum coherence and the dense coding capacity to this special case. Moreover, we will explore whether the quantitative relations between quantum coherence and the dense coding capacity, and that between quantum coherence and teleportation fidelity can be generalized to the general noisy maps.

In the present work, we will establish a complementarity relation between quantum coherence and the optimal dense coding capacity, and also relate quantum coherence to teleportation fidelity in the form of a complementarity relation. Here, quantum coherence is measured by the relative entropy of coherence.

Results

Relating quantum coherence to optimal dense coding and teleportation

In this section, we will investigate the relation between quantum coherence and the optimal dense coding, and that between quantum coherence and teleportation.

The definition of relative entropy of coherence Cre13 is

where is the relative entropy, is the set of all incoherent states and all density operators are of the form13

with {|i〉}i = 1,…,d being a particular basis of the d-dimensional Hilbert space . In the definition of relative entropy of coherence, the minimum is attained if and only if δ = ρdiag with ρdiag being the diagonal part of ρ. Cre satisfies the four postulates given in ref. 13 which are the conditions that a measure of quantum coherence should satisfy. Based on the definition, we can establish the complementarity relation between local quantum coherence and the optimal dense coding.

Relating quantum coherence to optimal dense coding

For a bipartite quantum state ρAB on two d-dimensional Hilbert spaces with ρB = trA(ρAB) being the reduced density matrix of the subsystem B, we have the following theorem.

Theorem 1 The sum of the optimal dense coding capacity of the state ρAB and quantum coherence of the reduced state ρB is always smaller than 2log2d, i.e.,

where χ(ρAB) is the optimal dense coding capacity of the state ρAB.

Proof. The d2 signal states generated by mutually orthogonal unitary transformations with equal probabilities will yield the maximal χ47,48. The mutual orthogonal unitary transformations are given as

where integers m and n range from 0 to d − 1. The ensembles generated by the unitary transformations with equal probabilities pm,n can be denoted as . The average state of the ensembles is

Here, is the d-dimensional identity matrix in the subsystem B. Accordingly, the capacity of the optimal dense coding can be given as47

Based on the result in ref. 47, i.e., , we have

For the reduced state ρB of the subsystem B, , and . Therefore, , from which we have

Now, we consider the sum of the optimal dense coding capacity of the whole system AB and quantum coherence of the subsystem B

where the first inequality is attained because of the fact given in Eq. (8), and the second inequality is obtained due to S(ρAB) ≥ 0. This completes the proof.

For the particular case that the shared entangled state is the Bell state, χ(ρAB) = 2 and Cre(ρB) = 0, and the sum of them equals to 2, which just equals to the right hand side of Eq. (3).

The inequality given in Eq. (3) indicates that the greater local quantum coherence is, the smaller capacity of the optimal dense coding will be. In other words, if the system AB is used to perform dense coding as much as possible, quantum coherence of the subsystem B would pay for the dense coding capacity of the whole system. The physical reason is that dense coding is based on entanglement, and would be impossible without the assistance of entangled states. The results given in ref. 20 show that entanglement of the whole system and quantum coherence of a subsystem are complementary to each other. That is, an increase in one leads to a decrease in the other. For example, for a Bell state, an incoherent state of the subsystem B will be acquired if qubit A is traced over. On the contrary, creating a superposition on a subsystem to have maximum coherence on it will exclude entanglement between subsystems.

In ref. 25, the task of incoherent quantum state merging is introduced and the amount of resources needed for it is quantified by an entanglement-coherence pair. It is found that the entanglement-coherence sum is non-negative, in other words, no merging procedure can gain entanglement and coherence at the same time. From the results given in this paper, the sum of the optimal dense coding capacity and quantum coherence is upper bounded by a definite value, i.e., there is a trade-off between the dense coding capacity and quantum coherence. It should be noted that dense coding is based on entanglement, and the former would be impossible when the latter is absent. In this sense, the result given in Eq. (3) is consistent with those presented in ref. 25.

