State Convertibility under Genuinely Incoherent Operations

The coherent superposition of states is one of the characteristic features that results in nonclassical phenomena [1,2]. Quantum coherence constitutes a powerful physical resource for implementing various tasks such as quantum algorithms [3,4,5,6,7,8,9,10,11], quantum metrology [12,13,14,15,16,17,18,19,20], quantum channel discrimination [21,22,23,24,25,26,27], witnessing quantum correlations [28,29,30,31,32,33,34], and quantum phase transitions and transport phenomena [35,36,37,38,39,40]. The resource theory of quantum coherence has been flourishing in recent years, it not only establishes a rigorous framework to quantify coherence but also provides a platform to understand quantum coherence from a different perspective [23,41].

Any quantum resource theory is described by two fundamental ingredients, namely, the free states and the free operations [42]. For the resource theory of coherence, the free states are quantum states that are diagonal in a prefixed reference basis. The free operations are not uniquely specified. Motivated by different physical considerations, several free operations are presented, such as incoherent operations (IOs) [41], maximally incoherent operations (MIOs) [43], strictly incoherent operations (SIOs) [44,45], dephasing-covariant incoherent operations (DIOs) [46,47,48], and genuinely incoherent operations (GIOs) [49].

Two fundamental problems in coherence resource theory are state convertibility and resource quantification [23,42]. The state convertibility problem is asking whether for two coherent states there exists a free operation converting one quantum state into the other. The goal of resource quantification is to quantify the amount of coherence in a quantum state. Recalling that coherent states cannot be created from incoherent states via free operations, it is intuitive to assume that (1) C(ρ)≥C(Φ(ρ)) C(\rho) \ge C(\Phi (\rho)) for any quantum state ρ and any free operation Φ. Quantifiers having this property are also called coherence monotones.

Both problems mentioned above—state convertibility and resource quantification—are in fact closely connected. A state ρ can be converted into σ via free operations if and only if (2) C(ρ)≥C(σ) C(\rho) \ge C(\sigma) holds true for all coherence monotones [50]. On the other hand, the fact that Eq. (2) holds for some coherence monotone C does not guarantee that the transformation ρ → σ is possible via free operations. The aim of state convertibility is to find a complete set of coherence monotones {C_i} which can completely classify state transformation, that is, (3) ρ→σ⇔Ci(ρ)≥Ci(σ) \rho \to \sigma \Leftrightarrow {C_i}(\rho) \ge {C_i}(\sigma) for all i.

The study of state convertibility is moving ahead since the question is proposed [41], it is completely answered in pure states or one-qubit case under IOs, SIOs, or MIOs [44,46,51,52,53,54,55,56,57,58,59,60,61,62,63]. The convertibility between mixed states seems to have remained unexplored territory. The difficulty lies in the complexity of pure state decomposition which results in an infinite number of measure conditions for characterizing convertibility of mixed states [57]. We investigate convertibility for coherent states under GIOs. In fact, GIOs are at the core of the resource theory of quantum coherence from both physical realization and dissecting the structure of SIOs and IOs [64]. Note that there is a hierarchical relationship between IOs, SIOs, MIOs, and GIOs [23], (4) GIOs⊆SIOs⊆IOs⊆MIOs. {\rm{GIOs}} \subseteq {\rm{SIOs}} \subseteq {\rm{IOs}} \subseteq {\rm{MIOs}}.

For any 𝒪 ∈ {IOs, SIOs, MIOs}, we define ρ→𝒪σ \rho \buildrel {\cal O} \over \longrightarrow \sigma if there exists Φ ∈ 𝒪 such that Φ(ρ) = σ. It is evident if ρ→GIOσ \rho \buildrel {{\rm{GIO}}} \over \longrightarrow \sigma , then ρ→𝒪σ \rho \buildrel {\cal O} \over \longrightarrow \sigma .

The complete set of coherence monotones for characterizing state convertibility under GIOs is found. In fact, convexity of the robustness of coherence is a good candidate. Moreover, it is also key to convert off-diagonal part of coherent states under more general free operations. Our results induce a useful tool for deciding maximally coherent states in the set of all states with fixed diagonal elements. This produces the so-called majorization condition of determining convertibility from pure states to mixed states under SIOs.

The paper is organized as follows. In Section 2, we briefly present the resource theory of quantum coherence. In Section 3, we will give our main results. Section 4 is a summary of our findings. The appendix is the proof of our results.

