Polynomial measure of coherence

Coherence, the superposition of orthogonal quantum states, is indispensable in various quantum processes. Inspired by the polynomial invariant for classifying and quantifying entanglement, we first define polynomial coherence measure and systematically investigate its properties. Except for the qubit case, we show that there is no polynomial coherence measure satisfying the criterion that its value takes zero if and only if for incoherent states. Then, we release this strict criterion and obtain a necessary condition for polynomial coherence measure. Furthermore, we give a typical example of polynomial coherence measure for pure states and extend it to mixed states via a convex-roof construction. Analytical formula of our convex-roof polynomial coherence measure is obtained for symmetric states which are invariant under arbitrary basis permutation. Consequently, for general mixed states, we give a lower bound of our coherence measure.


I. INTRODUCTION
Coherence describes a unique feature of quantum mechanics -superposition of orthogonal states. The study of coherence can date back to the early development of quantum optics [1], where interference phenomenon is demonstrated for the wave-particle duality of quantum mechanics. In quantum information, coherence acts as an indispensable ingredient in many tasks, such as quantum computing [2], metrology [3], and randomness generation [4]. Furthermore, coherence also plays an important role in quantum thermodynamics [5][6][7], and quantum phase transition [8,9].
With the development of the quantum information theory, a resource framework of coherence has been recently proposed [10]. The free state and the free operation are two elementary ingredients in a quantum resource theory. In the resource theory of coherence, the set of free states is a collection of all quantum states whose density matrices are diagonal in a reference computational basis I = {|i }. The free operations are incoherent complete positive and trace preserving (ICPTP) operations, which cannot map any incoherent state to a coherent state. With the definitions of free states and free operations, one can define a coherence measure that quantifies the superposition of reference basis. Based on this coherence framework, several measures are proposed, such as relative entropy of coherence, l 1 norm of coherence [10], and coherence of formation [11,12]. Moreover, coherence in distributed systems [13,14] and the connections between coherence and other quantum resources are also developed along this line [15][16][17].
One important class of coherence measures is based on the convex-roof construction [11]. For any coherence measure of pure states C(|ψ ), the convex roof extension of a general mixed state ρ is defined as where the minimization is over all the decompositions {p i , |ψ i } of ρ = i p i |ψ i ψ i |. When C(|ψ ) = S(∆(|ψ ψ|)), where S is von Neumann entropy and ∆(ρ) = i |i i|ρ|i i| is the dephasing channel on the basis I, the corresponding measure is the coherence of formation. When C(|ψ ) = max i | i|ψ | 2 , the corresponding measure is the geometric coherence [16]. In general, the minimization problem in Eq. (1) is extremely hard. In particular, analytical formula of the coherence of formation is only obtained for qubit states. This is very similar to quantifying another well-known quantum resource, entanglement, where free states are separable states and free operations are local operations and classical communication [18]. In entanglement measures, convex-roof constructions have been widely studied [19,20]. Similarly, the minimization problem is generally hard. Fortunately, there are two solvable cases, concurrence [21,22] and three-tangle [23]. Both of them are related to a very useful class of functions, referred as polynomial invariant [24]. A polynomial invariant is a homogenous polynomial function of the coefficients of a pure state, P h (|ψ ), which is invariant under stochastic local operations and classical communication (SLOCC) [25]. Denote h to be the degree of the polynomial function, for an N -qudit state |ψ , where κ is an arbitrary scalar and L ∈ SL(d, C) ⊗N is a product of invertible linear operators representing SLOCC.
For an entanglement measure of pure states, one can add a positive power m to the absolute value of the polynomial invariant, where the overall degree is hm. Polynomial invariants are used to classify and quantify various types of entanglement in multi-qubit [26,27] and qudit systems [28]. Specifically, the convex-roof of concurrence can be solved analytically in the two-qubit case [22], and the three-tangle for three-qubit is analytically solvable for some special mixed states [29][30][31]. Recently, a geometric approach [32] is proposed to analyse the convex-roof extension of polynomial measures for the states of more qubits in some specific cases. Inspired by the polynomial invariant in entanglement measure, we investigate polynomial measure of coherence in this work. First, in Sec. II, after briefly reviewing the framework of coherence measure, we define the polynomial coherence measure. Then, in Sec. III, we show a no-go theorem for polynomial coherence measures. That is, if the coherence measure just vanishes on incoherent states, there is no such polynomial coherence measure when system dimension is larger than 2. Moreover, in Sec. IV, we permit some superposition states to take zero-coherence, and we find a necessary condition for polynomial coherence measures. In Sec. V, we construct a polynomial coherence measure for pure states, which shows similar form with the G-concurrence in entanglement measure. In addition, we derive an analytical result for symmetric states and give a lower bound for general states. Finally, we conclude in Sec. VI.

