New Journal of Physics

The open access journal at the forefront of physics

Deutsche Physikalische Gesellschaft (I) DPG IOP Institute Of PhySiCS

PAPER • OPEN ACCESS

Polynomial measure of coherence

To cite this article: You Zhou et al 2017 New J. Phys. 19 123033

View the article online for updates and enhancements.

Related content

- The role of quantum coherence in dimer and trimer excitation energy transfer Charlotta Bengtson and Erik Sjoqvist

- Measures and applications of quantum correlations

Gerardo Adesso, Thomas R Bromley and Marco Cianciaruso

- Absolutely separating quantum maps and channels

S N Filippov, K Yu Magadov and M A Jivulescu

This content was downloaded from IP address 80.82.77.83 on 28/12/2017 at 22:57

New Journal of Physics

The open access journal at the forefront of physics

Dautsdie PhyilbUiicha GwalUdiaft DPG

IOP Institute of Physics

Published in partnership with: Deutsche Physikalische Gesellschaft and the Institute of Physics

Cross Mark

OPEN ACCESS

RECEIVED

28 April2017

REVISED

3 October 2017

ACCEPTED FOR PUBLICATION

9 October 2017

PUBLISHED

13December2017

Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.

Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation andDOI.

Polynomial measure of coherence

You Zhou, Qi Zhao, Xiao Yuan and Xiongfeng Ma©

Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, People's Republic of China

E-mail: xma@tsinghua.edu.cn, zhaoq14@mails.tsinghua.edu.cn andyxbdwl@gmail.com

Keywords: coherence, polynomial measure, convex roof, resource framework

Abstract

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.

1. 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 [5-7], 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, l1 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 [15-17].

One important class of coherence measures is based on the convex-roof construction [ 11 ]. For any coherence measure of pure states C (|y)), the convex-roof extension of a general mixed state p is defined as

C(p) = min £ptC(lyi)), (1.1)

{pi,iy)} i i

where the minimization is over all the decompositions {pi, |yi)} of p = £ipi\yi){yi |.When C (iy)) = S (A(|y) {y |)),where S is von Neumann entropy and A(p) = £i|i){i | p|i){i | is the dephasing channel on the reference basis I, the corresponding measure is the coherence of formation. When C (|y)) = maxi| {iy) |2, the corresponding measure is the geometric coherence [16]. In general, the minimization problem in equation (1.1) is extremely hard. In particular, analytical formula of the coherence of formation

© 2017 The Author(s). Published by IOP Publishing Ltd on behalf of Deutsche Physikalische Gesellschaft

Table 1. Criteria for a coherence measure.

(C1) C (6) = 0 if 6 £ 1; (C1') C (6) = 0 iff 6 £ 1

(C2) Monotonicitywith post-selection: for any incoherent operation Ficptp(p) = Y^nKn pK^,C (p) ^ pnC (pn), where pn = Kn pK^/pn

and pn = Tr(Kn pKl) (C3) Convexity: -£epeC(pe) > C(£epepe)

is only obtained for qubit states. The efficient calculation for this class of coherence measure is still an open problem.

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, Ph (\y)), 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 \y),

Ph (kL\y)) = khPh (\y)), (1.2)

where k is an arbitrary scalar and L £ SC(d, C)0N 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,

Ehm (\y)) = \Ph (\y))\m, (1.3)

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 [29-31]. Recently, a geometric approach [32] is proposed to analyze the convex-roof extension ofpolynomial measures for the states ofmore qubits in some specific cases.

Inspired by the polynomial invariant in entanglement measure, we investigate polynomial measure of coherence in this work. First, in section 2, after briefly reviewing the framework of coherence measure, we define polynomial coherence measure. Then, in section 3, we show a no-go theorem for polynomial coherence measures. That is, if the coherence measure only vanishes on incoherent states, there is no such polynomial coherence measure when system dimension is larger than 2. Moreover, in section 4, we permit some superposition states to take zero-coherence, and we find a necessary condition for polynomial coherence measures. In section 5, 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 section 6.

2. Polynomial coherence measure

Let us start with a brief review on the framework of coherence measure [ 10] .Ina d-dimensional Hilbert space Hd, 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, S = ^d=1 pi\i) (i\. Denote the set of the incoherent states to be 1. The incoherent operation can be expressed as an ICPTP map FICPTP (p) = ^nKn pK^, in which each Kraus operator satisfies the condition Kn pKTr(Kn pK^) £ 1 if p £ 1. 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 pn = Tr (Kn pKl).