Throughout the paper, we consider the d dimensional Hilbert space H and adopt the computational basis {|i〉}i=1d \{|i\rangle \} _{i = 1}^d as the incoherent basis [41]. Thus all diagonal density operators in this basis constitute the set of all incoherent states denoted as I. IOs are specified by a set of Kraus operators {K_j} such that KjρKj†/Tr(KjρKj†)∈I {K_j}\rho K_j^\dagger /Tr\left({{K_j}\rho K_j^\dagger} \right) \in {\rm{I}} for all Φ(ρ)=∑jKjρKj† \Phi \left(\rho \right) = \sum\nolimits_j {{K_j}\rho K_j^\dagger} . Such operation elements {K_j} are called incoherent. An incoherent operation is strictly incoherent if both K_j and Kj† K_j^\dagger are incoherent. The MIOs are known as incoherent states preserving operations. GIOs are operations that fix all incoherent states, that is, (5) Φ(ρ)=(ρ) \Phi \left(\rho \right) = \left(\rho \right) for any incoherent state ρ ∈ I. Since GIOs do not allow for transformations between different incoherent states, notably, for example, between the energy eigenstates (when coherence is measured with respect to the eigenbasis of the Hamiltonian of the system), they capture the framework of coherence in the presence of additional constraints, such as energy conservation. For other important type of incoherent operations, we refer the reader to the review article [23].

In order to characterize conversion of coherent states under GIOs, we need a key measure originated from the task of maximizing the mean value of an observable [65]. Let |ψ+〉=1d∑i=1d|i〉 |{\psi ^ +}\rangle = {1 \over {\sqrt d}}\sum\nolimits_{i = 1}^d {|i\rangle} , it is well-known that |ψ⁺〉〈ψ⁺| is a maximally coherent state under IOs, that is, a state from which all other states can be created via IOs [41]. It is easy to see that U|ψ⁺〉〈ψ⁺|U^† is maximally coherent under IOs for any diagonal unitary matrix U. Let Ω be the set of convex hull of U|ψ⁺〉〈ψ⁺|U^†. For every M ∈ Ω, define (6) CMGIOs(ρ)=maxΦ∈GIOs tr(Φ((ρ)M)−1d. C_M^{{\rm{GIOs}}}(\rho) = \mathop {\max}\limits_{\Phi \in {\rm{GIOs}}} {\rm{tr}}(\Phi ((\rho)M) - {1 \over d}.

In the following, we list some elementary properties of CMGIOs(⋅) C_M^{{\rm{GIOs}}}(\cdot) and discuss its relationship with other coherence measures (see appendix for the proof). (i)

CMGIOs(ρ)≥0 C_M^{{\rm{GIOs}}}\left(\rho \right) \ge 0 for every quantum state ρ and CMGIOs(ρ)=0 C_M^{{\rm{GIOs}}}\left(\rho \right) = 0 if ρ ∈ I;

Specially, if M = |ψ⁺〉〈ψ⁺|, then combining Theorem 1 with Theorem 2 of [67], we have (12) C|ψ+〉〈ψ+|GIOs(ρ)=CROC(ρ)d C_{|{\psi ^ +}\rangle \langle {\psi ^ +}|}^{{\rm{GIOs}}}\left(\rho \right) = {{{C_{{\rm{R}}OC}}\left(\rho \right)} \over d}

For general M ∈ Ω, there exist a probability distribution {p_i} and diagonal unitary matrices {U_i} such that M=∑ipiUi|ψ+〉{ψ+|Ui† M = \sum\nolimits_i {{p_i}{U_i}|{\psi ^ +}\rangle \langle{\psi ^ +}|U_i^\dagger} . That is, M is a convexity of maximally coherent states. In this sense, we say CMGIOs C_M^{{\rm{GIOs}}} is a convexity of the robustness of coherence. It is found that such measures play a key role in studying state convertibility under GIOs.

Now, we are in a position to give our main result.

Theorem 1 tells convertibility between pure states is impossible except for diagonal-unitary equivalent states. A parallel result in multipartite entanglement is almost all n-qubit pure states with n ≥ 3 can neither be reached nor be converted into any other LU-inequivalent state via deterministic LOCC [68]. On the other hand, deterministic convertibility between incoherent-unitary inequivalent pure states is possible under IOs, DIOs, SIOs, and MIOs [46,51]. Thus, compared with other free operations in the coherence resource theory, GIOs are more matching to LOCC in multipartite entanglement theory from the point of state convertibility.

