Quantitative Wave-Particle Duality Relations from the Density Matrix Properties

Quantitative wave-particle duality relations from the density matrix properties

Marcos L. W. Basso,1 Diego S. S. Chrysosthemos,1 and Jonas Maziero1, ∗ 1Departamento de Física, Centro de Ciências Naturais e Exatas, Universidade Federal de Santa Maria, Avenida Roraima 1000, Santa Maria, RS, 97105-900, Brazil We derive upper bounds for Hilbert-Schmidt’s quantum coherence of general states of a d-level quantum system, a qudit, in terms of its incoherent uncertainty, with the latter quantified using the linear and von Neumann’s entropies of the corresponding closest incoherent state. Similar bounds are obtained for Wigner-Yanase’s coherence. The reported inequalities are also given as coherence-populations trade-off relations. As an application example of these inequalities, we derive quantitative wave-particle duality relations for multi-slit interferometry. Our framework leads to the identification of predictability measures complementary to Hilbert-Schmidt’s, Wigner-Yanase’s, and l1-norm quantum coherences. The quantifiers reported here for the wave and particle aspects of a quanton follow directly from the defining properties of the quantum density matrix (i.e., semi- positivity and unit trace), contrasting thus with most related results from the literature.

Keywords: Wave-particle duality; Quantum coherence; Predictability measures

I. INTRODUCTION

Quantum Information Science (QIS) is a rapidly developing interdisciplinary field harnessing and instigating some of the most advanced results in physics, information theory, computer science, mathematics, material science, engineering, and artificial intelligence [1–11]. Nowadays we know of several aspects of quantum systems that contrast them from the classical ones. Of particular interest in QIS is to investigate how these quantum features can be harnessed to devise more efficient protocols for information processing, transmission, storage, acquisition, and protection. A few examples of connections among quantum features and advantages in QIS protocols are as follows. Of special relevance for the whole of QIS, and in particular for the development of a quantum internet, is the use of quantum entanglement as a channel for quantum teleportation [12, 13]. As for one of the most advanced branches of QIS, its is known that quantum nonlocality and quantum steering are needed for device independent and semi-independent quantum communication, respectively [14, 15]. By its turn, the use of quantum squeezing and quantum discord was related with increased precision of measurements in quantum metrology [16, 17]. Besides, quantum contextuality and quantum coherence (QC) were connected with the speedup of some algorithms in quantum computation [18, 19]. Quantum coherence, a kind of quantum superposition [20], is directly related to the existence of incompatible observables in Quantum Mechanics; and is somewhat connected to most of the quantumnesses mentioned above. Therefore, it is a natural research program trying to understand QC from several perspectives. Recently, researchers have been developing a resource theory framework to quantify QC, the so called resource theory of coherence (see e.g. d Refs. [21–24] and references therein). In this resource theory, given an orthonormal reference basis {|βni}n=1, with d = dim H, for a system with state space H, the free states are incoherent mixtures of these base states:

d X ι = ιn|βnihβn|, (1) n=1

where {|ι i}d is a probability distribution. A geometrical way of deﬁning functions to quantify coherence is via the arXiv:1809.06354v5 [quant-ph] 21 Jul 2020 n n=1 minimum distance from ρ to incoherent states:

D CD(ρ) = min D(ρ, ι) =: D(ρ, ιρ ), (2) ι

D where ιρ is the closest incoherent state to ρ under the distance measure D. If CD does not increase under incoherent operations, which are those quantum operations mapping incoherent states to incoherent states, then its dubbed a coherence monotone. In Quantum Mechanics [25], the more general description of a system state is given by its density operator ρ = P m pm|ψmihψm|, where {pm} is a probability distribution and {|ψmi} are state vectors [26]. Because of this ensemble

∗Electronic address: [email protected] 2 interpretation, the density operator is required to be a positive (semi-deﬁnite) linear operator, besides having trace equal to one [27]. If we consider, for instance, two-level systems whose density operator represented in the orthonormal 2 basis {|βni}n=1 reads hβ1|ρ|β1i hβ1|ρ|β2i ρ1,1 ρ1,2 ρ = =: ∗ , (3) hβ2|ρ|β1i hβ2|ρ|β2i ρ1,2 1 − ρ1,1 these properties impose a well known restriction on the oﬀ-diagonal elements of ρ, its coherences, by the product of its diagonal elements, its populations:

2 ρ1,1(1 − ρ1,1) ≥ |ρ1,2| . (4)

The product of the populations of ρ can be seen as the incoherent uncertainty we have about measurements of an 2 observable with eigenvectors {|βni}n=1, since it is independent of the whole ensemble coherences. On the other hand, the presence of non-null off-diagonal elements of ρ implies that one or more members of the ensemble are a coherent 2 superposition of the base states {|βni}n=1. It is an interesting mathematical, physical, and possibly practical problem to derive quantum coherence–incoherent uncertainty trade-off relations regarding general-discrete quantum systems. In this article, we obtain such trade-offs for one-qudit (d-level) quantum systems. We start considering Hilbert-Schmidt’s coherence (HSC) function [28], that has a convenient algebraic structure but is known not to be a coherence monotone [29]. We also regard Wigner- Yanase’s coherence (WYC) [30], which is a coherence monotone. To quantify the incoherent uncertainty of a state ρ, √ we employ linear entropy and von Neumann’s entropy of its closest incoherent state or of the diagonal of ρ. It is worthwhile mentioning that the relation between “quantum” and “classical” uncertainties, and other complementarity relations, have been investigated in other contexts elsewhere [31–36]. Wave-particle duality has been at the center stage of conceptual discussions in Quantum Mechanics since its early days [37]. Recently, several authors have investigated about possible definitions of predictability and visibility quantifiers for d-slits interferometers. For a review of the literature, see e.g. Ref. [38]. Dürr’s [39] and Englert et al.’s [40] criteria can be taken as a standard for checking for the reliability of newly defined predictability measures P (ρ) and interference pattern visibility quantifiers W (ρ). For P , these required properties can be restated as follows: P1 P must be a continuous function of the diagonal elements of the density matrix. P2 P must be invariant under permutations of the paths’ indexes.