The result given in Theorem 1 can also be extended to the case of dense coding by using unital memoryless noise quantum channels. The unital noisy channels acting on Alice’s and Bob’s systems are described by the completely positive map , where corresponds to trace preservation, and guarantees the unital property, i.e., Λ(I) = I. Here, Ki denotes the Kraus operators. In ref. 33, the authors found that the encoding with the equally probable operators Um,n, as given in Eq. (4), is optimal for the states of which the von Neumann entropy after the channel action is independent of unitary encoding. In other words, the states satisfy

where . The corresponding dense coding capacity can also be given by , where is the average of the ensemble after encoding with the equally probable unitaries Um,n and after the channel action. That is, is the average state of the ensemble . Based on the fact that 33, . Following the proof process of Theorem 1, one can easily obtain , which indicates our result in Eq. (3) applying to the case of dense coding by using unital memoryless noise quantum channels.

Now, we consider an example of two-sided depolarizing channel33. Alice firstly prepares the bipartite state ρAB, and sends one part of it, i.e., B, via a noisy channel ΛB to the receiver, Bob, so as to establish the shared state for dense coding. Subsequently, Alice does the local unital encoding and then sends her part of the state, i.e., A, via the noisy channel ΛA to Bob. The two-sided d-dimensional depolarizing channel is defined as

with the probability parameters qμν = 1 − (d2 − 1)p/d2 for μ = ν = 0, otherwise qμν = p/d2. The operators Vμν read

It is proved that the von Neumann entropy of a state, which is sent through the two-sided depolarizing channels, is independent of any local unitary transformations that were performed before the action of the channel, i.e., the condition given in Eq. (10) is satisfied33.

Specific to the case that Alice and Bob have the two-sided 2-dimensional depolarizing channel for the transfer of the qubit states, the initial resource state is chosen as |ϕ〉AB = cos θ|Φ+〉AB + sin θ|Ψ+〉AB, where θ ∈ (0, π), and , are the Bell states. After sending the qubit B to Bob via the depolarizing channel, Alice implements the local unital encoding and then sends the qubit A to Bob via the depolarizing channel too. The dense coding capacity χ(ΛAB(ρAB)) and the relative entropy of coherence Cre(ΛB(ρB)) can be straightforwardly calculated, however, the expressions of them are analytically messy, and thus we have chosen to simply plot the exactly numerical results. In Fig. 1, we plot the evolutions of χ(ΛAB(ρAB)) + Cre(ΛB(ρB)), χ(ΛAB(ρAB)) and Cre(ΛB(ρB)) as functions of the state parameter θ and the noise parameter p. From Fig. 1(a), it is found that χ(ΛAB(ρAB)) + Cre(ΛB(ρB)) ≤ 2 is always satisfied, which indicates the result given in Theorem 1 is validated. This can be appreciated in Fig. 1(b,c), where χ(ΛAB(ρAB)) reaches its maximum value while Cre(ΛB(ρB)) gets its minimum value, or vice versa. The underlying physical mechanism is that the dense coding capacity is much greater when the two-qubit state is much more entangled, while the coherence of the subsystem is much smaller. This physical explanation is verified in Fig. 2, where we plot χ(ΛAB(ρAB)) + Cre(ΛB(ρB)), χ(ΛAB(ρAB)) and Cre(ΛB(ρB)) versus θ for p = 0. For the particular cases of θ = π/4 and 3π/4, and , respectively. The subsystem B has the maximum value of coherence Cre(ρB) = 1 when the two-qubit state is the product state and is useless for dense coding. On the contrary, for the cases of θ = 0 and π/2, |ϕ〉AB = |Φ〉AB and |Ψ〉AB, respectively, and the dense coding capacity gets its maximum value χ(ρAB) = 2 for both of them. At these points, the two-qubit states are maximally entangled, and the subsystem has no coherence.

Figure 1
figure 1

(a) The sum of the relative entropy of coherence for subsystem B Cre(ΛB(ρB)) and the dense coding capacity χ(ΛAB(ρAB)), (b) Cre(ΛB(ρB)), and (c) χ(ΛAB(ρAB)) as functions of the state parameter θ and the noise parameter p.