Generally speaking, a coherence measure C (p) maps a quantum state p to a non-negative number. There are three criteria for C (p), as listed in table 1 [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 (p) ^ C (FICPTP (p)). 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 ofpolynomial coherence measure, drawing on the experience ofpolynomial invariant for entanglement measure. Denote a homogenous polynomial function ofdegree-h, constructed by the coefficients of a pure state \y) = ^d=1 ai\i) in the computational basis, as

Ph(|y>) = E Xklh...kd n (2.1)

h>h>--¿d '=1

where ki are the non-negative integer power of ai, Ek' = h, and Xktk2■■■kd 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,

Cp (|y>) = |Ph (iy>)|m, (2.2)

where the overall degree is hm, and the subscriptp is the abbreviation for polynomial.

A polynomial coherence measure for pure states Cp (|y>) can be extended to mixed states by utilizing the convex-roof construction,

Cp(p) = min EP'Cp(|y'>), (2.3)

(p,-,iy>) '

where the minimization runs over all the pure state decompositions of p = E^^yXy | with E;p; = 1 and pi ^ 0,and Cp (|y>) is the pure-state polynomial coherence measure as shown in equation (2.2).

A legal polynomial measure, in the form of equation (2.1), should satisfy the monotone criteria, showed in table 1, (C1) (or (C1')), (C2), and (C3). Note thatifthe pure-state measure equation (2.2) satisfies (C1) (or (C1')), (C 2), the mixed-state measure via the convex-roof construction equation (2.3) would satisfy (C1) (or (C1')), (C 2), and (C3) [11]. This is because the convex-roof construction guarantees the convexity of the measure. Thus, without loss of generality, we only need to focus on the polynomial coherence measure on pure states in the following sections.

3. No-go theorem

The simplest example of polynomial coherence measure is the l1-norm for d = 2 on pure state. For a pure qubit state, |y> = a|0> + 3|1>, the l1-norm coherence measure takes the sum ofthe absolute value of the off-diagonal terms in the densitymatrix,

Qi (|y>) = |a3*| + |a*3| = 2|ab|. (3.1)

By the definition of equation (2.2), C/1 is the absolute value of a degree-2 homogenous polynomial function with a power m = 1. Meanwhile, this coherence measure on pure state satisfies the criteria (C1'), (C 2) [10]. Thus its convex-roof construction turns out to be a polynomial coherence measure satisfying all the criteria. Note that when the function equation (3.1) is extended to d > 2, it cannot be expressed as the absolute value of a homogenous polynomial function. Thus, when d > 2, the l1-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 Cp (|y>) and two orthogonal pure states |y1> and | y2>, there exists two complex numbers a and /3 such that

Cp (ay> + 3|^2>) = 0 (3.2)

where |a|2 + |/|2 = 1. That is, there exists at least one zero-coherence state in the superposition of |y1> and | y2>.

Proof. Since m > 0, the roots of Cp (|y>) = 0 inequation (2.2) are the same with the ones of |Ph (|y>)| = 0 in equation (2.1). That is, we only need to prove lemma for the case of m = 1. Since Ph (|y>) is a homogenous polynomial function ofthe coefficients of | y>, one can ignore its global phase. Thus, any pure state in the superposition of |y 1> and |y2> can be represented by

|y> = lyK^), (3.3)

V1 + | w |2

where the global phase is ignored, u is a complex number containing the relative phase, and | y> ^ | y2>, as | w| ^ ¥.

First, if Cp (| y2>) = 0, the lemma holds automatically. When Cp (| y2>) > 0, Cp (| y>) can be written as,

| y> + w | y2>

Cp ( | y>) =

= (1 + | w |2 )-h/2 | Ph ( | y> + w | y2>) | , (3.4)

since Ph a homogenous polynomial function of degree h. Note that the condition Cp (\y2)) > 0, i.e.,

lim (1 + M2)-h/2\Ph(ly) + w\y2))\ > 0, (3.5)

guarantees that the coefficient of wh in Ph (\y1) + w\y2)) = 0 is non-zero. Then, there are h roots ofthe homogenous polynomial function of u,

Ph (\y1) + w\y2)) = 0, (3.6)

denotedby (z1, z2 ... , zjJ.Thus, Cp (\y)) can be expressed as

Cp(\y)) = A(1 + \w\2)-V2n \w - Zi\, (3.7)

where A > 0 is some constant. In summary, we find at least one u, a = (1 + \w\2 )-^2 and b = w(1 + \w\2)-1/2, such that Cp (\y)) = 0. □

Theorem 1. There is no polynomial coherence measure in Hd with d ^ 3 that satisfies the criterion (C1').