P3 If ρj,j = 1 for some j, then P must reach its maximum value.

d P4 If {ρj,j = 1/d}j=1, then P must reach its minimum value.

P5 If ρj,j > ρk,k for some (j, k), the value of P cannot be increased by setting ρj,j → ρj,j − and ρk,k → ρk,k + , for ∈ R+ and 1. P6 P must be a convex function, i.e., P (ωξ + (1 − ω)η) ≤ ωP (ξ) + (1 − ω)P (η), for 0 ≤ ω ≤ 1 and for ξ and η being valid density matrices. One can also write down a list of required properties for the functions to be used to quantify the wave aspect W of a quanton in a d-slit interferometer [39, 40]: W1 W must be a continuous function of the elements of the density matrix. W2 W must be invariant under permutations of the paths’ indexes.

W3 If ρj,j = 1 for some j, then W must reach its minimum value.

d W4 If ρ is a pure state and {ρj,j = 1/d}j=1, then W must reach its maximum value.

W5 W cannot be increased when decreasing |ρj,k| by an inﬁnitesimal amount, for j 6= k. W6 W must be a convex function, i.e., W (ωξ + (1 − ω)η) ≤ ωW (ξ) + (1 − ω)W (η), for 0 ≤ ω ≤ 1 and ξ and η are well deﬁned density matrices. 3

There are convincing arguments indicating quantum coherence as a good measure for the wave aspect of a quanton [41]. And we’ll show that our coherence-incoherent uncertainty trade-off relations can be applied to obtain quantitative wave-particle duality relations, with the associated predictability and visibility measures satisfying the criteria listed above. The framework devised in this article shows that the semi-positiveness and unit trace of the quantum density matrix leads to the identification of predictability measures complementary to HSC, to WYC, and to l1-norm quantum coherence (L1C). We organized the remainder of this article in the following manner. In Sec.II, we present the Gell-Mann matrix basis (GMB), defined using any vector basis of Cd, and discuss the representation of general and diagonal matrices in the GMB. We obtain trade-off relations between quantum coherence and incoherent uncertainty measuring the first with Hilbert-Schmidt coherence in Sec.IIIA. We obtain analogous relations using Wigner-Yanase’s coherence in Sec.IIIB. We show how to write these inequalities as coherence-populations trade-off relations in Sec.IV. In Sec. VA we report quantitative visibility-predictability duality relations identified directly from our coherence-incoherent uncertainty inequalities. In Secs.VB andVC we apply the positivity and unit trace of the density matrix to identify predictability measures complementary to WYC and L1C, respectively. We give our conclusions in Sec.VI.

dxd II. GELL-MANN BASIS FOR C

d d Let {|βmi}m=1 be any given vector basis for C . Using this basis, we can deﬁne the generalized Gell-Mann’s matrices as [42]:

j = 1, ··· , d − 1 and 1 ≤ k < l ≤ d. (8)

d One can easily see that these matrices are Hermitian and traceless. Besides, if we use Γ0 for the d × d identity matrix, it is not diﬃcult to verify that under the Hilbert-Schmidt’s inner product,

† hA|Bihs := Tr(A B), (9) with A, B ∈ Cd×d, the set ( ) Γd Γd Γτ √0 , √j , √k,l , (10) d 2 2 with τ = s, a, forms an orthonormal basis for Cd×d [42]. So, any matrix X ∈ Cd×d can be decomposed in this basis, called hereafter of Gell-Mann’s basis (GMB), as follows:

Tr(X) 1 X 1 X X = Γd + hΓd|XiΓd + hΓτ |XiΓτ . (11) d 0 2 j j 2 k,l k,l j k,l,τ

d×d We observe that the most general decomposition in GMB of a matrix Xd ∈ C which is diagonal in the basis d {|βmi}m=1 shall be given by:

Tr(Xd) 1 X X = Γd + hΓd|X iΓd, (12) d d 0 2 j d j j i.e., only the diagonal elements of the GMB can have non-null components in this decomposition. 4

III. TRADE-OFF RELATIONS BETWEEN QUANTUM COHERENCE AND INCOHERENT UNCERTAINTY

A. Upper bound for Hilbert-Schmidt’s coherence

The Hilbert-Schmidt’s coherence (HSC) of a quantum state ρ is deﬁned as [28]

2 Chs(ρ) := min ||ρ − ι||hs, (13) ι with the Hilbert-Schmidt’s norm of a matrix A ∈ Cd×d being deﬁned as p ||A||hs := hA|Aihs, (14) and here the minimization is taken over the incoherent states of Eq. (1). For general one-qudit states, using the decompositions in GMB:

1 1 X 1 X ρ = Γd + hΓd|ρiΓd + hΓτ |ρiΓτ , (15) d 0 2 j j 2 k,l k,l j k,l,τ 1 1 X ι = Γd + hΓd|ιiΓd, (16) d 0 2 j j j the analytical formulas for the HSC and for the associated closest incoherent state were obtained in Ref. [28] and read, respectively:

1 X C (ρ) = hΓτ |ρi2, (17) hs 2 k,l k,l,τ 1 1 X ιhs = Γd + hΓd|ρiΓd. (18) ρ d 0 2 j j j

The main tool we use to obtain some of the results reported in this article is a condition for matrix positivity. The eigenvalues of a matrix A ∈ Cd×d, let us call them a, can be obtained from [43]: d 0 = det A − aΓ0 (19) X = sgnd(j1, j2, ··· , jd)(A1,j1 − aδ1,j1 )(A2,j2 − aδ2,j2 ) ··· (Ad,jd − aδd,jd ) (20)