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Figure 2
figure 2

The sum of the relative entropy of coherence for subsystem B Cre(ρB) and the dense coding capacity χ(ρAB) (Red line), Cre(ρB) (Black line), and χ(ρAB) (Blue line) versus the state parameter θ for a fixed value of p = 0.

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The relation between quantum coherence and dense coding has been given in Eq. (3), and in the following, we will relate quantum coherence to teleportation.

Relating quantum coherence to teleportation

For an arbitrary two-qubit mixed state ρAB with ρA = trB(ρAB) being the reduced state of the subsystem A, we have the following theorem.

Theorem 2 For any two-qubit state

where is the binary entropy, F(ρAB) is the teleportation fidelity of the state ρAB and Cre(ρA) denotes quantum coherence of the subsystem A. Here, we just consider the case where the state ρAB is useful for teleportation, which means F(ρAB) ≥ 2/3.

Proof. In the proof, the subscripts are omitted in the case that it does not cause confusion. For a two-qubit state, the relation between the teleportation fidelity F(ρ) and negativity N(ρ) is 3F(ρ) − 2 ≤ N(ρ)49, while negativity is related to concurrence C(ρ) as N(ρ) ≤ C(ρ)50. Combining the two relations, one can obtain 3F(ρ) − 2 ≤ N(ρ) ≤ C(ρ). F(ρ) ≥ 2/3 leads to all of them being larger than 0, so the square of them also obey the rules, i.e., [3F(ρ) − 2]2 ≤ N2(ρ) ≤ C2(ρ). Subsequently, the following expression exists

The last inequality can be acquired based on the fact that concurrence C(ρ) for two-qubit state runs from 0 to 1.

As known to all, h(x) is a monotonically decreasing function in the interval [1/2, 1], thus one can obtain

where EF(ρ) is the entanglement of formation of the state ρAB.

For any bipartite state ρAB, entanglement of formation and quantum coherence obey the relation20

Combining Eq. (15) with (16), and specializing to the two-qubit state, i.e., dA = 2, it is easy to complete the proof.

The inequality given in Eq. (13) indicates that the greater the teleportation fidelity is, the smaller local quantum coherence will be. That is to say, quantum coherence of the subsystem should pay for teleportation fidelity of the whole system. The reason for this result is that teleportation relies on entanglement. However, quantum coherence of the subsystem and entanglement of the whole system are complementary to each other.

For the particular case that the Bell state is utilized to perform teleportation, F(ρAB) = 1 leads to while Cre(ρA) = 0. Thus, equals to 1.

Now, we investigate the example of two-qubit state |ϕ〉AB = cos θ|Φ〉AB + sin θ|Ψ〉AB with θ ∈ (0, π), which is distributed to Alice and Bob through the 2-dimensional depolarizing channels. According to the Eq. (11), one can obtain the output state ΛAB(ρAB), which will be considered as the resource state for implementing teleportation. The unknown state of qubit a to be teleported is assumed to be |ψ〉a = cos(α/2)exp(iβ/2)|0〉 + sin(α/2)exp(−iβ/2)|1〉, where α ∈ (0, π), β ∈ (0, 2π). Bob can get the teleported state ρout after a series of teleportation procedures, and ρout can be expressed as . In the expression, tra,A is the partial trace over the qubits a and A, and both of them are in Alice’s side. is the unitary operator51, and denotes the controlled-k operation with i being the controlled qubit and j being the target qubit. The Hadamard operation on qubit a is denoted as . The teleportation fidelity F(α, β) is the overlap between the unknown input state |ψ〉 and the teleported state ρout

In order to get rid of α and β on the teleportation fidelity, the average teleportation fidelity is given

where 4π is the solid angle. Henceforth, it means the average teleportation fidelity as we refer to the teleportation fidelity. After straightforward calculation, the teleportation fidelity reads

