Generalized Coherence Concurrence and Path Distinguishability

Generalized Coherence Concurrence and Path Distinguishability

Seungbeom Chin College of Information and Communication Engineering, Sungkyunkwan University, Suwon 16419, Korea
Abstract

We propose a new family of coherence monotones, named the generalized coherence concurrence (or coherence -concurrence), which is an analogous concept to the generalized entanglement concurrence. The coherence -concurrence of a state is nonzero if and only if the coherence number (a recently introduced discrete coherence monotone) of the state is not smaller than , and a state can be converted to a state with nonzero entanglement -concurrence via incoherent operations if and only if the state has nonzero coherence -concurrence. We apply the coherence concurrence family to the problem of wave-particle duality in multi-path interference phenomena. We obtain a sharper equation for path distinguishability (which witness the duality) than the known value and show that the amount of each concurrence for the quanton state determines the number of slits which are identified unambiguously.

PACS numbers

03.65.Ta, 03.67.Bg, 03.67.Mn

pacs:
03.65.Ta, 03.67.Bg, 03.67.Mn

I Introduction

The superposition principle is a key aspect of quantum physics, and representative quantum properties such as entanglement are understood best under the framework of superposition. We can say that a state is nonclassical (quantum) if and only if it is a superposition of some classical state Killoran et al. (2016). In the resource theory of coherence, classical states are in some fixed orthonormal basis set , and the coherence is a basis-dependent quantity.

Since the quantitative formulation for measuring the amount of coherence is presented in Baumgratz et al. (2014), the coherence resource theory flourished in diverse aspects, e.g., finding new measures and monotones of coherence Yuan et al. (2015); Winter and Yang (2016); Napoli et al. (2016); Piani et al. (2016); Tan et al. (2016); Chitambar and Gour (2016), understanding the relation of coherence with other correlationsStreltsov et al. (2015); Adesso et al. (2016); Roga et al. (2016); Ma et al. (2016); Marvian and Spekkens (2016); Marvian et al. (2016); Rana et al. (2016a, b), dynamics of coherenceBromley et al. (2015); Mani and Karimipour (2015); Puchała et al. (2016); Singh et al. (2016); Mondal et al. (2016), and quantum thermodynamical approachesLostaglio et al. (2015); Ćwikliński et al. (2015); Narasimhachar and Gour (2015) (see Streltsov et al. (2016) for an up-to-date review on the coherence resource theory).

One of the interesting topics in this area is to understand the connection between coherence and entanglement theory. In Streltsov et al. (2015) it was shown that a nonzero coherent state can be used to create entanglement. Killoran Killoran et al. (2016) generalized this process, and provided a framework for converting nonclassicality (including coherence) into entanglement. While discussing the conversion theorem, the authors presented an analogous concept to the Schmidt rank of entangled pure states, which is the of pure states. This concept was generalized to mixed state case in Chin (2017), where the of a mixed state was introduced along the Schmidt number of entangled mixed states. It is proved there that a state can be converted to an entangled state of nonzero -concurrence (the -th member of the generalized concurrence monotone family Gour (2005)) if and only if the coherence number of the state is not smaller than .

In this paper we introduce a set of new coherence monotones, which we call “generalized coherence concurrence” (or coherence -concurrence following), which is organized to witness the coherence number and also has the formal and operational similarity with the generalized entanglement concurrence. The coherence 2-concurrence from our construction is different from the coherence concurrence recently provided by Qi et al. (2016), and we compare the two monotones and -norm coherence monotone quantitatively. It will be seen that the coherence -concurrence of a mixed state , denoted as , is nonzero if and only if , and the state can be converted to a state with nonzero entanglement -concurrence via incoherent operations by adding an ancilla system set in a fixed incoherent state if and only if is nonzero.

The fact that the coherence concurrence family is well-ordered and has a hierarchy means that the monotones will be useful when we need to know for a quantum system how many bases are how intensively coherent to each other. As an example problem, we exploit the generalized coherence concurrence to understand wave-particle duality in the context of multi-slit interference experiments. It turns out that the concurrence family can capture the path distinguishability more accurately than -norm monotone used in Bera et al. (2015); Paul and Qureshi (2017).

