The classical correlation limits the ability of the measurement-induced average coherence

Coherence is the most fundamental quantum feature in quantum mechanics. For a bipartite quantum state, if a measurement is performed on one party, the other party, based on the measurement outcomes, will collapse to a corresponding state with some probability and hence gain the average coherence. It is shown that the average coherence is not less than the coherence of its reduced density matrix. In particular, it is very surprising that the extra average coherence (and the maximal extra average coherence with all the possible measurements taken into account) is upper bounded by the classical correlation of the bipartite state instead of the quantum correlation. We also find the sufficient and necessary condition for the null maximal extra average coherence. Some examples demonstrate the relation and, moreover, show that quantum correlation is neither sufficient nor necessary for the nonzero extra average coherence within a given measurement. In addition, the similar conclusions are drawn for both the basis-dependent and the basis-free coherence measure.

Quantum coherence originating from the quantum superposition principle is the most fundamental quantum feature of quantum mechanics. It plays an important role in various fields such as the thermodynamics 1-6 , the transport theory 7-10 , the living complexes [11][12][13] and so on. With the resource-theoretic understanding of quantum feature in quantum information, the quantification of coherence has attracted increasing interest in recent years [14][15][16][17][18][19] and has also led to the operational resource theory of the coherence 20 .
The quantitative theory also makes it possible to understand one type of quantumness (for example, the coherence) by the other type of quantumness such as the entanglement and the quantum correlation, vice versa [21][22][23][24][25][26][27][28][29][30][31] . For example, for a bipartite pure state, the maximal extra average coherence that one party could gain was shown to be exactly characterized by the concurrence assisted by the local operations and classical communication (LOCC) with the other party 21 . Ref. 22 showed that the maximal average coherence was bounded by some type of quantum correlation in some particular reference framework. In the asymptotic regime, ref. 23 showed that the rate of assisted coherence distillation for pure states was equal to the coherence of assistance under the local quantum-incoherent operations and classical communication. Quite recently, a unified view of quantum correlation and quantum coherence has been given in ref. 24. In addition, if only the incoherent operations are allowed, the state with certain amount of coherence assisted by an incoherent state can be converted to an entangled state with the same amount of entanglement 32 or a quantum-correlated state with the same amount of quantum correlation 33 .
In this paper, instead of the quantum correlation, we find, it is the classical correlation of a bipartite quantum state that limits the extra average coherence at one side induced by the unilateral measurement at the other side. We also find the necessary and sufficient condition for the zero maximal average coherence that could be gained with all the possible measurements taken into account. Besides, we show, through some examples, that quantum correlation is neither sufficient nor necessary for the extra average coherence subject to a given measurement. We have selected both the basis-dependent and the basis-free coherence measure to study this question and obtain the similar conclusions. In particular, one should note that all our results are valid for the positive-operator-valued measurement (POVM), even though we only consider the local projective measurement in the main text.

Results
The upper bound on the extra measurement-induced average coherence. Coherence measure. To begin with, let's first give a brief review of the measure of the quantum coherence 14 . If a quantum state δ can be written as (p1) Nonnegative-i.e., C(ρ) ≥ 0 and C(ρ) = 0 if and only if the quantum state ρ is incoherent.
Even though there are many good coherence measures such as the coherence measures based on l 1 -norm, trace norm, fidelity, the relative entropy and so on [14][15][16][17][18][19] , in this paper we will only employ the relative entropy to quantify the quantum coherence, i.e., where ρ σ ρ ρ ρ σ = − S ( ) Tr log Tr log is the relative entropy, ρ ρ ρ = − S ( ) Tr log is the von Neumann entropy and ρ * is the diagonal matrix by deleting all the off-diagonal entries of any ρ (we will use this notation throughout the paper). For simplicity, we will restrict ourselves in the computational basis throughout the paper. In contrast, the basis-free coherence (or the total coherence) 34 is quantified by quantifies the maximal coherence of a state with all the bases taken into account.
The Classical correlation as the upper bound. Now let's turn to our game sketched in Fig. 1. Suppose two players, Alice and Bob, share a two-particle quantum state ρ AB and Alice performs some projective measurement Π Π : { } i on her particle and sends her outcomes to Bob. Bob isn't allowed to do any operation. Based on Alice's outcomes, Bob will obtain the state . Thus in the computational basis, the measurement-induced average coherence (MIAC: Bob's average coherence induced by Alice's measurement Π) is given by Similarly, the measurement-induced average total coherence (MIATC: Bob's average total coherence induced by Alice's measurement Π) is with d denoting the dimension of Bob's space. With Alice's measurement Π, the Bob's average coherence is usually different from the coherence of ρ ρ = Tr It is obvious that  ∆ ≥ Π 0 P T / which is impied by the convexity of the coherence , that is, . Thus our main results can be given by the following theorems.
The inequality holds if all Bob's states ρ B and ρ i B have the same diagonal entries. The proof is completed.  Theorem 2: For a bipartite quantum state ρ AB , the extra MIAC ∆ Π P  is upper bounded by the classical correlation of ρ AB , that is, where the classical correlation is defined by and the corresponding probability where U A is unitary, |j〉 , j are the local computational basis.