(j1,j2,··· ,jd) d d d−1 d−1 d−2 d−2 2 = (−1) cda + (−1) cd−1a + (−1) cd−2a + ··· + c2a − c1a + c0. (21)

By Descartes rule of signs (see e.g. Ref. [44] and references therein), we see that for A to be a positive matrix, we d have to have non-negativity for all the coeﬃcients {cm ≥ 0}m=0. In this article, we shall look at the positivity of: X X cd−2 = sgnd(j1, j2, 3 ··· , d)A1,j1 A2,j2 + ··· + sgnd(j1, 2, ··· , d − 1, jd)A1,j1 Ad,jd

(j1,j2) (j1,jd) X X + sgnd(1, j2, j3, 4, ··· , d)A2,j2 A3,j3 + ··· + sgnd(1, j2, 3, ··· , d − 1, jd)A2,j2 Ad,jd

(j2,j3) (j2,jd) X + ··· + sgnd(1, 2, ··· d − 2, jd−1, jd)Ad−1,jd−1 Ad,jd (22)

(jd−1,jd) d−1 d X X = (Am,mAn,n − Am,nAn,m) (23) m=1 n=m+1 1 = (Tr(A))2 − Tr A2 ≥ 0. (24) 2 Using the orthonormality of GMB, i.e., the inner product between diﬀerent elements of the GMB is zero and d d d d τ τ hΓ0|Γ0i = d and hΓj |Γj i = hΓk,l|Γk,li = 2, the positivity condition for the coeﬃcient in Eq. (24) applied to the 5 density matrix of Eq. (15), as Tr(ρ) = 1, can be rewritten as

0 ≤ 1 − Tr(ρ2) (25) 1 1 X 1 X = 1 − − hΓd|ρi2 − hΓτ |ρi2. (26) d 2 j 2 k,l j k,l,τ

Now, if we use the formula for the HSC in Eq. (17), this inequality can be cast as a restriction to the HSC:

d − 1 1 X C (ρ) ≤ − hΓd|ρi2. (27) hs d 2 j j

If we utilize again the orthonormality of the GMB, the right hand side of this inequality is easily seen to be the incoherent uncertainty of the state ρ measured using the linear entropy of its closest incoherent state (Eq. (18)), i.e.,

2 d − 1 1 X S (ιhs) = 1 − Tr ιhs = − hΓd|ρi2. (28) l ρ ρ d 2 j j

Now, using − ln x ≥ 1 − x [45], we can get an upper bound for the linear entropy in terms of von Neumann’s entropy as follows:

Svn(x) := Tr(x(− ln x)) (29) 2 ≥ Tr(x(1 − x)) = Tr(x) − Tr(x ) = Sl(x) + Tr(x) − 1. (30)

hs Gathering the results above, as Tr(ιρ ) = 1, we have obtained the following quantum coherence–incoherent uncertainty trade-oﬀ relations:

hs hs Chs(ρ) ≤ Sl(ιρ ) ≤ Svn(ιρ ), (31) which are valid for any one-qudit state. The “veriﬁcation” of these entropic inequalities using random states is presented in Fig.1. We observe that the upper bound given by linear entropy is tight for qubits ( d = 2). However, as the dimension increases, and typicality is approached [46], the upper bounds get less and less tight.

B. Upper bound for Wigner-Yanase’s coherence

In this subsection we shall deal with Wigner-Yanase’s coherence (WYC) [30]:

d d X 1 X √ C (ρ) := I (ρ, |β ihβ |) := − Tr(([ ρ, |β ihβ |])2) (32) wy wy j j 2 j j j=1 j=1 d d ! 1 X X √ √ = hβ |ρ|β i + |hβ | ρ|β i|2 − 2hβ | ρ|β i2 , (33) 2 j j j k j j j=1 k=1 with [·, ·] being the commutator. Identifying the following relation between a diagonal element of ρ and the elements √ in the corresponding row of ρ:

d d X √ √ X √ 2 hβj|ρ|βji = hβj| ρ|βkihβk| ρ|βji = |hβj| ρ|βki| , (34) k=1 k=1 we can write

d d d X X √ 2 X √ 2 Cwy(ρ) = |hβj| ρ|βki| − hβj| ρ|βji (35) j=1 k=1 j=1 X √ 2 = |hβj| ρ|βki| . (36) j6=k 6

0.5 0.7

0.4 0.6 d = 2 d = 4 0.5 0.3 0.4

0.2 0.3

0.2 0.1 0.1

0.0 0.0

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.7 1.4

0.6 1.2 d = 2 d = 4 0.5 1.0

0.4 0.8

0.3 0.6

0.2 0.4

0.1 0.2

0.0 0.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 1: (color online) “Veriﬁcation” of the Hilbert-Schmidt quantum coherence–incoherent uncertainty trade-oﬀ relations of Eq. (31) for one thousand random density matrices generated for each value of the system dimension d. The random states hs were created using the standard method described in Refs. [47, 48]. The y axis is for Chs(ρ) and the x axis is for Sl(ιρ ) or hs hs hs hs Svn(ιρ ), with ιρ given in Eq. (18). The black lines stand for Chs(ρ) = Sl(ιρ ) and for Chs(ρ) = Svn(ιρ ).