This paper is organized as follows. In Section II, we briefly review the concept of the generalize entanglement concurrence and the coherence number. In Section III, we define the generalized coherence concurrence and show that it is a coherence monotone family. Then we derive the convertibility theorem of the generalized concurrence between coherence and entanglement, and compare the second member of the concurrence family, , with other coherence monotones such as the coherence concurrence recently presented in Qi et al. (2016) and -norm coherence . In Section IV, we calculate a new equation for path distinguiashbility using , and show that the generalized coherence concurrence provides further information on how many slits are distinguisable, which is by the hierarchy property of the coherence concurrence family. In Section V, we summarize our results and present some remaining issues.

Ii Review: The Generalized Entalgement Concurrence and Coherence Number

In this section we briefly review the concepts of the generalized entanglement concurrence and the coherence number.

The generalized entaglement concurrence

The generalized entanglement concurrence monotone is a set of entanglement monotones for -systems Gour (2005). This is the generalization of the -system entanglement concurrence Hill and Wootters (1997); Wootters (1998). With a -dimensional bipartite pure state (’s are the Schmidt coefficients of ), the -concurrence () of is defined as

 E(k)c(|ψ⟩) (1) Sk(λ) ≡∑i1

where , so is normalized as . Only when is maximally entangled is equal to . The -concurrence for a mixed state is defined by convex roof extension:

 E(k)c(ρ)≡min{pi,|ψi⟩}∑ipiE(k)c(|ψi⟩) (ρ=∑ipi|ψi⟩⟨ψi|,∑ipi=1,pi≥0). (3)

The -concurrence of is nonzero only when the Schmidt number of is not smaller than . All the -concurrences with consist in the generalized entanglement concurrence. This entanglement monotone family is worth investigating since it consists of continuous measures of all possible entanglement dimension, some of which might be useful resources for quantum computation Sentís et al. (2016).

The last member of the concurrence family is named -concurrence , i.e., . It has some convenient properties such as multiplicativity which are from the geometric mean form of the concurrence. The -concurrence also provides a lower bound for the whole -concurrence family:

 E(2)c(ρ)≥E(3)c(ρ)≥⋯≥E(d−1)c(ρ)≥E(d)c(ρ)=Gd(ρ). (4)

This is useful to analyze some entanglement system such as remote entanglement distribution (RED) protocols Gour (2005).

The pure state -concurrence can be rewritten in terms of that is not Schmidt-decomposed, i.e.,

 |ψ⟩=∑ijψij|ij⟩AB, (5)

which is given by Chin (2017)

 E(k)c(|ψ⟩) =d[1(dk)∑i1<⋯

Coherence number

The coherence rank of a pure state Killoran et al. (2016) is defined as

 rC(|ψ⟩)≡min{r∣∣∣|ψ⟩=r≤d∑j=1ψj|cj⟩}, (7)

where the set is the (classical) referential basis set that is relabeled, and . A pure state is nonclassical when . There exists a unitary operation on such that the Schmidt rank of equals the coherence rank of . And the Chin (2017) is a generalized concept of the coherence rank, in a similar manner to the Schmidt number Terhal and Horodecki (2000).

Definition 1.

The coherence number for a mixed state is defined as

 (8)

For pure states the coherence number is equal to the coherence rank. The logarithm of the coherence number is a discrete coherence monotone satisfying the axioms (C1-3) listed in Section III.

It is proved that the coherence number is a simple criterion for a state to be a source for nonzero entanglement -concurrences, i.e., can be converted to an entangled state with ( is an operation between the given system of and and an ancilla system) if and only if Chin (2017).

Iii Generalized Coherence Concurrence

The coherence resource theory resembles the entanglement resource theory in many aspects, which is considered as evidence that the quantum entanglement is a sort of derivative of coherence. Streltsov Streltsov et al. (2015) showed that a coherent state can be a resource of a bipartite entangled state through incoherent operations by attaching an ancilla system. A similar process is possible for quantum discord Ma et al. (2016).

In this section, we introduce a family of new coherence monotones that is designed to correspond to the generalized entanglement concurrence. Before tackling the main task, we summarize the axioms that the coherence monotones should fulfill Baumgratz et al. (2014):

(C1) Nonnegativity:

(a stronger condition: if and only if is incoherent)
(C2) Monotonicity: is non-increasing under the incoherent operations, i.e., for any incoherent operation , where admits a set of Kraus operators such that and for any (the set of incoherent density operators).
(C3) Strong monotonicity: is non-increasing under selective incoherent operations, i.e., with , for incoherent Kraus operators .
(C4) Convexity: .

The conditions (C1) and (C2) are the minimal requirements for a quantity to be a coherent monotone, and a quantity that fulfills (C3) and (C4) naturally fulfills (C2).