II. POLYNOMIAL COHERENCE MEASURE
Let us start with a brief review on the framework of coherence measure [10]. In a d-dimensional Hilbert space H d , the coherence measure is defined in a reference basis I = {|i } i=1,2,...,d . Thus, the incoherent states are the states whose density matrices are diagonal, Denote the set of the incoherent states to be I. The incoherent operation can be expressed as an ICPTP map Φ ICP T P (ρ) = n K n ρK † n , in which each Kraus operator satisfies the condition K n ρK † n /T r(K n ρK † n ) ∈ I if ρ ∈ I. That is to say, no coherence can be generated from any incoherent states via incoherent operations. Here, the probability to obtain the nth output is denoted by p n = Tr(K n ρK † n ). Generally speaking, a coherence measure C(ρ) maps a quantum state ρ to a non-negative number. There are three criteria for C(ρ), as listed in Table. I [10]. Note that the criterion (C1 ′ ) is a stronger version than (C1). Sometimes, a weaker version of (C2) is used, where the monotonicity holds only for the average state, C(ρ) ≥ C(Φ ICPTP (ρ)). In this work, we focus on the criterion (C2), since it is more reasonable from the physics point of view.  Next, we give the definition of the polynomial coherence measure, drawing on the experience of polynomial invariant for entanglement measure. Denote a homogenous polynomial function of degree-h, constructed by the coefficients of a pure state |ψ = d i=1 a i |i in the computational basis, as where k i are the nonnegative integer power of a i , k i = h, and χ k1k2···k d are coefficients. Then after imposing a proper power m > 0 on the absolute value of a homogenous polynomial, one can construct a coherence measure as, where the overall degree is hm, and the subscript p is the abbreviation for polynomial. A polynomial coherence measure for pure states C p (|ψ ) can be extended to mixed states by utilizing the convex-roof construction, where the minimization runs over all the pure state decompositions of ρ = i p i |ψ i ψ i | with i p i = 1 and p i ≥ 0, and C p (|ψ ) is the pure-state polynomial coherence measure as shown in Eq. (6). Note that if the pure-state measure Eq. (6) satisfies the coherence measure criteria listed in Table I, the mixed-state measure via the convex-roof construction Eq. (7) would also satisfy these criteria [11].

III. NO-GO THEOREM
The simplest example of the polynomial coherence measure is the l 1 -norm for d = 2 on pure state. For a pure qubit state, |ψ = α|0 + β|1 , the l 1 -norm coherence measure takes the sum of the absolute value of the off-diagonal terms in the density matrix, By the definition of Eq. (6), C l1 is the absolute value of a degree-2 homogenous polynomial function with a power m = 1. Meanwhile, this coherence measure satisfies the criteria (C1 ′ ), (C2), and (C3) [10]. Then its convex-roof construction via Eq. (8) turns out to be a polynomial coherence measure satisfying these criteria. Note that when the function Eq. (8) is extended to d > 2, it cannot be expressed as the absolute value of a homogenous polynomial function. Thus, when d > 2, the l 1 -norm coherence measure is not a polynomial coherence measure. Surprisingly, for d > 2, there is no polynomial coherence measure that satisfies the criterion (C1 ′ ). In order to show this no-go theorem, we first prove the following Lemma. Lemma 1. For any polynomial coherence measure C p (|ψ ) and two orthogonal pure states |ψ 1 and |ψ 2 , there exists two complex numbers α and β such that where |α| 2 + |β| 2 = 1. That is, there exists at least one zero-coherence state in the superposition of |ψ 1 and |ψ 2 .
Proof. Since m > 0, the roots of C p (|ψ ) = 0 in Eq. (6) are the same with the ones of |P h (|ψ )| = 0 in Eq. (5). That is, we only need to prove Lemma for the case of m = 1. Since P h (|ψ ) is a homogenous polynomial function of the coefficients of |ψ , one can ignore its global phase. Thus, any pure state in the superposition of |ψ 1 and |ψ 2 can be represented by where the global phase is ignored, ω is a complex number containing the relative phase, and |ψ → |ψ 2 , as |ω| → ∞.