However, the expression of relative entropy of coherence Cre(trB[ΛAB(ρAB)]) is analytically messy. Alternatively, we plot the evolution of h(F) + Cre(ρA), h(F) and Cre(ρA) as functions of the state parameter θ and the noise parameter p in Fig. 3. In this paragraph, and Cre(TrB[ΛAB(ρAB)]) are denoted by h(F) and Cre(ρA) for the sake of simplicity in the case that it does not cause confusion. From the figure, it is found that h(F) and Cre(ρA) compensate each other. For a fixed value of p, the relative entropy of coherence Cre(ρA) increases when h(F) decreases with the increasing of θ, or vice verse. These results can be observed much more clearly from Fig. 4, where the evolutions of h(F) + Cre(ρA), h(F) and Cre(ρA) versus θ for a fixed value of p = 0 are plotted. The underlying physical mechanism for these results is that the resource state changes from the maximally entangled state |Φ〉AB to the product state when θ ranges from 0 to π/2. The maximally entangled state can be used for teleportation with the fidelity getting the maximum value 1, however, the relative entropy of coherence of the subsystem A equals to zero. On the contrary, the product state cannot be used for teleportation while Cre(ρA) = 1.

Figure 3
figure 3

(a) The sum of h(F) and the relative entropy of coherence for the subsystem A Cre(ρA), (b) h(F), and (c) Cre(ρA) as functions of the state parameter θ and the noise parameter p. In the plot, we only consider the case of F > 2/3.

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Figure 4: The sum of h(F) and the relative entropy of coherence for the subsystem A Cre(ρA) (Red line), h(F) (Blue line), and Cre(ρA) (Black line) versus the state parameter θ for a fixed value of p = 0.
figure 4

In the plot, we only consider the case of F > 2/3.

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As proved in ref. 20, the relative entropy of coherence is unitary invariant by using the different bases, the results given in Eqs (3) and (13) hold for all local bases.

From the results given in Eqs (3) and (13), it is found that there is trade-off between local quantum coherence and the optimal dense coding capacity or the teleportation fidelity. In general, the relation among coherence, discord and entanglement has been given by use of quantum relative entropy, where quantum coherence is found to be a more ubiquitous manifestation of quantum correlations19. For two-qubit states with maximally mixed marginals, the pairwise correlations between local observables are complementary to the coherence of the product bases they define52. Furthermore, the results in refs 19,52 also indicate that the existence of correlations, particularly entanglement, together with the purity of the global state, implies that the reduced states are highly mixed, and thus have low coherence in any basis. Combing with the fact that dense coding and teleportation rely on quantum correlations, especially entanglement, our complementarity relations between local quantum coherence and dense coding capacity or teleportation fidelity can be easily understood. Therefore, our results in the present paper are harmonious with those given in refs 19 and 52.

Discussion

In this paper, we relate the relative entropy of coherence to quantum dense coding and teleportation. Firstly, we establish a complementarity relation between the optimal dense coding capacity of a bipartite system and local quantum coherence. The inequality indicates that smaller local quantum coherence will bring about the greater capacity of optimal dense coding. It is also found that the relation can be applied to the case of dense coding by using unital memoryless noisy quantum channels. Secondly, an inequality in the form of complementarity relation between teleportation fidelity for a two-qubit system and local quantum coherence of its subsystem is given. From the inequality, it is found that the greater the teleportation fidelity is, the smaller local quantum coherence will be. Our results in this paper give a clear quantitative analysis between quantum coherence and some specific quantum information protocols.

In the subsection of relating quantum coherence to optimal dense coding, it is found that the result given in Theorem 1 can also be extended to the case of dense coding by using unital memoryless noise quantum channels. In general, our results given in Eqs (3) and (13) can be generalized to general noisy maps. A noisy map can be described by a completely positive trace preserving linear map with the Kraus operators Ki satisfying . If ρAB, ρA and ρB are respectively substituted by ΛAB(ρAB), trB(ΛAB(ρAB)) and trA(ΛAB(ρAB)), the results given in Eqs (3) and (13) are still tenable. Actually, in the subsection of relating quantum coherence to teleportation, we have considered the distribution of two-qubit state through 2-dimensional depolarizing channels, and found that the Eq. (13) is still satisfied.

Additional Information

How to cite this article: Pan, F. et al. The complementarity relations of quantum coherence in quantum information processing. Sci. Rep. 7, 43919; doi: 10.1038/srep43919 (2017).

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