The generalized coherence concurrence as a coherence monotone family

As mentioned in Section II, the entanglement -concurrence of a mixed state is nonzero if and only if the Schmidt number of is not smaller than . Since the coherence number (rank) is the corresponding quantity in coherence theory to Schmidt number (rank) in entanglement theory, we expect that if there exists a coherence monotone that corresponds to the entanglement -concurrence, the monotone would have a similar relation with coherence number to the relation of entanglement -concurrence with Schmidt number. We name it coherence -concurrence, which consists in the generalized coherence concurrence family). The definition of the coherence -concurrence is as follows:

Definition 2.

The coherence -concurrence for a pure state is defined as

 C(k)c(|ψ⟩) =d(1(dk)∑i1

and for a mixed state is defined by convex roof extension:

 C(k)c(ρ)≡min{pa,|ψa⟩}∑apaCk(|ψa⟩) (ρ=∑apa|ψa⟩⟨ψa|,∑apa=1,pa≥0). (10)

A family of generalized coherence concurrence consists of with .

The normalization factor is multiplied so that the generalized coherence concurrence has a similar inequality order to the generalized entanglement concurrence, i.e.,

 C(2)c(ρ)≥C(3)c(ρ)≥⋯≥C(d−1)c(ρ)≥C(d)c(ρ), (11)

which is straightforward by Maclaurin’s inequality.

Theorem 1.

is a coherence monotone that satisfies (C1) to (C4).

Proof.

The fulfillment of (C1) and (C4) is clear by definition, the strong condition of (C1) does not hold though (see Theorem 2 below). (C2) is satisfied if (C3) and (C4) are satisfied, so what remains to be proven is (C3). It is proved in Appendix A that (C3) is fulfilled for pure states, i.e.,

 ∑npnC(k)c(|ψn⟩)≤C(k)c(|ψ⟩). (12)

Then also satisfies (C3) for the convex roof extension of the quantity to mixed states (see Appendix A.1 of Qi et al. (2016)).

The conversion of concurrence from coherence into entanglement

In this subsection, we examine the entanglement convertibility theorem of the generalized concurrence monotone and show that the generalized coherence concurrence is an entanglement-based monotone Streltsov et al. (2015). The referential entanglement monotone is the generalized entanglement concurrence as expected.

To obtain the convertibility theorem, we first clarify the relation between the coherence -concurrence and the coherence number, which is basically identical to the relation between the entanglement -concurrence and the Schmidt number.

Theorem 2.

For a state , the coherence -concurrence is nonzero if and only if the coherence number is not smaller than .

Proof.

: Supposing is the optimal decomposition of for , the condition means that there exists at least one decomposing pure state that satisfies . So by Definition 2, which goes to by Definition 1.
: Suppose . Then there exists a decomposition that satisfies . So we have . ∎

Now with Theorem 2 we can obtain the conversion theorem for each coherence -concurrence quite simply:

Theorem 3.

A state can be converted to a state of nonzero entanglement -concurrence via an incoherent operation by appending an ancillar system which is set in a referential incoherent state if and only if the coherence -concurrence is nonzero.

Proof.

This statement is derived using the coherence number, which links the generalized concurrences for coherence and entanglement theory. Theorem 3 of Chin (2017) and Theorem 2 gives

 ∃ΛSA:E(k)c(ΛSA[ρ⊗|1⟩⟨1|A])≠0 ⟺rC(ρ)≥k ⟺C(k)c(ρs)≠0. (13)

Now we consider what happens when the incoherent operation on the bipartite system is a unitary operation that transforms the coherence rank of a pure state to the Schmidt rank of the bipartite pure state,

 U≡d∑i=1d∑j=i|i⟩⟨i|S⊗|i⊕(j−1)⟩⟨j|A, (14)

where represents an addition modulo . This is often called the generalized CNOT operation. And goes to under ().

Then we have

 E(k)c(|ψ⟩SA) =d[1(dk)∑i1<⋯ik|ψi1⋯ψik|2]1k =C(k)c(|ψ⟩S). (15)

for a pure state , and this equality holds for a mixed state by the definition of convex roof extension:

 E(k)c(ΛSAu[ρs⊗|1⟩⟨1|A])=C(k)c(ρs). (16)

So we can say that the generalized coherence concurrence is a kind of entanglement-based coherence monotone (Eq. (12) of Streltsov et al. (2015)).