Proof. In the following proof, we focus on the case of d ^ 4 and leave d = 3 in appendix A. With d ^ 4, we can decompose Hd into two orthogonal subspaces Hdl © Hd2 in the computational basis, i.e., Hd1 = (\i)i=1,-,d1} and Hd2 = (\i)i=di+1,—,d} with the corresponding dimensions d1and d2 = d — d1 both larger than 2.

Suppose there exist a polynomial coherence measure Cp (\y)) such that the criterion (C1') listed in table 1 can be satisfied. Then, there are exactly d zero-coherence pure states \i) (i = 1, • • •, d), which form the reference basis. One can pick up two coherent states, \y1) £ Hd1 and \y2) £ Hd2.Thatis, Cp (\y1)) > 0 and Cp (\y2)) > 0. Since two subspaces Hdl and Hd2 are orthogonal, any superposition of these two states, a\y1) + b\y2) with \a\2 + \b\2 = 1, should not equal to any ofthe reference basis states, i.e., a\y1) + b\y2) ^ \i), " i = 1, •••, d. Thus, due to the criterion (C1'), we have

Cp(a\y) + b\^2)) > 0. (3.8)

On the other hand, for the polynomial coherence measure Cp (\y)), lemma 1 states that provided any two orthogonal pure states \y 1), \y2), there exists at least a pair of complex numbers, a and b, such that a\y1) + b\y2) is a zero-coherence state, i.e.

Cp (a\y) + bW2» = 0. (3.9)

Therefore, it leads to a contradiction. □

4. Necessary condition for polynomial coherence measure

In theorem 1, we have shown a no-go result of polynomial coherence measure for d ^ 3 when the criterion (C1') in table 1 is considered. Thus, we release (C1) to (C1') and study polynomial coherence measure with the criteria (C1'), (C2), and (C3) in the following discussions. 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 Cp (\y)) = 0, for all \y) whose support does not span all the reference bases (i}.

Theorem 2. For any \y) £ Hd, the value of a polynomial coherence measure Cp (\y)) should vanish if the rank of the correspondingdephasedstate A(\y) (y \) is less than d, i.e. rank(A(\y) (y \)) < d.

Proof. Suppose there exists \y1) £ Hd such that Cp (\y1)) > 0 and rank(A(\y1) (y1\)) = d1 < d .Withoutloss ofgenerality, \y1) is assumed to be in the subspace Hd1 = spanned{\1), \2), •••, \d1)}.Define the complementary subspace to be Hd2 = spanned(\d1 + 1), \d1 + 2), •••, \d)}, where d2 = d — d1 > 0. We prove this theorem by two steps.

Step 1: we show that if d1 ^ d/2, then Cp (\ y 1)) > 0 leads to a contradiction to lemma 1. Now that d1 ^ d/2 ^ d2, there exists a relabeling unitary Ut that transforms the bases in Hd1 to parts ofthe bases in Hd2. Forinstance, Hdt = spanned(\1), \2)} and Hd2 = spanned(\3), \4), \5)},then Ut canbe chosen as \1)(3\ + \3)(1\ + \2)(4\ + \ 4) (2 \ + \5)(5\. Infact, Ut and U are both incoherent operation, since they just exchange the index ofthe reference bases. Assume that Ut maps \y1) toanew state \y2) = Ut\y1) £ Hd2,thenwe have (y1\y2) = 0. Due to the criterion (C2), it is not hard to show that an incoherent unitary transformation does not change the coherence,

Cp ( | = Cp ( I ^2». (4.1)

Define another incoherent operation, composed by two operators P1 = Ed=1 ¿)(i| and P2 = Ef=d+1 *')(*'| that project states to Wd1 and Wd2, respectively,

Ficptp (p) = E Pi PP?> (4.2)

i = 1,2

which represents a dephasing operation between the two subspaces. Then, for any superposition state, a | y1) + bly2) with | a |2 + | b|2 = 1, its coherence measure should not increase under the ICPTP operation with post-selection, as required by (C2) in table 1,

Cp (a | y) + b| y 2)) = | a 12Cp ( | y)) + | b 12Cp ( |

= Cp( |i1)) > 0, (4.3)

where the last equality comes from equation (4.1). Therefore, Cp (a | y1) + b| i2)) > 0 forany a and b .This leads to a contradiction to lemma 1.