√ √ √ √ √ √ Now, by using hΓs | ρi = 2 Re(( ρ) ) and hΓa | ρi = −2 Im(( ρ) ), where Re(( ρ) ) and Im(( ρ) ) are the k,l √ k,l k,l k,l k,l k,l real and imaginary part of ( ρ)k,l respectively, we get 1 X √ C (ρ) = hΓτ | ρi2. (37) wy 2 k,l k,l,τ

Pd d A quantum state, with spectral decomposition ρ = m=1 rm|rmihrm|, is a positive matrix [45], i.e., {rm ≥ 0}m=1. √ Pd √ √ So, ρ = m=1 rm|rmihrm| is also a positive matrix. If we apply the positivity condition of Eq. (24) to ρ decomposed in GMB as √ √ Tr( ρ) 1 X √ 1 X √ ρ = Γd + hΓd| ρiΓ + hΓτ | ρiΓτ , (38) d 0 2 j j 2 k,l k,l j k,l,τ the following inequality is obtained: √ √ 0 ≤ (Tr( ρ))2 − Tr(( ρ)2) (39) √ 1 1 X √ 1 X √ = (Tr( ρ))2 1 − − hΓd| ρi2 − hΓτ | ρi2. (40) d 2 j 2 k,l j k,l,τ 7

0.5 0.7 d = 2 d = 4 0.4 0.6

0.5 0.3 0.4

0.2 0.3

0.2 0.1 0.1

0.0 0.0

0.0 0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5 2.0 2.5

0.5 d = 2 0.7 d = 4

0.4 0.6

0.5 0.3 0.4

0.2 0.3

0.2 0.1 0.1

0.0 0.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Figure 2: (color online) “Veriﬁcation” of the upper bounds for Wigner-Yanase quantum coherence in Eqs. (42) and (43) for ﬁve thousand random density matrices generated for each value of the system dimension d. The method used to create the random states is the same as the one mentioned in Fig.1. The black lines stand for Cwy(ρ) = Υ and for Cwy(ρ) = Ω.

Using WYC in Eq. (37) and the linear and von Neumann’s entropies of √ √ Tr( ρ) 1 X √ ρ := Γd + hΓd| ρiΓ , (41) diag d 0 2 j j j we obtain the following upper bounds for WYC:

√ √ 2 Cwy(ρ) ≤ Sl( ρdiag) + (Tr( ρdiag)) − 1 =: Υ (42) √ √ √ ≤ Svn ρdiag + Tr( ρdiag) Tr( ρdiag) − 1 =: Ω . (43) √ In the particular case of pure states, ρ = |ψihψ| ∴ ρ = ρ, we have Cwy(ρ) = Chs(ρ). Hence, in this case, the inequalities above are equivalent to the ones obtained for Hilbert-Schmidt’s coherence in Eq. (31), i.e.,

hs hs Cwy(|ψihψ|) ≤ Sl ιρ ≤ Svn ιρ . (44) The upper bounds for WYC were also “veriﬁed” using random states, as shown in Fig.2. Here the restrictiveness of those upper bounds is also seen to diminish with the increase of the system dimension. 8

IV. COHERENCE-POPULATIONS TRADE-OFF RELATIONS

In this section, we start rewriting the upper bound for Hilbert-Schmidt’s coherence given in Eq. (27) by expressing d the components of the so called Bloch’s vector corresponding to the diagonal elements of Gell-Mann’s basis, hΓj |ρi with j = 1, ··· , d − 1, in terms of the density matrix populations, ρm,m = hβm|ρ|βmi with m = 1, ··· , d. For that purpose, after some algebraic manipulations, one can infer that for m = 2, ··· , d − 1:

r d−1 d d−1 1 m − 1 X hΓj |ρi X ρ = − hΓd |ρi + δ . (45) m,m d 2m m−1 p n,j j=1 2j(j + 1) n=m For m = d and m = 1 we can use this same expression for the populations, but without the last and second terms, respectively. By inverting the expressions in Eq. (45) iteratively, we obtain the general expression we need to rewrite the trade-oﬀs in Eq. (31) in terms of ρ’s populations:

s j ! 2 X hΓd |ρi = 1 − (d − n + 1)δj,n ρ . (46) d−j (d − j + 1)(d − j) d−n+1,d−n+1 n=1 d As examples, let us start considering qubit and qutrit systems. For d = 2, hΓ1|ρi = 1 − 2ρ2,2 and, from Eq. 2 d (27), we get Chs(ρ)√ = 2|ρ1,2| ≤ 2ρ1,1ρ2,2, which is equivalent to Eq. (4). For d = 3, hΓ1|ρi = 1 − ρ3,3 − 2ρ2,2, d hΓ2|ρi = (1 − 3ρ3,3)/ 3 and

Chs(ρ) ≤ 2(ρ1,1ρ2,2 + ρ1,1ρ3,3 + ρ2,2ρ3,3). (47) As this same pattern appears also for d = 4 and for d = 5, one would conjecture that for any one-qudit state the following inequality will be satisﬁed:

d−1 d X X Chs(ρ) ≤ 2 ρm,mρn,n. (48) m=1 n=m+1 We could not give limitations for the Wigner-Yanase’s coherence of a state ρ directly in terms of the density matrix populations. Notwithstanding, relations identical to the ones above shall follow for this quantum coherence measure √ if we replace ρ by ρ in Eqs. (45) and (46) and on the right hand side of Eq. (48), i.e.,

d−1 d X X √ √ Cwy(ρ) ≤ 2 ( ρ)m,m( ρ)n,n. (49) m=1 n=m+1 A simpler and general proof of this kind of inequality can be given as follows. It is known from Matrix Analysis [49] that all principal sub-matrices of a positive semi-deﬁnite matrix are also positive semi-deﬁnite. In particular, Aj,j Aj,k A ≥ O ⇒ ≥ O ∀j, k. It follows then that Ak,j Ak,k

X 2 X 2 X 2 X X 2 X |Aj,k| = |Aj,k| − Aj,j ≤ Aj,jAk,k − Aj,j = Aj,jAk,k, (50) j6=k j,k j j,k j j6=k from which we can obtain the inequalities in Eqs. (48) and (49). In the next section, we will use one of these two expressions for giving closed formulas for measures of path information and interference pattern visibility.