IV. NECESSARY CONDITION FOR POLYNOMIAL COHERENCE MEASURE
From Theorem 1, we have shown a no-go result of the polynomial coherence measure for d ≥ 3 when the criterion (C1 ′ ) in Table I is considered. In the following discussions, we study the polynomial coherence measure with the criteria (C1), (C2), and (C3). Then, there will be some coherent states whose coherence measure is zero. This situation also happens in entanglement measures, such as negativity, which remains zero for the bound entangled states [33]. Here, we focus on the pure-state case and employ the convex-roof construction for general mixed states. As presented in the following theorem, we find a very restrictive necessary condition for polynomial coherence measures that C p (|ψ ) = 0, for all |ψ whose support does not span all the reference basis {i}.

V. G-COHERENCE MEASURE
From Theorem 2, we can see that only the states with a full support on the computational basis could have positive values of a polynomial coherence measure. Here, we give an example of polynomial coherence measure satisfying this condition, which takes the geometric mean of the coefficients, for |ψ = d i=1 a i |i , Note that it is a degree-d homogenous polynomial function modulated by a power m = 2/d. This definition is an analogue to the G-concurrence in entanglement measure, which is related to the geometric mean of the Schmidt coefficients of a bipartite pure state [34]. Hence we call the coherence measure defined in Eq. (23) G-coherence measure. Since the geometric mean function is a concave function [35], following Theorem 1 in Ref. [36], we can quickly show that the G-coherence measure satisfies the criteria (C1), (C2) and (C3). When d = 2, the G-coherence measure becomes the l 1 -norm measure on pure state. When d > 2, according to Theorem 2, there is a significant amount of coherent states whose C G is zero. For instance, in the case of d = 3, the state 1 √ 2 (|0 + |1 ) has zero G-coherence and this state cannot be transformed to a coherent state |ψ , where rank(∆(|ψ ψ|)) = 3, via a probabilistic incoherent operation [12]. Now we move onto the mixed states with the convex-roof construction. In fact, searching for the optimal decomposition in Eq. (7) is generally hard. However, like the entanglement measures, there exist analytical solutions for the states with symmetry [37,38]. Here, we study the states related to the permutation group G s on the reference basis. A element g ∈ G s is defined as 24) and the order (the number of the elements) of G s is d!. The corresponding unitary of g is denoted as U g = k |i k k|.
Then we have the following definition.