Comparison of C(2)c with the coherence concurrence Cc and l1-norm coherence Cl1

In the entaglement theory, the 2-concurrence in the generalized concurrence family is equal to the entanglement concurrence presented by Hill and Wootters (1997); Wootters (1998). In the coherence theory, we can compare 2-concurrence with the coherence concurrence recently presented by Qi et al. (2016), which is defined as

 Cc(|ψ⟩)=2∑j

for a pure state and for a mixed state is the convex roof extension of the pure state monotone.

We have

 C(2)c(|ψ⟩) =d(d2)12(∑j

and the following relation between and holds:

Theorem 4.
 1d−1Cc(ρ)≤C(2)c(ρ)≤√d2(d−1)Cc(ρ). (19)
Proof.

For pure state case, the right inequality is trivial and the left inequality comes from the Newton’s inequality of elementary symmetric polynomials. By convex roof extension, we obtain Theorem 4. ∎

This relation is interesting for some reasons. First, it supports the claim in Rana et al. (2016a, b) that is analogous to entanglement negativity. Indeed, considering is equal to for pure states, we have

 1d−1Cl1(|ψ⟩)≤C(2)c(|ψ⟩)≤√d2(d−1)Cl1(|ψ⟩). (20)

There exists the same form of inequality between the 2-concurrence and the negativity in entanglement theory Eltschka and Siewert (2014). Second, combining Theorem 4 with Eq. (III), imposes an uppper bound for the whole coherence -concurrence family, i.e.,

 √d2(d−1)Cc(ρ) ≥C(2)c(ρ) ≥C(3)c(ρ)≥⋯≥C(d−1)c(ρ)≥C(d)c(ρ). (21)

As an additional discussion, is in general not smaller than -norm coherence monotone, i.e.,

 Cc(ρ)≥Cl1(ρ)≡2∑j

Qi et al. (2016), but there exists a necessary and sufficient condition for of a mixed state to be equal to .

Theorem 5.

For , and coincide if and only if the state satisfies

 ρijρjkρki|ρijρjkρki|=1 (no summation over i,j,k and i≠j≠k) (23)

for all non-zero components of . For , they always coincide.

The proof is given in Appendix B. We can see that the equality always holds for a real symmetric state .

Iv C(k)c and Path Distinguishability

We expect that the generalized coherence concurrence is useful for some quantum systems about which we want to know both how many bases are coherent with each other and how much they are coherent to each other. As an example, here we try to delve into the path distinguishability problem in multi-slit experiments using .

Wave-particle duality is an ironic but intriguing property of quantum theory. There have been efforts to understand the complementary principle in the context of two-path interference experiments Wootters and Zurek (1979); Greenberger and Yasin (1988); Englert (1996), in which the duality is quantitatively expressed, e.g., as the Englert-Greenberger-Yasin (EGY) relation. The investigation has gone further to multi-path cases Jaeger et al. (1995); Dürr (2001); Bimonte and Musto (2003); Englert et al. (2008).

Based on the natural idea that coherence is a representative wave-like property, a new duality relation for general -slit interference was obtained using Bera et al. (2015); Paul and Qureshi (2017):

 DQ+1d−1Cl1≤1, (24)

where is a path distinguishability, (representative of particle-like property) based on UQSD (the unambiguous quantum state discrimination). The inequality is saturated when the quanton (wave-particle-like quantum system) is pure.

But as is non-zero if and only if the coherence number is not smaller than 2, what we can say with is just whether the quanton has wave-like property or not. We claim that the generalized coherence concurrence provides further information on how many slits are unambiguously identified.

We first show that coherence 2-concurrence gives a sharper bound for and then discuss the relation between the amount of for each and the number of completely distinguishable slits.

Path distinguishability revisited with C(2)c

We tackle the problem by first considering -slit interference of pure quantons, the state of which is expressed with basis states as

 |Ψ⟩=∑ici|ψi⟩, (25)

where the bases are orthonormal since they represent well seperated different slits and . To measure the quanton, we need to let a detector interact and correlate as follows:

 |Ψ⟩↦∑ici|ψi⟩⊗|0⟩D↦∑ici|ψi⟩⊗|i⟩D. (26)

The states are normlized, . But as ’s are not necessarily orthogonal, we express them with an orthonormal basis set and as

 |i⟩D=ϕi|ϕi⟩+∑a√paqia|a⟩≡ϕi|ϕi⟩+|qi⟩, (27)

where , only when , and for all and . The normalization condition gives . And the reduced density matrix of the quanton is given by