V. QUANTITATIVE COMPLEMENTARITY RELATIONS FOR d-SLITS INTERFEROMETERS

A. Complementarity relations with Hilbert-Schmidt’s coherence

Here we use HSC written in terms of the density matrix elements:

d d 2 X 2 X 2 Chs(ρ) := min ||ρ − ι||hs = min |(ρ − ι)j,k| = min |ρj,k − ιjδj,k| (51) ι ι ι j,k=1 j,k=1 X 2 = |ρj,k| . (52) j6=k 9

In the sequence we show that W := Chs satisﬁes the properties listed in the introduction. One can easily see from the expression for Chs that it has the properties W 1 and W 2, i.e., Chs is continuous and invariant under paths’ indexes exchanges. Besides:

2 W3 If ρj,j = 1 for any j, then all the other populations and all the coherences are null (by the restrictions |ρj,k| ≤ ρj,jρk,k∀j, k). Therefore Chs = 0, which is the required minimum, since we have Chs ≥ 0. ρj,j ρj,k W4 If ρj,j = 1/d ∀j, we shall have from ≥ O that |ρj,k| ≤ 1/d ∀j 6= k. So, for the maximum value for ρk,j ρk,k 2 the density matrix coherences ρj,k = 1/d ∀j 6= k, we see that T r(ρ ) = 1 and that Chs reaches its maximum value, (d − 1)/d.

2 2 W5 If we set ρj,k → ρ˜j,k = ρj,k − , with Re(ρj,k) Re() ≥ 0 and Im(ρj,k) Im() ≥ 0, then |ρ˜j,k| ≈ |ρj,k| − 2 2(Re(ρj,k) Re() + Im(ρj,k) Im()) ≤ |ρj,k| , which implies that C˜hs ≤ Chs.

W6 For 0 ≤ ω ≤ 1 and ξ and η valid density operators, we verify that Chs is convex as follows:

Chs(ωξ + (1 − ω)η) − ωChs(ξ) − (1 − ω)Chs(η) X 2 X 2 X 2 = |(ωξ + (1 − ω)η)j,k| − ω |ξj,k| − (1 − ω) |ηj,k| (53) j6=k j6=k j6=k X 2 2 = ω(ω − 1) (Re(ξj,k) − Re(ηj,k)) + (Im(ξj,k) − Im(ηj,k)) (54) j6=k ≤ 0. (55)

Let us deﬁne

max Sτ := max Sτ (ρ), with τ = l, vn. (56) ρ

For d-dimensional density matrices, ρ = Id/d gives the maximum for the entropies:

max max Sl = (d − 1)/d and Svn = ln d. (57) Now, we can rewrite the inequalities in Eqs. (31) as

max τ max Chs(ρ) + Sτ − Sτ (ιρ) ≤ Sτ . (58) By deﬁning the Hilbert-Schmidt’s predictability measures

d−1 d d − 1 X X P l (ρ) := Smax − S (ιhs) = − 2 ρ ρ , (59) hs l l ρ d m,m n,n m=1 n=m+1 d vn max hs X Phs (ρ) := Svn − Svn(ιρ ) = ln d + ρn,n ln ρn,n, (60) j=1 we obtain the coherence-predictability complementarity relations:

τ max Chs(ρ) + Phs(ρ) ≤ Sτ . (61)

τ One could put the upper bounds for these inequalities in the usual form, equal to one, by normalizing Chs and Phs. In the sequence we shall verify that the predictability measures introduced here satisfy the criteria listed in Sec.I:

hs d τ P1 As Sτ (ιρ ) is a continuous function of {ρj,j}j=1, so is Phs(ρ).

l l P2 It is straightforward to see in the right hand side of Eqs. (59) and (60) that Phs and Phs are invariant under exchange of paths’ labels.

2 P 2 P3 If ρj,j = 1 for some j then ρk,k = 0 ∀k 6= j and Sl = 1 − ρj,j − k6=j ρk,k = 0 and Svn = −ρj,j ln ρj,j − P τ max k6=j ρk,k ln ρk,k = 0. So Phs := Sτ . 10

d Pd 2 max Pd max τ P4 If {ρj,j = 1/d}j=1 then Sl = 1 − j=1(1/d ) = Sl and Svn = − j=1(1/d) ln(1/d) = Svn . So Phs = 0. τ P5 Once Phs is invariant under exchange ρj,j ↔ ρk,k∀j, k we can, without loss of generality, consider ρ1,1 > ρ2,2, ρ1,1 → ρ1,1 − , and ρ2,2 → ρ2,2 + for > 0 and 1. Thus

d ˜l max 2 2 X 2 Phs = Sl − 1 + (ρ1,1 − ) + (ρ2,2 + ) + ρj,j (62) j=3 d max X 2 2 = Sl − (1 − ρj,j) − 2(ρ1,1 − ρ2,2) + O( ) (63) j=1 l ≤ Phs (64) and

d ˜vn max X Phs = Svn + (ρ1,1 − ) ln(ρ1,1 − ) + (ρ2,2 + ) ln(ρ2,2 + ) + ρj,j ln ρj,j (65) j=3 d max X 2 = Svn + ρj,j ln ρj,j − (ln ρ1,1 − ln ρ2,2) + (ρ1,1 − )(−/ρ1,1) + (ρ2,2 + )(/ρ2,2) + O( ) (66) j=1 vn 2 = Phs − (ln ρ1,1 − ln ρ2,2) + O( ) (67) vn ≤ Phs . (68) Above we used ln(1 ± x) ≈ ±x for x > 0 and x 1.