Step 2: we show that if d/2 < d1 < d ,then Cp ( | y1)) > 0 also leads to a contradiction. Now that 0 < d2 < d/2 < d1 < d,forany | y2) G Wd2,wehave Cp ( | y2)) = 0 due to the above proof in Step 1.

Similar to the proof of lemma 1, we only need to consider the case of m = 1 and we can get the coherence measure for the superposition state of | y1) G Wd1 and | y2) G Hd2 as (1 + | w|2 )-h/2 ph ( | y1) + w| y2)) | .Since

Cp( |y2)) = lim (1 + | w|2 |Ph( |y1) + w| y2)) | = 0, (4.4)

the largest degree of u in the polynomial Ph (| y1) + w| y2)), denoted by p, is smaller than the degree h.

When ¡1 = 0, i.e., the polynomial is a constant, we denote its absolute value by k. Then the coherence measure becomes,

Cp ( | y)) = k (1 + | w |2 )-^2. (4.5)

We show that the constant k = 0 in appendix B.Asa result, Cp ( | y1)) = 0. This leads to a contradiction to our assumption that Cp ( | y1)) > 0.

When 0 < m < d,i.e., Ph (| y1) + w| y2)) is a non-constant polynomial of u, there exists at least one root |z| < ¥,such that Ph ( | y1) + z| y2)) = 0.Then,wecan find that the coherence measure of the state

| yr) = ( | yi) + z| y2)V V1+

| z |2 is Cp ( | yr)) = 0. Next, we apply the ICPTP operation described in equation (4.2) on | yr) and obtain,

Cp( |yr)) Cp( |y1)) + —LI— Cp( |y2))

1 |z| 2

1 +1-2Cp( |y1)) + | 2

1 + | z | 2 1 + | z | 2

Cp (|y1)), (4.6)

1 + | z |2

where we use Cp (| y2)) = 0 in the equality. Combing the fact that Cp (| yr)) = 0, we can reach the conclusion that Cp (| y1)) = 0. This leads to a contradiction to our assumption that Cp (| y1)) > 0.

5. 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 ofthe coefficients, for | y> = Ed=1 ai| '>,

Cg( |y>) = d| «1 «2 ■•• «d |2/d. (5.1)

Note that it is a degree-d homogenous polynomial function modulated by a power m = 2/d. This definition is an analog to the G-concurrence in entanglement measure, which is related to the geometric mean of the Schmidt coefficients of abipartite pure state [34]. Hence we call the coherence measure defined in equation (5.1) G-coherence measure. Since the geometric mean function is a concave function [35], following theorem 1 in [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 l1-norm measure on pure state. When d > 2, according to theorem 2, there is a significant amount of coherent states whose G-coherence is zero. For instance, in the case of d= 3, the state(| 0> + |1>) has zero G-coherence and this state cannot be transformed to a

coherent state | y>, where rank(A( | y> (y | )) = 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 equation (2.3) is generally hard. However, like the entanglement measures, there exist analytical solutions for the states with symmetries [37,38]. Here, we study the states related to the permutation

group Gs on the reference basis. A element g £ Gs is defined as

g = i1 2 .■. d ] (5.2)

\h i2 ... id)

and the order (the number of the elements) of Gs is d!. The corresponding unitary of g is denoted as Ug = £k\4)(k\. Then we have the following definition.

Definition 1. A state p is a symmetric state if it is invariant under all the permutation unitary operations, i.e.

"g £ Gs, UgpUl = p.

Denote the symmetric state as ps and the symmetric state set as S. Given the maximally coherent state \ Yd) = £\i), it is not hard to show the explicit form of symmetric states,

ps = p\Yd)(Yd \ + (1 — p) i, (5.3)

which is only determined by a single parameter, the mixing probability p £ [0, 1]. Apparently, the symmetric state ps is a mixture of the maximally coherent state \ Yd) and the maximally mixed state 1/d .The state \ Yd) 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 CG (ps) in theorem 3, following lemmas 2 and 3. First, we consider a map

l(p) = G £ Ug pU1 . (5

\ Gs\ g

It uniformly mixes all the permutation unitary Ug on a state p, which is an incoherent operation by definition.

Lemma 2. The map A(p) defined in equation (5.4) satisfies two properties, " p,

(1) A(p) £ S,i.e. the output state is a symmetric state, as defined in definition 1;

(2) (Yd\ p\Yd) = (Yd\A(p)\Yd), i.e. the map A(p) does not change the overlap with the maximally coherent state \Yd).