For one-parameter maximally mixed states [52,67] (14) ρp=p|ψ+〉〈ψ+|+1−pdI, {\rho _p} = p|{\psi ^ +}\rangle \langle {\psi ^ +}| + {{1 - p} \over d}I, Theorem 1 shows that (15) ρp→GIOρq⇔q≤p. {\rho _p}\mathop \to \limits^{{\rm{GIO}}} {\rho _q} \Leftrightarrow q \le p.

Based on Theorem 1, we can provide a nice majorization condition that determines the convertibility from pure states to mixed states under SIOs, IOs, and MIOs.

By the hierarchical relationship SIOs ⊆ IOs ⊆ MIOs, there exists some IO or MIO Φ with Φ(|ψ〉〈ψ|) = σ if (|ψ₁|², ⋯ , |ψ_d|²)^t ≺ (σ₁₁, ⋯ , σ_dd)^t.

For |ψ+〉=∑i=1d1d|i〉 |{\psi ^ +}\rangle = \mathop \sum \nolimits_{i = 1}^d {1 \over {\sqrt d}}|i\rangle , it is evident that (17) (1d,⋯, 1d)t≺(σ11,⋯, σdd)t {({1 \over d}, \cdots,{1 \over d})^t} \prec {({\sigma _{11}}, \cdots,{\sigma _{dd}})^t} for any quantum state σ. A direct consequence of Theorem 2 is that |ψ⁺〉〈ψ⁺| is maximally coherent under IOs which is an important conclusion of [41].

It is well-known that convertibility between pure states is completely characterized by majorization relation [51]. Theorem 2 can be regarded as an extension when the output state is mixed. Although a structural characterization of coherence conversion for the output mixed state is provided in terms of a finite number of measure conditions [57], such conditions are somewhat hard to verify because pure state decomposition is involved. In comparison, Theorem 2 is more handy because we need only to check a majorization relation.

The core for the proof of Theorem 2 is to find maximally coherent states (MCS) in the set of all states S with fixed diagonal elements, here an MCS means a state from which all other states of S can be created via GIOs.

We remark that the existence of MCS in a particular set of states S has independent meaning, because one may not be able to prepare all states of choice in many situations. Suppose we are bound to a particular set of states S, can we find a notion of maximally coherent state in S. By Theorem 1, a natural choice of S is the set of all states with fixed diagonal elements, that is, S = {(ρ_ij) : ρ_ii = p_i, i = 1, 2, ⋯ , d}, here {p_i} is a fixed probability distribution. In fact, there exists an MCS in S. Our result reads as follows.

By the hierarchical relationship GIOs⊆SIOs⊆IOs⊆MIOs,GIOs⊆SIOs⊆DIOs⊆MIOs, \matrix{{{\rm{GIOs}} \subseteq {\rm{SIOs}} \subseteq {\rm{IOs}} \subseteq {\rm{MIOs,}}} \cr {{\rm{GIOs}} \subseteq {\rm{SIOs}} \subseteq {\rm{DIOs}} \subseteq {\rm{MIOs,}}} \cr} we can obtain |ψ〉=∑i=1dpi|i〉 |\psi \rangle = \mathop \sum \nolimits_{i = 1}^d \sqrt {{p_i}} |i\rangle is also maximally coherent in S under SIOs, DIOs, IOs, and MIOs.

Theorem 1 and Theorem 3 show that coherent mixed states cannot be converted into pure states in general. This is a parallel result of no-go theorem of purification for coherent mixed states of discrete-variable and Gaussian systems [60,69]. It shows a strong limit on the efficiency of perfect coherent purification under GIOs.

We also remark that parallel discussion of Theorem 3 in quantum entanglement is the existence of a maximally entangled state within a given set of states with fixed spectrum. This is the Problem 5 in the Open Quantum Problems List maintained by the Institute for Quantum Optics and Quantum Information (IQOQI) in Vienna [70,71]. It is newly shown that maximally entangled mixed states for a fixed spectrum do not always exist [72].

By Theorem 1, if diagonal elements of ρ and σ are not completely equal in the same position, then both ρ ↛ σ and σ ↛ ρ under GIOs hold true. However, exact conditions for realizing conversion between off-diagonal parts of coherent states can also be found.