 ρs =trD(|Ψ⟩⟨Ψ|) =∑i,j(∑apaciqiac∗jqj∗a+δijcic∗j|ϕi|2)|ψi⟩⟨ψj| =∑i|ciϕi|2|ψi⟩⟨ψi|+∑a(pa∑i|ciqia|2)|ψa⟩⟨ψa|, (28)

where is normalized. This is one way of pure state decompostion for , so we have

 C(k)c(ρs)≤∑a(pa∑i|ciqia|2)C(k)c(|ψa⟩). (29)

by the definition of . We can adjust so that the inequality is saturated. A special case is when

 |i⟩D=|a⟩, (30)

,i.e., are the same for all . Then Eq. (IV) becomes pure,

 ρs=∑i,jcic∗j|ψi⟩⟨ψj|, (31)

which means that the measurement plays no role for the quanton system.

To obtain the path distinguishability, we divide measurement operations into two groups as

 ^Am|i⟩D∝ |ϕi⟩, ^Bm|i⟩D∝ |qi⟩ (32)

so that and represent successful and failure transformations for distinguishability respectively with the restriction . The success and failure probability are defined as

 Pd=∑i|ci|2∑m⟨i|^A†m^Am|i⟩D, Qd=∑i|ci|2∑m⟨i|^B†m^Bm|i⟩D.(P+Q=1) (33)

Then using Cauchy-Schwarz inequality (see Eq. (9) of Qiu (2002)), we have

 Q2d≥dd−1∑i≠j|ci|2|cj|2 ⟨i|∑m^B†m^Bm|i⟩D ×⟨j|∑m^B†m^Bm|j⟩D (34)

Eq. (IV) gives

 ⟨i|∑m^B†m^Bm|i⟩D =∑a,b√papbqi∗aqib⟨a|∑m^B†m^Bm|b⟩ =∑a,b√papbqi∗aqib⟨a|(I−∑m^A†m^Am)|b⟩ =∑apa|qia|2, (35)

and Eq. (IV) is rewritten as

 Qd ≥[dd−1∑i≠j|ci|2|cj|2∑a,b(pa|qia|2)(pb|qjb|2)]12 ≥∑apa[dd−1∑i≠j|ci|2|cj|2|qia|2|qja|2]12 ≥C(2)c(ρs). (36)

The second inequality comes from Eq. (46) and the third from Eq. (29). So the success probability is bounded by

 Pd≤1−C(2)c, (37)

and the path distinguishability , the upper bound of , is given by

 DQ=1−C(2)c≤1−1d−1Cl1 (38)

This is a tighter upper bound than that presented in Bera et al. (2015).

Now we move on to mixed quanton case, in which the quanton system has some degree of interation with the environment. The mixed state density matrix is expressed as

 ρsd=∑xλx∑i,jχixχj∗x|ψi⟩⟨ψj|⊗|i⟩⟨j|D. (39)

After partial-tracing the detector, the reduced density matrix is given by

 ρs= ∑a,xpaλx∑i,jχixqiaχj∗xqj∗a|ψi⟩⟨ψj| +∑i(∑x|χix|2)|ϕi|2|ψi⟩⟨ψj|, (40)

This is a pure state decompostion of and a similar ineqaulity to Eq. (29) holds. Since appears with probability , the failure probability for the mixed state is bounded below as

 Qd ≥∑apa[dd−1∑i≠j∑xλx|χix|2∑yλy|χiy|2|qia|2|qja|2]12 ≥∑a,xpaλx[dd−1∑i≠j|χixqia|2|χjxqja|2]12 ≥C2c(ρs). (41)

So for mixed states is also given by

 DQ=1−C(2)c, (42)

which is more accurate result than the inquality for mixed states given in Bera et al. (2015).

Summarizing, we obtained the same form of the path distinguishability for both pure and mixed quanton systems with . On the other hand, -norm presents less tight bound for pure systems and inequality for mixed states.

C(k)c and the number of distinguishable slits

Now we think of a quanton state with , or equivalently. In this case we can express the detector states, without loss of generality, as

 |1⟩D=ϕ1|ϕ1⟩+|q1⟩,|2⟩D=ϕ2|ϕ2⟩+|q2⟩,⋯, |k+1⟩D=|ϕ(k+1)⟩,⋯,|d⟩D=|ϕd⟩, (43)

which we can see from Eq. (IV). With measurement operators and in (IV), we can not receive confusing information from to -th slit. So we can state that if for the quanton state then there exist -slits that we can identify unambiguously.