Proof. For any Ug, with g' £ Gs,

Ug'A(p) Ug, = -L £ (Ug, Ug) p (Ug, Ug)t \ Gs\ g

= — £ uxi pU],

\Gs\ Y gg g g

= A(p). (5.5)

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 Gs. By definition 1, we prove that A(p) £ S.

The overlap between the output state A(p) and the maximally coherent state \ Yd) is given by,

(Yd\A(p) \ Yd) = (Yd \ -L £ Ug pUg\Yd) \ Gs\ g

= G £(Yd\ UJ—1 pUg—1\Yd)

\Gs\ g g

= (Yd\ p\Yd). (5.6)

where in the second line we use the relation Ug = Ug—1 and the last line is due to the fact that \Yd) £ S and Ug—1\Yd) = \Yd). □

Then, we define the following function for a symmetric state ps,

C g (ps) = min ( cg (\y))\A(\y)(y \) = ps}. (5.7)

Since the state ps in equation (5.3) only has one parameter p, it can be uniquely determined by its overlap with the maximally coherent state K = (Yd \ ps\Yd) = p-——- + 1 .Thus, ps linearly depends on K. According to lemma 2, A(\y)(y \) is a symmetric state and the overlap does not change under the map A. Hence, the constraint A(\y)(y \) = ps inequation (5.7) is equivalent to \ (Yd\ y)\2 = (Yd\ ps\Yd). Following the derivations of the G-concurrence [39], we solve the minimization problem and obtain an explicit form of

Cg(K) K> CG(K)

Figure 1. Illustration for the two functions CG (K ) and CG(K ) in d = 4case.When 0 < K < = 0.75, CG (K ) = 0; when

——— = 0.75 ^ K ^ 1, Cg (K) is a concave function following the form in equation (5.8), represented bythe dashed blue line. Thus the minimization result via equation (5.13), CG(K) is the linear function 1 — 4(1 — K), when = 0.75 ^ K ^ 1, described by the red line.

Со (Гs),

0 0 ^ K ^

С о (K ) =

d (abd-1)(2/d) d-1 ^ K ^ 1

: = (VK - V d - W1 - K ), Vd

0 ^ K ^

d , (5.8)

b = — ( 4K + - k

yfd Vd - 1

Details can be found in appendix C. Here, we substitute CG (K) for CG (ps) without ambiguity. When ——1 ^ K ^ 1, CG (K) is a concave function [39].Weshow CG (K) in the case of d = 4in figure 1. Moreover, following the results of [38], we have the following lemma.

Lemma 3. The convex-roof of the G-coherence measure CG for a symmetric state ps is given by,

Cg (ps) = min E PCg ( IУ))

p IУ» i '

= min E qCg(pS), (5.9)

where Ei pi I У)(У I = Ps, E, q p] = Ps,and ps, î S.

'jij'}'' 'j

d ^ = rnirn Wjijro4>j,

Proof. Denote Z1 = min[ft, | y)) ( | y)) and Z2 = min^.,^) ЕдСо (p5).Nowwe prove the lemma by

showing that both of them equal to,

Z3 = min JepCg( |y,>) EpA( |y'>(y«|) = Psl. (5.10)

(p,-> | y'» [ ' ' J

Z1 = Z3: for a decomposition, ps = E' p | y >(y | , after applying the map A on both sides, we have

E p A( | y >( y | ) = A(ps) = ps. (5.11)

Here, we use the fact that ps is a symmetric state, which is invariant under the map A. That is, any decomposition satisfies the constraint E' pi| ^'X^d = ps as required forZ1 also satisfies the constraint E' p A(| y>(y | ) = ps as required for Z3. Thus, we have Z3 ^ Z1. On the other hand, the constraint E' p' A(| | ) = ps in equation (5.10) is also a pure-state decomposition of the state ps, since every component in A( | y > (y | ) is a pure state Ug|y> with probability p'/| Gs|. Thus we also have Z1 ^ Z3. Consequently, Z1 = Z3.

Z2 = Z3: In fact, the constraint in equation (5.10) is on A(\yi)(yi\) £ S,thuswe can solve the minimization problem of equation (5.10) in two steps. First, given A(\yi) (yi \) £ S, we minimize CG (\yi)), which turns out to be the same as the definition of CG (A(\yi) (yi \)) in equation (5.7). Next, we optimize the decomposition of ps in the symmetric state set S, which turns out to be the same as the definition of Z2. Thus we have Z2 = Z3. □

Theorem 3. For a symmetric state ps £ S in Hd, the G-coherence measure is given by

CG(ps) = max(1 — d(1 — K), 0}, (5.12)

where K = (Yd\ ps\Yd) is the overlap between ps and the maximally coherent state \ Yd).