Definition 1.
A state ρ is a symmetric state if it is invariant under all the permutation unitary operations, i.e., ∀g ∈ G s , U g ρU † g = ρ.
Denote the symmetric state as ρ s and the symmetric state set as S. Given the maximally coherent state |Ψ d = 1 √ d i |i , it is not hard to show the explicit form of symmetric states, which is only determined by a single parameter, the mixing probability p ∈ [0, 1]. Apparently, the symmetric state ρ s is a mixture of the maximally coherent state |Ψ d and the maximally mixed state I/d. The state |Ψ d is the only pure state in set S. Borrowing the techniques used in quantifying entanglement of symmetric states [38,39], we obtain an analytical result C G (ρ s ) in Theorem 3, following Lemma 2 and Lemma 3. First, we consider a map It uniformly mixes all the permutation unitary U g on a state ρ, which is an incoherent operation by definition. (1) Λ(ρ) ∈ S, i.e., the output state is a symmetric state, as defined in Definition 1; (2) Ψ d |ρ|Ψ d = Ψ d |Λ(ρ)|Ψ d , i.e., the map Λ(ρ) does not change the overlap with the maximally coherent state |Ψ d .
Proof. For any U g ′ with g ′ ∈ G s , The last equality is due to the fact that by going through all permutations g, the joint permutation g ′ g also traverses all the permutations in the group G s . By Definition 1, we prove that Λ(ρ) ∈ S. The overlap between the output state Λ(ρ) and the maximally coherent state |Ψ d is given by, where in the second line we use the relation U † g = U g −1 and the last line is due to the fact that |Ψ d ∈ S and Then, we define the following function for a symmetric state ρ s , Since the state ρ s in Eq. (25) only has one parameter p, it can be uniquely determined by its overlap with the maximally coherent state K = Ψ d |ρ s |Ψ d = p d−1 d + 1 d . Thus, ρ s linearly depends on K. According to Lemma 2, Λ(|ψ ψ|) is a symmetric state and the overlap does not change under the map Λ. Hence, the constraint Λ(|ψ ψ|) = ρ s in Eq. (29) is equivalent to | Ψ d |ψ | 2 = Ψ d |ρ s |Ψ d . Following the derivations of the G-concurrence [39], we solve the minimization problem and obtain an explicit form ofC G (ρ s ), Details can be found in Appendix C. Here, we substituteC G (K) forC G (ρ s ) without ambiguity. When d−1 d ≤ K ≤ 1, C G (K) is a concave function [39]. We showC G (K) in the case of d = 4 in Fig. 1. Moreover, following the results of Ref. [38], we have the following lemma.
Lemma 3. The convex-roof of the G-coherence measure C G for a symmetric state ρ s is given by, where i p i |ψ i ψ i | = ρ s , j q j ρ s j = ρ s , and ρ s j ∈ S.
. Now we prove the lemma by showing that both of them equal to, Z 1 = Z 3 : For a decomposition, ρ s = i p i |ψ i ψ i |, after applying the map Λ on both sides, we have Here, we use the fact that ρ s is a symmetric state, which is invariant under the map Λ. That is, any decomposition satisfies the constraint i p i |ψ i ψ i | = ρ s as required for Z 1 also satisfies the constraint i p i Λ(|ψ i ψ i |) = ρ s as required for Z 3 . Thus, we have Z 3 ≤ Z 1 . On the other hand, the constraint i p i Λ(|ψ i ψ i |) = ρ s in Eq. (32) is also a pure-state decomposition of the state ρ s , since every component in Λ(|ψ i ψ i |) is a pure state U g |ψ i with probability p i /|G s |. Thus we also have Z1 ≤ Z3. Consequently, Z1 = Z3. Z 2 = Z 3 : In fact, the constraint in Eq. (32) is on Λ(|ψ i ψ i |) ∈ S, thus we can solve the minimization problem of Eq. (32) in two steps. First, given Λ(|ψ i ψ i |) ∈ S, we minimize C G (|ψ i ), which turns out to be the same as the definition ofC G (Λ(|ψ i ψ i |)) in Eq. (29). Next, we optimize the decomposition of ρ s in the symmetric state set S, which turns out to be the same as the definition of Z 2 . Thus we have Z 2 = Z 3 .
Theorem 3. For a symmetric state ρ s ∈ S in H d , the G-coherence measure is given by where K = Ψ d |ρ s |Ψ d is the overlap between ρ s and the maximally coherent state |Ψ d .
Proof. According to Lemma 3, the G-coherence measure for a symmetric state is given by C G (ρ s ) = min {qj ,ρ s j } j q jCG (ρ s j ) with j q j ρ s j = ρ s . Since the symmetric state linearly depends on the overlap K, this minimization is equivalent to,  The dependence ofC G (K) and C G (K) on K in the case of d = 4 are plotted in Fig. 1. Furthermore, we can give a lower bound of the G-coherence measure C G for any general mixed state ρ, with the analytical solution for ρ s in Theorem 3.

VI. CONCLUSION AND OUTLOOK
In this paper, we give the definition of polynomial coherence measure C p (ρ), which is an analog to the definition of polynomial invariant in classifying and quantifying the entanglement resource. First, we show that there is no polynomial coherence measure satisfying criterion (C1 ′ ) in Table. I, when the dimension of the Hilbert space d is larger than 2. That is, there always exist some pure states |ψ = |i (i = 1, ..., d) possessing zero-coherence when d ≥ 3. Then, we find a very restrictive necessary condition for polynomial coherence measures -the coherence measure should vanish if the rank of the corresponding dephased state ∆(|ψ ψ|) is smaller than the Hilbert space dimension d. Meanwhile, we give an example of polynomial coherence measure C G (ρ), called G-coherence measure. We derive an analytical formula of the convex-roof of C G for symmetric states, and also give a lower bound of C G for general mixed state. In addition, we should remark that the symmetry consideration in our paper is also helpful to understand and bound the other coherence measures, especially the ones built by the convex-roof method.
In entanglement quantification, the polynomial invariant is an entanglement monotone if and only if its degree η ≤ 4 in the multi-qubit system [40,41]. Here, the quantification theory of coherence shows many similarities to the one for entanglement. Following the similar approaches in our paper, some results can be extended to the entanglement case. For example, one can obtain some necessary conditions where a polynomial invariant serves as an entanglement monotone, in more general multi-partite system H = H d l ⊗N , whose local dimension d l > 2 [28]. After finishing the manuscript, we find that a coherence measure similar to C G (ρ) is also put forward in Ref. [42], dubbed generalized coherence concurrence, by analog to the generalized concurrence for entanglement [34]. However, the analytical solutions and its relationship with polynomial coherence measure are not presented in Ref. [42].