Theorem 3:
The extra MIATC ∆ Π T  for a bipartite quantum state ρ AB is upper bounded by the classical correlation

T i A
The equality holds for the pure ρ AB . Proof. From the classical correlation, we have (16) saturates for the pure quantum state ρ AB . The proof is finished.  All the above three theorems hold for any projective measurement, so if we specify the particular measurement such that the maximal extra MIAC or MIATC can be achieved, the three theorems are also valid, which can be given in a rigorous way as: For a bipartite state ρ AB with the reduced density matrix ρ B , the maximal extra MIAC and the maximal extra MIATC satisfy Proof: It is obvious from theorem 1, 2 and 3.  i . Similar to the proof of theorem 4, one can find that  which corresponds to a product state ρ AB . The proof is finished.  Examples. The above theorems mainly show that, even though the coherence is the quantum feature of a quantum system, in the particular game as sketched in Fig. 1, the extra average coherence obtained by Bob with the assistance of Alice's measurement is well bounded by the classical correlation of their shared state, instead of the quantum correlation. However, one can find that the necessity for all the attainable bounds is to share the pure states which happen to own the equal quantum and classical correlations. Therefore, one could think that the classical correlation is trivial in contrast to the quantum correlation (e.g., quantum correlation serves as a tight upper bound, but is less than classical correlation).
The following examples show that it is not the case.

Example 1. The extra average coherence could be induced in classical-classical states. Suppose a bipartite state is given by
, the reduced quantum state 2 is incoherent. So the classical correlation is equal to the total correlation, i.e., . The extra MIAC and the extra MIATC subject to the measurement Π can be calculated as If the subsystem A is measured by the projective measurement Π + + − − : { , } , subsystem B will collapse to the state  Since there is no quantum correlation subject to subsystem A, the corresponding classical correlation is directly determined by the total correlation as i A given by Eq. (32). This example shows that an improper measurement could induce no extra average coherence even though quantum correlation is absent.  . It is easy to demonstrate that a ± = b ± for cot2 θ = cos φ which further leads to

Example 3. No extra average coherence could be induced in the quantum-classical state. Suppose the quantum-classical state is given by
Thus there is no extra average coherence can be gained in terms of this measurement constraint, that is, Similar to the second example, an improper measurement could induce no extra average coherence even though quantum correlation is present.  where s x = sin(x) and = s x cos( ) x . In addition, it is obvious that the reduced quantum states . So the extra average coherence can be directly given by the MIAC or MIATC as ( 2 ) cos 2 2( )( ) cos 2 sin

Discussion
Before the end, we would like to emphasize that all the results in the paper are valid for the POVMs, since it was shown 38 that the classical correlations always attained by the rank-one POVM. In addition, we have claimed that Bob isn't allowed to do any operation, which is mainly for the basis-dependent coherence measure. In fact, when we consider the basis-free coherence measure, it is equivalent to allowing Bob to select the optimal unitary operations on his particle. In this case, theorem 3 implies that for pure states the extra MIATC is the exact quantum entanglement of their shared state (von Neumann entropy of the reduced density matrix). Thus the coherence also provides an operational meaning for the pure-state entanglement under LOCC.
To sum up, we employ the basis-dependent and basis-free coherence measure to study the extra average coherence induced by a unilateral quantum measurement. Despite that the coherence is the most fundamental quantum feature, we find that the extra average coherence is limited by the classical correlation instead of the quantum correlation. In addition, we find the necessary and sufficient condition for the zero maximal average coherence. We also show that the quantum correlation is neither sufficient nor necessary for the extra average coherence by some examples.

Methods
Proof of Theorem 2. We will give the main proof the theorem 2. in the main text. Following Eq. (9), we have where the second inequality holds due to the optimal Ω { } i A implied in Eq. (12). So Eq. (11) is satisfied. In addition, Eq. (11) saturates if both Eqs (9) and (15)