l P6 The convexity of the predictability measure Phs is veriﬁed as follows:

l l l Phs(ωξ + (1 − ω)η) − ωPhs(ξ) − (1 − ω)Phs(η) (69) d − 1 X = − (ωξ + (1 − ω)η )(ωξ + (1 − ω)η ) (70) d j,j j,j k,k k,k j6=k     d − 1 X d − 1 X − ω − ξ ξ − (1 − ω) − η η (71)  d j,j k,k  d j,j k,k j6=k j6=k X X 2 = ω(1 − ω) (ξj,j − ηj,j)(ξk,k − ηk,k) = ω(ω − 1) (ξj,j − ηj,j) (72) j6=k j6=k ≤ 0. (73)

vn For Hilbert-Schmidt’s predictability quantiﬁer Phs , convexity follows from the concavity of von Neumann’s entropy [50]. With this we have shown that although Hilbert-Schmidt distance does not provide a coherence monotone, it can be used, in conjunction with the positivity of the density matrix, to provide bona ﬁde measures for the particle and wave aspects of a quanton in d-slits interferometry. In order to exemplify the application of our complementarity relations, we shall regard a generalized Werner’s state of a ququart (d = 4)[51]:

ρ = (1 − w)I4 + w|ψihψ|, (74) w,a 4 √ √ with |ψi = a|β0i + 1 − a|β1i. The Hilbert-Schmidt coherence function, the linear- and von Neumann-Hilbert- Schmidt predictability measures, their sum, and the associated upper bounds are shown graphically in Fig.3 as a function of a for some values of w. l vn This ﬁgure shows that Phs and Phs reach the respective upper bounds for ρw=1,a=1 = |β0ihβ0| and for ρw=1,a=0 = l |β1ihβ1|. Besides, the equality in the complementarity relation Chs + Phs ≤ (d − 1)/d is obtained for all ρw=1,a while vn the inequality Chs + Phs ≤ ln d is saturated only for ρw=1,a=1 and ρw=1,a=0. 11

0.7 0.7 0.7 w = 0 0.6 0.6 w = 0.25 0.6 w = 0.5 0.5 0.5 w = 0.75 0.5 l w = 1 hs 0.4 0.4 0.4 l hs hs

Upper bound P + P C hs

0.3 0.3 C 0.3

0.2 0.2 0.2

0.1 0.1 0.1

0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 a a a 1.4 1.4 1.4 1.2 1.2 1.2

1.0 1.0 1.0 vn 0.8 0.8 hs 0.8 hs vn hs P + C P

0.6 0.6 hs 0.6 C

0.4 0.4 0.4

0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 a a a

Figure 3: (color online) Hilbert-Schmidt coherence function Chs (upper-left and lower-left plots), linear-Hilbert-Schmidt pre- l vn l dictability Phs (upper-center plot), von Neumann-Hilbert-Schmidt predictability Phs (lower-center plot), and the sums Chs+Phs vn (upper-right plot) and Chs + Phs (lower-right plot) of the modiﬁed Werner’s state of Eq. (74) as a function of a for some values of w.

B. Complementarity relations with Wigner-Yanase’s coherence

√ X √ 2 X 2 ( ρ)j,j = rk|Uk,j| ≥ rk|Uk,j| = ρj,j. (75) k k So

d X √ 2 Cwy(ρ) = 1 − hβj| ρ|βji (76) j=1 d X 2 ≤ 1 − hβj|ρ|βji = Sl(ρdiag) ≤ Svn(ρdiag), (77) j=1 with ρdiag = diag(ρ1,1, ρ2,2, ··· , ρd,d) and the linear and von-Neumann entropies are deﬁned in Sec.IIIA. Thus we identify the complementarity relations:

τ max Cwy(ρ) + Phs(ρ) ≤ Sτ , (78) with τ = l, vn and with the predictability measures complementary to Wigner-Yanase’s coherence being the same hs max as for Hilbert-Schmidt’s coherence, since ρdiag = ιρ . These functions appear, together with Sτ , in Sec.VA. We observe that other candidate predictability measures Pwy could be defined using the inequalities in Eqs. (42) and (43). But we do not include such functions here because for them we succeed in verifying axioms P1-P6 only in terms √ of changes in ρdiag, which still lacks physical significance. In the sequence we verify that Wigner-Yanase’s coherence satisfies the properties listed in the Introduction for a measure of the wave character of a quanton: 12

1.4 0.7 0.7 w = 0 1.2 w = 0.25 0.6 0.6 w = 0.5 1.0 0.5 0.5 w = 0.75 l hs vn w = 1 hs 0.8 0.4 0.4 wy

P + Upper bound P + C wy wy 0.6 C 0.3 0.3 C

0.2 0.2 0.4

0.1 0.1 0.2

0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 a a a

l vn Figure 4: (color online) Wigner-Yanase’s coherence (left plot), Cwy + Phs (center plot), and Cwy + Phs (right plot) for the modiﬁed Werner’s state of Eq. (74) as a function of a for some values of w.

√ d W1 Continuity if Cwy follows from the continuity of {( ρ)j,j}j=1.

Pd √ 2 W2 Once j=1(( ρ)j,j) does not change under |βji ↔ |βki, Cwy is invariant under paths’ indexes exchanges. √ W3 If ρ = 1 for some j, then by Tr(ρ) = 1 we have to have ρ = 0 ∀k 6= j, which implies that ρ = |β ihβ | = ρ. j,j √ √ k,k 1 1 Therefore ( ρ)j,j = 1 and ( ρ)k,k = 0 ∀k 6= j. So Cwy = 1 − 1 = 0, which is its minimum value.