Proof. According to lemma 3, the G-coherence measure for a symmetric state is given by CG (ps) = min(q.,ps} £ q CG (ps) with £j qj ps. = ps. Since the symmetric state linearly depends on the overlap K, this minimization is equivalent to,

Cg (K) = min < £ qjCG (Kj)

(qj,Kj} \ y 1

(5.13)

Then, according to the explicit expression of CG (K) in equation ( ): When 0 < K < , CG (K) = 0. Thus,

CG(K) < CG(K) = 0. When ^ < K < 1, fortunately, CG(K) is a concave function. It is not hard to find that

the optimization result is a straight line connecting the point (, 0} and (1, 1} on the (K, CG (K)} plane. Consequently, CG (ps) shows the form in equation (5.12). □

The dependence of CG (K) and CG(K) on K in the case of d = 4 are plotted in figure 1. Furthermore, we can give a lower bound of the G-coherence measure CG for any general mixed state p, with the analytical solution for ps in theorem 3.

Corollary 1. For a mixed state p,

CG(p) ^ max[1 — d(1 — K), 0], (5.14)

where K = (Yd\ p\Yd).

Proof. Since A is an incoherent operation, we have,

Cg (p) ^ Cg (A(p)). (5.15)

From lemma 2, we know that the overlap K = (Yd \ p\Yd) = (Yd\A(p)\Yi) and A(p) £ S. Following theorem 3, the corollary holds. □

In fact, the tightness of the bound depends on the overlap. Thus, we can enhance the bound by pre-treating the state by a certain ICPTP x that can increase the overlap, i.e.

Cg(p) ^ Cg(c(p)) ^ Cg(A(c(p))) ^ max[1 — d(1 — K'), 0], (5.16)

where K' = (Yd\ c(p)\Y,) > K = (Yd\ p\Yd).

6. Conclusion and outlook

In this paper, we give the definition of polynomial coherence measure Cp (p), 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 the criterion (C1') in table 1, when the dimension of the Hilbert space d is larger than 2. That is, there always exist some pure states \y) ^ \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 A (\ y)(y \) is smaller than the Hilbert space dimension d. Meanwhile, we give an example of polynomial coherence measure CG (p), called G-coherence measure. We conjecture that there are not too many polynomial coherence measures, due to the restrictive condition given by theorem 2; and we suspect that all the polynomial measures would share similar structure as the G-coherence. Moreover, we derive an analytical formula of the convex-roof of CG for symmetric states, and also give a lower bound of CG for general mixed states. In addition, we should remark that the symmetry consideration in our paper is also helpful to understand and bound 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 h ^ 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 = Hd®N, whose local dimension di > 2 [28]. Moreover, polynomial coherence measure (especially G-coherence) defined here may serve as an important quantifier when studying the relation and conversion between the two important quantum resources, coherence and entanglement.

After finishing the manuscript, we find that a coherence measure similar to CG (p) is also put forward in [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 [42].

Acknowledgments

We acknowledge J Ma and T Peng for the insightful discussions. This work was supported by the National Natural Science Foundation of China Grants No. 11674193.

Appendix A. Proof of theorem 1 for d = 3

In the main part, theorem 1 for the case of d ^ 4 has been proved. Here we prove the d = 3 case. First, a lemma that is an extension of lemma 1 follows.

Lemma 4. For any polynomial coherence measure Cp (\y)), and any two pure quantum states \y1), \y2) satisfying \ (y2 \ y1) \ < 1, there is at least one zero-coherence state in the superposition space of them.

Proof. Like in lemma 1, without loss of generality, we just need to consider the scenario of power m = 1. First, if Cp (\y2)) = 0, the lemma holds automatically. So we focus on the Cp (\y2)) ^ 0 case in the following.