V Conclusions

In this study, we introduced a family of new coherence monotones, generalized coherence concurrence, which has a close interrelationship with the coherence number and the generalized entanglement concurrence. We then compared the coherence 2-concurrence with the coherence concurrence of Qi et al. (2016) and -norm coherence, which supports the assumption that the operational role of -norm in coherence quantitative theory is that of negativity in entanglement theory. An example for the applications of to quantum systems was path distinguishability problems. We obtained a sharper equation for the path distinguishability, and gave the relation between the coherence number of the system and the number of identifiable slits.

One of the remaining problems is to measure as the path distinguishability in the experimental interference pattern, as in Paul and Qureshi (2017). We also guess the other members of the coherence family have accurate relations with some quantities in multi-slit interference. Comparison of the coherence monotones obtained from the differential Chernoff bound Calsamiglia et al. (2008); Biswas et al. (2017) might present a clue to this problem. In a broader sense, the relation of four quantities— the coherence number, the Schmidt number, the generalized entanglement and coherence concurrence— would have an interesting structure to pursue further in both mathematical and practical directions. We expect the generalized coherence concurrence will be useful for any quantum phenomena in which not only the amount but also the order of coherence is crucial.

Acknowledgements.
The author is grateful to Prof. Jung-Hoon Chun for his advice during the research. This was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education(NRF-2016R1D1A1B04933413).

Appendix A Proof of (12)

For a pure state , we have

 ∑npnC(k)c(|ψn⟩)(|ψn⟩≡Kn|ψ⟩⟨ψ|K†nKn|ψ⟩12) =d∑n((d−k)!d! ×∑i1≠⋯≠ik∣∣(∑j1Ki1j1nψj1)2⋯(∑jkKikjknψjk)2∣∣)1k (44)

Then

 1d(d!(d−k!))1k∑npnC(k)c(|ψn⟩) =∑n(∑i1≠⋯≠ik∣∣∑j1≠⋯≠jk(Ki1j1nψj1)2⋯(Kikjknψjk)2∣∣)1k (45)

from the fact that is expressed as where is a re-ordered vector of the referential basis index Winter and Yang (2016). Using the relation

 Sk(λ1)1k+Sk(λ2)1k≤Sk(λ1+λ2)1k (46)

of elementary symmetric polynomials Gour (2005), we have the following inequality:

 ∑n(∑i1≠⋯≠ik∣∣∑j1≠⋯jk(Ki1j1nψj1)2⋯(Kikjknψjk)2∣∣)1k ≤∑n(∑j1≠⋯≠jk∣∣ψ2j1⋯ψ2jk∣∣∑i1≠⋯≠ik∣∣Ki1j1n⋯Kikjkn∣∣2)1k ≤(∑j1≠⋯≠jk∣∣ψ2j1⋯ψ2jk∣∣∑n1,i1∣∣Ki1j1n1∣∣2⋯∑nk,ik∣∣Kikjkn∣∣2)1k. (47)

Since

 ∑n,i|Kijn||Kijn|=∑n|(cjn)2⟨sj|(∑i|i⟩⟨i|)|sj⟩|=1, (48)

we finally have

 ∑npnC(k)c(|ψn⟩)≤C(k)c(|ψ⟩). (49)

Appendix B Proof of Theorem 5

1. case:
a.

The hermiticity of a quantum state is explicitly expressed as

 ρij=|ρij|eiθij,with|ρij|=|ρji|,θij=−θji. (50)

The condition (5) restricts the phases of ’s as

 θij+θjk+θki=0. (51)

Then the general solution of (50) and (51) is given with new real variables by

 θij=θi−θj (52)

(See, e.g., p40 of Siegel (1999)). So it is always possible to decompose with

 |~ψa⟩=∑i~ψia|i⟩=|~ψia|eiθi|i⟩ (53)

as

 ρjk=∑a~pa~ψja(~ψka)∗=(∑a~pa|~ψja~ψka|)ei(θj−θk). (54)

Then under this decomposition is

 Cl1(ρ)=2∑j

and we have

 Cc(ρ)≤∑a~paCc(|~ψa⟩)=Cl1(ρ). (56)

Since the inequality always holds, we have .

b.

Suppose that the decomposition of as gives the minimal value for . Then we have

 Cc(ρ)=2∑j

On the other hand, expressed with this decomposition is

 Cl1(ρ)=2∑j