√ d W4 If ρ is a pure state then ρ = ρ. Therefore, if {ρj,j = 1/d})j=1 then Cwy = (d − 1)/d, which is its maximum value. W5 We can diminish |ρ | inﬁnitesimally by taking ρ → (1 − )ρ , with ∈ and 1. By noticing that √ √j,k j,k j,k √R+ √ ρ = P ( ρ) ( ρ) , we see that this change leads to ρ → P ((1 − )( ρ) )( ρ) , which is equivalent j,k l j,l l,k √ j,k l j,l l,k to multiplying the j-th row of ρ by 1 − . Thus it follows that

X √ 2 X X √ 2 C˜wy = |(1 − )( ρ)j,l| + |( ρ)k,l| (79) l6=j k6=j l6=k X √ 2 X X √ 2 ≈ (1 − 2) |( ρ)j,l| + |( ρ)k,l| (80) l6=j k6=j l6=k X √ 2 = Cwy(ρ) − 2 |( ρ)j,l| (81) l6=j

≤ Cwy(ρ). (82)

W6 Convexity of Cwy follows from the convexity of Wigner-Yanase’s skew information Iwy [52]. In Fig.4 we exemplify the application of the complementarity relations of Eq. (78) for the state of Eq. (74). The general aspects of the obtained results are similar to those described above for the Hilbert-Schmidt coherence.

C. Complementarity relations with l1-norm coherence

In this subsection we derive quantitative complementarity relations for d- dimensional systems by applying l1-norm coherence [29]:

d d X X Cl (ρ) = min ||ρ − ι||l = min |(ρ − ι)j,k| = min |ρj,k − ιjδj,k| (83) 1 ι 1 ι ι j,k=1 j,k=1 X = |ρj,k|. (84) j6=k

2 Here we will use again ρ ≥ O ⇒ |ρj,k| ≤ ρj,jρk,k ∀j, k to obtain X √ Cl1 (ρ) ≤ ρj,jρk,k ≤ d − 1, (85) j6=k 13 from which follows the complementarity relation:

Cl1 (ρ) + Pl1 (ρ) ≤ d − 1, (86) with the l1-norm predictability deﬁned as: X √ Pl1 (ρ) := d − 1 − ρj,jρk,k (87) j6=k X √ = d − 1 − 2 ρj,jρk,k. (88) j

l 2 It worthwhile mentioning that for d = 2 we can write Phs(ρ) = (ρ1,1 − ρ2,2) , which is similar to predictability 2 measure used in [39, 40]. But we notice that P = (f(ρ1,1) − f(ρ2,2)) is also a bona-ﬁde measure of√ predictability, with f being any monotonic increasing function of the probabilities ρj,j, j = 1, 2. Hence, for f(x) = x the l1-norm √ √ 2 predictability is a generalization of two dimensional function P = ( ρ1,1 − ρ2,2) .

Next we verify that Cl1 satisfy the axioms for a measure of the wave aspect of a quanton: W1 Continuity follows from the continuity of the absolute value function.

W2 Invariance under paths’ indexes exchange follows directly from the analytical expression for Cl1 . 2 W3 If ρj,j = 1 for some j, then ρk,k = 0 ∀k 6= j and, by |ρj,k| ≤ ρj,jρk,k, ρj,k = 0 ∀j 6= k. Therefore Cl1 = 0. d 2 W4 If {ρj,j = 1/d}j=1, then the same inequality used to prove W3 leads to |ρj,k| ≤ 1/d. The equality gives Tr(ρ ) = 1

and Cl1 = d − 1, which is its maximum value.

W5 For ∈ C, || 1, Re(ρj,k) Re() > 0, Im(ρj,k) Im() > 0, we set ρ˜j,k = ρj,k − . Then q ∗ ∗ |ρ˜j,k| = (ρj,k − )(ρj,k − ) (89) q 2 ∗ ≈ |ρj,k| − 2 Re(ρj,k ) (90) ∗ 2 ≈ |ρj,k| 1 − Re(ρj,k )/|ρj,k| , (91)

˜ ∗ which gives Cl1 = Cl1 − Re(ρj,k )/|ρj,k| ≤ Cl1 .

W6 For 0 ≤ ω ≤ 1 and ξ and η valid density operators, we verify convexity of Cl1 as follows: X Cl1 (ωξ + (1 − ω)η) = |(ωξ + (1 − ω)η)j,k| (92) j6=k X = |ωξj,k + (1 − ω)ηj,k| (93) j6=k X ≤ (|ωξj,k| + |(1 − ω)ηj,k|) (94) j6=k

= ωCl1 (ξ) + (1 − ω)Cl1 (η). (95)

At last we verify that Pl1 satisﬁes the axioms listed in Sec.I for a measure of predictability:

P1 Continuity of Pl1 follows from the continuity of the square root. P √ P2 The sum j6=k ρj,jρk,k warrants invariance under paths’ indexes exchanges.

P3 If ρj,j = 1 for some j, then ρk,k = 0 ∀k 6= j. Thus Pl1 = d − 1 − 0, which is its maximum value. d P √ P4 If {ρj,j = 1/d}j=1, then j6=k ρj,jρk,k = d − 1, and Pl1 = 0, which is its minimum value. 14

3.0 3.0 3.0

2.5 w = 0 2.5 2.5 w = 0.25 w = 0.5 2.0 2.0 2.0

w = 0.75 1 l 1 1 l w = 1 l

P + 1.5

1.5 P 1.5 C 1 Upper bound l C 1.0 1.0 1.0

0.5 0.5 0.5

0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 a a a

Figure 5: (color online) l1-norm coherence (left plot), l1-norm predictability (center plot), and Cl1 + Pl1 (right plot) for the modiﬁed Werner’s state of Eq. (74) as a function of a for some values of w.