Letusdenote (y1\ y2) = kei0 with k < 1.Then, after ignoring the global phase, any superposition state of \y1) and \y2) can be represented by

\y) = , (A.1)

where u is a complex number and the normalization factor Z(w) = \\y1) + w\y2)\= yj1 + \w\2 + 2\w\k cos(0 + 0 ') with w = \w\ei0'. Similar to lemma. 1, we can factorize Cp (\y)) as

Cp (\y)) =

. jyx> + w\y2)

A' Z (w)

\ph (\y) + w\y 2)) \

■nh=1\w — Zi\, (A. 2)

where A ' is a constant and zi (i = 1, 2, •••, h) are the roots of the polynomial function Ph (\y1) + w\y2)).Thus we can find at least one root in this Cp (\y2)) ^ 0 case, or equivalently, a zero-coherence state. □

With the help of lemma 4, now we prove theorem 1 for d = 3 case. First, similar to the main part, we can choose two states with non-zero coherence as,

№1) = ^ (\1) + \2)),

\^2) = (\2) + \3)). (A.3)

Even though these two states share overlap with each other, any superposition state a\y1) + b\y2) should not equal to the pure state\i) (i = 1, 2, •••, d) in the computational basis. As required by the criterion (C1') in table. 1, \i) (i = 1, 2, ..., d) are the only zero-coherence pure state. Thus, Cp (a\y1) + b\y2)) > 0. Nonetheless, it is contradict to lemma. 4. Consequently, there is no polynomial coherence measure satisfying the criterion (C1') for d = 3 case.

Appendix B. Proof for k = 0 in equation (4.5)

In the main part, the coherence measure for the superposition state of | y1) G Wd1 and | y2) G Hd2 shows,

Cp ( | y)) = k (1 + | w |2 )—h/2.

If k > 0, the coherence measure strictly decreases with the increasing of | w| . That is, for any superposition state | y) = (| y1) + w| y2))/+ | w|2 with | w| > 0, we have Cp (| y)) < Cp (| y 1)). We denote the state coefficients by a = (1 + | w|2and b = w (1 + | w|2here. In the following, we show that there exists a state | y) = a| y1) + b| y2) with a < 1 (or equivalently | w| > 0), such that Cp (| y)) ^ Cp (| y 1)). As a result, this contradiction leads to k = 0.

From [12,43], weknowthat | Y) = Ed=1 Y| i) can transform to | F) = Ed=1 F| i) via incoherent operation, if (| Y1 | 2, ..., | Yd |2 / ismajorizedby ( | F1 |2, ..., | Fd |2 )f. Then combing the criteria (C2) and (C3) in table. 1,we obtain that the coherence measure is non-increasing after incoherent operation. Thus, C ( | Y)) ^ C ( | F)) forany coherence measure.

Inourcase, first, we denote | y1) = Ed=1 ai | i) with " i, | ai | > 0.Andchoose | y2) = -i=Ed=d +1 i).Then we

V d2 1

can build a state | y) = a | y1) + b| y2) that satisfies a < 1 and Cp ( | y)) ^ Cp ( | y1)), with the help of the aforementioned majorization condition. To be specific, if a satisfying,

a2 | aj | 2 ^ bVd2, (B.1)

where | aj |2 is the minimal value in {| ai |2},then (a21 a12, a21 a2 |2, ..., a2 | ad1 |2, b2/d2, .••, b2/d2 )t ismajorizedby ( | a1 |2, | a2 | 2, ..., | ad1 | 2, 0, ..., 0/ .Thus, Cp ( | y)) ^ Cp ( | y^). In fact, a = (d2 | a, |2 + 1)-1/2 < 1, when the inequality is saturated in equation (B.1).

Appendix C. Derivation of equation (5.8)

As mentioned in the main part, the constraint for the pure state | y) = Ei ai| i) in equation (5.7) is the overlap K = | (Yd| y) | 2,i.e.

| E ai | = JdK, (C. 1)

and the coefficients a{ of the state should also satisfy the normalization condition,

E | ai |2 = 1. (C.2)

When 0 ^ K ^ we can always set one of the coefficients aj = 0with j G {i}, and let the corresponding CG equal to 0. Thus CG(K) = 0 in this Kdomain.

On the other hand, all the coefficients ai ^ 0,when ^ K ^ 1. In this Kdomain, we should minimize

Cg (| y)) = d (P | ai | }d under the constraints in equations (C.1) and (C.2). Note that Ei | ai | ^ | Eiai | and the equality can be reached when the coefficients share the same phase. Thus the constraint in equation (C.1) can be replaced by,

E | ai | = VdK. (C.3)

In fact, the function optimized here is the same to the one in [39] for the G-concurrence, after substituting the Schmidt coefficients for the state coefficients | ai|. Thus, utilizing the same Lagrange multipliers in Supplemental Material of [39], we can obtain equation (5.8) in the main part. And we can show that CG (K) is a concave function, when ^ K ^ 1, by directly following the derivation there.

