Quantum coherence arising from quantum superposition is a fundamental feature of quantum mechanics. It is a key component in various quantum information and is primarily accountable for the advantage offered by quantum tasks versus classical ones.^[1,2] Coherence as an important physical resource is used in thermodynamics,^[3–5] biomolecular networks,^[6–11] nanoscale physics,^[12,13] and quantum metrology.^[14,15] But the rigorous characterization of coherence in the framework of resource theories has been a rather recent development,^[16] and a subsequent stream of work has identified some coherence measures.^[17–22] Examples of coherence measures include the -norm and the relative entropy of coherence.^[16]

Nonclassical correlations have long been regarded as a key quantum resource. Entanglement was the first concept to be known about. Both coherence and entanglement capture the quantumness of a physical system, and it is well known that entanglement stems from the superposition principle, which is also the essence of coherence.^[23] It is then legitimate to ask what relations they are. Recently, the relation between coherence and entanglement was studied in Refs. [19], [24], [25], and [26]. Streltsov et al.^[19] utilized a common frame to quantify quantumness in terms of coherence and entanglement. In particular, they showed that any nonzero amount of coherence of a system A can be converted into entanglement between A and an initially incoherent ancilla B, by means of incoherent operations. Then we ask whether any nonzero amount of coherence can be converted to entanglement via incoherent operations. If not, what conditions should be satisfied under which the coherence can be converted into entanglement via incoherent operations? We do the work in this paper.

The rest of this paper is organized as follows. In Section 2, we briefly review the resource theory of coherence and entanglement. In Section 3, we study the necessary conditions of the initial state of converting coherence into entanglement via incoherent operation. In Section 4, we study the necessary conditions of the coherence consumption of converting coherence into entanglement via incoherent operation, and demonstrate the reasons for those conditions. Section 5 contains the concluding remarks.

We first review the information-theoretic definitions of coherence and entanglement. The resource theory of coherence characterizes the free resources, i.e., incoherent states and incoherent operations, and the criteria identifying coherence measures.^[16] The framework has been extended to the multipartite scenario.^[19,21,24] In an N-partite system, the incoherent states can be represented as , where , is the pre-fixed local basis of the i-th subsystem, , and . The set of incoherent states is the set of quantum states with diagonal density matrices with respect to this basis. Any other state has nonzero coherence. An incoherent operation is any quantum operation that maps an incoherent state to an incoherent state. A measure of coherence is a nonnegative function which vanishes for incoherent states and is a non-increasing monotone under incoherent operations, . This ensures that the coherence cannot be increased through incoherent operations. A notable coherence measure is the relative entropy of coherence, which is given by , where denotes the set of incoherent states. For a quantum state , the closest incoherent state is , where is the matrix containing only the leading diagonal elements of in the reference basis, so the relative entropy of coherence^[14] is , where is the von Neumann entropy.

The theory of entanglement has been the cornerstone of major developments in quantum information theory and has triggered the advancement of modern quantum technologies. An N-partite state is disentangled if it can be written in the separable form, . For pure quantum states, the separable states are in the form of . If a state of the quantum system cannot be written in these forms, it is called entangled state. To compare with the relative entropy of coherence, we focus on the relative entropy of entanglement, , where SE represents the sets of separable states.

Considering the unitary operation 1 where and are the reference bases of subsystems A and B, respectively. It can be seen by inspection that this unitary is incoherent (i.e., the state is incoherent for any incoherent state . Note that for two qubits this unitary is equivalent to the generalized controlled-not gate (CNOT) , with A serving as the control qubit and B the target qubit.^[19] It can convert any pure incoherent state into another pure incoherent state, . It is well known that the state is entangled for any coherent state .^[2] But if the target qubit is also a coherent state, the state is always entangled anyway.

Using the CNOT gate as an incoherent operation, we analyze the conditions under which a product state can be converted into an entangled state via the incoherent operation. For some product pure states, the evolved states under the CNOT gate are as follows. We perform the CNOT gate on the incoherent control qubit and the coherent target qubit, the evolved state is 2a 2b where . We can see that the evolved state is also a separable state when the control qubit is an incoherent state. We perform the CNOT gate on the coherent control qubit A and the coherent target qubit B, the evolved state is 3 where , and . For a bipartite pure state, it is well known that the relative entropy of entanglement is equal to the entropy of the subsystem, . We calculate the entanglement of the evolved state, 4 We can see that the entanglement is zero when . If we set , the evolved state is a product state, 5 In terms of the equation . Then we conclude that the entanglement cannot be generated when the target qubit is a maximally coherent state.

We now analyze the conditions of converting coherence into entanglement via the CNOT gate when the initial states are the mixed states. As is well known, a product state can be converted into an entangled state via the CNOT gate if and only if the control qubit is coherent.^[19] The two-qubit CNOT gate can be rewritten as 6 with is the identity operator. We perform the CNOT gate on the state , the state evolves into a final state, 7

We now consider a general single-qubit state as the control qubit and another state as the target qubit, where , and being the j-th Pauli matrix, with , and . For convenience, we assume its Bloch vectors and . In the reference basis , the initial product state of the two qubits is 8 where , , , , , , , , , .

We perform the CNOT gate on the control qubit and the target qubit separately, the initial state of the qubits becomes 9

If , the final state is a separable state, 10 That is to say, if the control qubit is an incoherent state, the entanglement cannot be generated.

If and , the control qubit is a coherent state, and the target qubit is an incoherent state. We find that the final state is a two-qubit X-state. A two-qubit X-state is entangled if and only if either of the following two conditions is satisfied: 11 and these two conditions cannot hold simultaneously.^[27,28] Here, the condition becomes or . After calculation, we find that when the following condition is satisfied: 12 the final state is an entangled state. Obviously, if or , the condition is fulfilled.

If , and , the final state is identical to the initial state. Namely, when the control qubit is a coherent state and the target qubit is a maximally coherent state, the final state is a product state. The coherence cannot be converted into entanglement.

In order to demonstrate the variety of coherence of a quantum system under a unitary operation, we first prove the following theorem and define the concept of total coherence consumption.

In this paper, we discuss the conditions of converting coherence into entanglement, demonstrating the formal potential of coherence for entanglement generation. According to the analysis of entanglement of the evolved state, we find the necessary conditions of converting coherence into entanglement via incoherent operations: firstly, the control qubit is a coherent state, secondly, the target qubit is not the maximally coherent state. In addition to these basic conditions, due to the methods of measuring entanglement and the property of logarithm function, there are some additional conditions for converting coherence into entanglement. According to the notion of coherence consumption, we derive the necessary conditions of converting coherence into entanglement via incoherent operations, that is, the total coherence consumption is always positive, the conditions have been proven as valid conditions to determine quantum correlation.^[32] Finally, we provide a hypothesis for demonstrating the process of converting coherence into entanglement. Although converting coherence into entanglement via an incoherent operation may be not necessarily useful in practical applications, it, as a cheaper scheme for creating entanglement, might be available. But we hope our work can provide some clues to demonstrating the process of entanglement generation.