For any 𝒪 ∈ {GIOs, DIOs MIOs}, we define (19) CM𝒪(ρ)= maxΦ∈𝒪 tr(Φ(ρ)M)−1d C_M^{\cal O} \left(\rho \right) = \mathop {\max}\limits_{\Phi \in {\cal O}} {\rm{tr}}\left({\Phi \left(\rho \right)M} \right) - {1 \over d} for M ∈ Ω. By the hierarchical relationship between GIOs, DIOs, and MIOs [23], we know that each CM𝒪(⋅) C_M^{\cal O}(\cdot) is a coherence measure. Based on this, we actually have the following result.

Imaginarity as resource is a hot topic and recently receives much attention (see [73] and the references therein). For any coherence measure C, (21) C(ρ)=C(ρ*) C\left(\rho \right) = C\left({{\rho ^*}} \right) is an axiomatic assumption proposed in [73] for studying coherence and imaginarity of quantum states, here ρ^* is the complex conjugate of ρ. The intuition tells us (21) is right. Actually, the author has checked that all existing important coherence measures such as the l₁-norm of coherence, the relative entropy of coherence [41], the Tsallis relative entropy of coherence [74], the robustness of coherence, the geometric coherence [75], the coherence weight [76], and coherence measures from the convex roof construction [77] satisfying C(ρ) = C(ρ^*). From the point of state convertibility, we need only to prove (22) ρ→Φ1ρ*,ρ*→Φ2ρ, \rho \mathop \to \limits^{{\Phi _1}} {\rho ^*},\,{\rho ^*}\mathop \to \limits^{{\Phi _2}} \rho, Φ₁, Φ₂ ∈ {GIOs, DIOs, MIOs}. By Theorem 4, we need to check CM𝒪(ρ)=CM𝒪(ρ*) C_M^{\cal O} \left(\rho \right) = C_M^{\cal O} \left({{\rho ^*}} \right) . However, we can find that CMGIOs(⋅) C_M^{{\rm{GIOs}}}(\cdot) has a distinguished property (23) CMGIOs(ρ)≠CMGIOs(ρ*) C_M^{{\rm{GIOs}}}(\rho) \ne C_M^{{\rm{GIOs}}}({\rho ^*}) for some ρ and M ∈ Ω (see the appendix for an example). This shows the peculiarity of CMGIOs(⋅) C_M^{{\rm{GIOs}}}(\cdot) and the necessity of assumption C(ρ) = C(ρ^*).

Among the most fundamental questions in quantum coherence theory is state convertibility, it is aimed to study whether incoherent operations can introduce an order on the set of coherent states, that is, whether, given two coherent states ρ and σ, either ρ can be transformed into σ or vice versa. Since the question of state convertibility in coherence resource theory is proposed [41], understanding the exact conditions for existence of incoherent transformations between coherent states has attracted a lot of work [23]. In this work, we have determined the exact conditions for coherence conversion under GIOs. Our conditions show that coherence measures from convexity of the robustness of coherence are central. Based on these conditions, maximally incoherent states in a particular set are classified. This induces the majorization condition of determining the convertibility from pure states to mixed states under SIOs. Furthermore, conditions of conversion between off-diagonal parts of coherent states are also characterized. The study of state convertibility for general resource theory has also been discussed recently [50,78].

There still exist some interesting open questions. First, note that the existence proof of our Theorem 1 is not constructive, given two states satisfying coherence order, a problem is how to construct desired GIOs realizing convertibility. Second, can we offer an efficient algorithm to compute CMGIOs(⋅) C_M^{{\rm{GIOs}}}(\cdot) ? Note that CMGIOs(⋅) C_M^{{\rm{GIOs}}}(\cdot) is a generalization of quantum coherence fraction which quantifies the closeness between a given state and the set of maximally coherent states [67]. Therefore an efficient algorithm of CMGIOs(⋅) C_M^{{\rm{GIOs}}}(\cdot) is also efficient for quantum coherence fraction which is key in the framework of coherence theory.

Proofs of all results in this paper are given in Appendix A.

Before giving the proof of our main results, we firstly recall some fundamental properties of GIOs. In fact, the notion of GIOs is equivalent to the Schur channels [79,80,81]. Suppose Φ is trace-preserving completely positive maps on density operators, the following statements are equivalent: (1)

Φ is a GIO, that is, a Schur channel;