ORCIDiDs

Xiao Yuan https:/orcid.org/0000-0003-0205-6545 Xiongfeng Ma https://orcid.org/0000-0002-9441-4006

References

[1] Glauber R J1963 Phys. Rev. 130 2529-39

[2] Nielsen M A and Chuang IL 2010 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)

[3] Braunstein S L, Caves C M and Milburn G 1996 Ann. Phys., NY247135-73

[4] Ma X, Yuan X, Cao Z, Qi B and Zhang Z 2016 NPJ Quantum Inf. 2 16021

[5] ÄbergJ2014 Phys. Rev. Lett. 113 150402

[6] LostaglioM, Jennings D and Rudolph T 2015 Nat. Commun. 6 6383

[7] Lostaglio M, Korzekwa K, Jennings D and Rudolph T 2015 Phys. Rev. X 5 021001

[8] gakmak B, Karpat G and Fanchini F F 2015 Entropy 17 790-817

[9] Karpat G, ^akmak B and Fanchini F 2014 Phys. Rev. B 90 104431

[10] Baumgratz T, Cramer M and Plenio M B 2014 Phys. Rev. Lett. 113 140401

[11] Yuan X, Zhou H, Cao Z and Ma X 2015 Phys. Rev. A 92 022124

[12] Winter A and Yang D 2016 Phys. Rev. Lett. 116120404

[13] Chitambar E and Hsieh M H 2016 Phys. Rev. Lett. 117 020402

[14] Streltsov A, Rana S, Bera M N and Lewenstein M 2017 Phys. Rev. X 7 011024

[15] Ma J, Yadin B, Girolami D, Vedral V and Gu M 2016 Phys. Rev. Lett. 116160407

[16] Streltsov A, Singh U, Dhar H S, Bera M N and Adesso G 2015 Phys. Rev. Lett. 115 020403

[17] Chitambar E, Streltsov A, Rana S, Bera M, Adesso G and Lewenstein M 2016 Phys. Rev. Lett. 116 070402

[18] Horodecki R, Horodecki P, Horodecki M and Horodecki K 2009 Rev. Mod. Phys. 81 865-942

[19] Bennett C H, DiVincenzo D P, Smolin J A and Wootters W K1996 Phys. Rev. A 54 3824

[20] Uhlmann A1998 Open Syst. Inf. Dyn. 5 209-28

[21] Hill S and Wootters W K1997 Phys. Rev. Lett. 78 5022

[22] Wootters W K1998 Phys. Rev. Lett. 80 2245

[23] Coffman V, Kundu J and Wootters W K 2000 Phys. Rev. A 61052306

[24] Eltschka C and Siewert J 2014 J. Phys. A: Math. Theor. 47 424005

[25] Dür W, Vidal G and Cirac J12000 Phys. Rev. A 62 062314

[26] Osterloh A and Siewert J 2005 Phys. Rev. A 72 012337

[27] Djokovic D Z and Osterloh A 2009 J. Math. Phys. 50 033509

[28] GourG and Wallach NR 2013 Phys. Rev. Lett. 111 060502

[29] Lohmayer R, Osterloh A, Siewert J and Uhlmann A 2006 Phys. Rev. Lett. 97 260502

[30] Jung E, Hwang M R, Park D and Son J W 2009 Phys. Rev. A 79 024306

[31] Siewert J and Eltschka C 2012 Phys. Rev. Lett. 108 230502

[32] Regula B and AdessoG 2016 Phys. Rev. Lett. 116 070504

[33] Horodecki P 1997 Phys. Lett. A 232 333-9

[34] GourG 2005 Phys.Rev. A 71 012318

[35] Boyd SandVandenbergheL 2004 Convex Optimization (Cambridge: Cambridge University Press)

[36] Du S, Bai Z and Qi X 2015 Quantum Inf. Comput. 15 1307-16

[37] Terhal B M and Vollbrecht K G H 2000 Phys. Rev. Lett. 85 2625

[38] Vollbrecht K G H and Werner R F 2001 Phys. Rev. A 64 062307

[39] Sentis G, Eltschka C, Gühne O, Huber M and Siewert J 2016 Phys. Rev. Lett. 117 190502

[40] Verstraete F, Dehaene J and De Moor B 2003 Phys. Rev. A 68 012103

[41] Eltschka C, Bastin T, Osterloh A and Siewert J 2012 Phys. Rev. A 85 022301

[42] Chin S 2017 J. Phys. A: Math. Theor. 50 475302

[43] Du S, Bai Z and Guo Y 2015 Phys. Rev. A 91 052120