P5 In view of P2, we set ρ1,1 > ρ2,2, ρ1,1 → ρ1,1 − , and ρ2,2 → ρ2,2 + with ∈ R+ and 1. So

d ˜ p p p X √ Pl1 = d − 1 − 2 ρ1,1 − ρ2,2 + − 2 ρ1,1 − ρk,k k=3 d d−1 d p X √ X X √ − 2 ρ2,2 + ρk,k − 2 ρj,jρk,k (96) k=3 j=3 k=j+1 d √ √ X √ ≈ d − 1 − 2 ρ1,1ρ2,2 (1 − /2ρ1,1) (1 + /2ρ2,2) − 2 ρ1,1 (1 − /2ρ1,1) ρk,k k=3 d d−1 d √ X √ X X √ − 2 ρ2,2 (1 + /2ρ2,2) ρk,k − 2 ρj,jρk,k (97) k=3 j=3 k=j+1 r r d ρ1,1 ρ2,2 1 1 X √ ≈ Pl1 − − − √ − √ ρk,k (98) ρ2,2 ρ1,1 ρ2,2 ρ1,1 k=3

< Pl1 . (99)

∂ P6 We will prove convexity through the positivity of the Hessian matrix (Hn,m) = (∂n∂mf), with ∂n := and ∂xn P √ P ∗ d f = α − j6=k xjxk where α is a constant, i.e., we will verify that hy|H|yi = n,m ynymHn,m ≥ 0 ∀|yi ∈ C . Once the diagonal and oﬀ-diagonal elements of H are given, respectively, by: ∂ ∂ f = 1 P px /x3 and √ m m 2 j6=m j m ∂n∂mf = −1/2 xnxm, we shall have: r X 2 1 X xn X ∗ −1 hy|H|yi = |ym| 3 + ynym √ (100) 2 x 2 xnxm m n6=m m n6=m 2 1/2 2 1/2 ∗ ∗ ! 1 X |ym| xn |yn| xm y ym + y yn = + − n m (101) 3/2 3/2 √ 4 xnxm m6=n xm xn 2 2 ∗ ∗ 1 X 1/2 1/2 |ym| |yn| ynym + ymyn = xn xm 2 + 2 − (102) 4 x x xnxm m6=n m n 2 1 X 1/2 1/2 ym yn = xn xm − ≥ 0. (103) 4 xm xn m6=n

In Fig.5 we instantiate the application of the inequality of Eq. (86) for the quantum state in Eq. (74). We observe that although Cl1 does not reach the upper bound and Pl1 does reach this value only for ρw=1,a=0 and for ρw=1,a=1, the coherence-predictability relation is saturated for all ρw=1,a. 15

VI. CONCLUSIONS

Quantum coherence (QC) is an important resource in Quantum Information Science [19, 53–68]. In this article we proved upper bounds for Hilbert-Schmidt’s QC of a general one-qudit state ρ by its associated incoherent uncertainty measured using the linear entropy and von Neumann’s entropy of the closest incoherent mixture. Similar bounds were √ obtained for Wigner-Yanase QC in terms of entropies of the diagonal part of ρ. We also wrote these inequalities with the upper bound given in terms of the populations of the density matrix or of its square root. We have presented numerical examples of the proven inequalities using random quantum states. These examples showed that the given upper bounds are tight for qubits and that they have their restrictiveness progressively weakened as the system dimension grows. So, in the future it would be interesting to investigate if the positivity of coefficients in Eq. (21) others than the one considered in this article (see e.g. [69, 70]) may be used to obtain similar but more generally stronger upper bounds for quantum coherence. We showed that our inequalities can be used to derive quantitative wave-particle duality relations. In our formalism these relations appear naturally, with the predictability measures defined by the inequalities themselves, which, by its turn, follows directly from the positivity of the density matrix (akin to what was implicitly done for 2-slits interferometers in Ref. [71]). Finding another applications for the reported inequalities is another natural continuation for the present research. One possibility for investigation is regarding coherence generation via quantum operations with restrictions on the possible density matrix populations changes [72, 73]. Other promising candidate area for application of quantum coherence–incoherent uncertainty trade-off relations reported here is quantum thermodynamics [74–77]. In this scenario, if the reference basis is the energy basis, restrictions on populations changes shall be related to restrictions on energy changes. And these restrictions may be useful for analyzing thermodynamical processes that consume or create quantum coherence.

Acknowledgments

This work was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), process 88882.427924/2019-01, by the Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul (FAPERGS), and by the Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ), process 465469/2014-0.

[1] L. Fallani and A. Kastberg, Cold atoms: A ﬁeld enabled by light, EPL 110, 53001 (2015). [2] M. Simmons, A new horizon for quantum information, npj Quantum Information 1, 15013 (2015). [3] S. Pirandola and S. L. Braunstein, Physics: Unite to build a quantum Internet, Nature News 532, 169 (2016). [4] M. Schleier-Smith, Editorial: Hybridizing Quantum Physics and Engineering, Phys. Rev. Lett. 117, 100001 (2016). [5] Institute of Physics Publications, The age of the qubit, (2011). [6] J. F. Fitzsimons, E. G. Rieﬀel, and V. Scarani, The Quantum Frontier, arXiv:1206.0785. [7] G. Adesso, R. L. Franco, and V. Parigi, Foundations of quantum mechanics and their impact on contemporary society, Phil. Trans. R. Soc. A 376, 20180112 (2018). [8] J. Stuhler, Quantum optics route to market, Nature Physics 11, 293 (2015). [9] J. Preskill, Quantum information and physics: some future directions, J. Mod. Opt. 47, 127 (2000). [10] R. Cowen, The quantum source of space-time, Nature 527, 290 (2015). [11] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum Machine Learning, Nature 549, 195 (2017). [12] S. Popescu, Bell’s inequalities versus teleportation: What is nonlocality?, Phys. Rev. Lett. 72, 797 (1994). [13] P. S. D. Cavalcanti and I. Šupić, All Entangled States can Demonstrate Nonclassical Teleportation, Phys. Rev. Lett. 119, 110501 (2017). [14] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014). [15] C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, One-sided Device-Independent Quantum Key Distribution: Security, feasibility, and the connection with steering, Phys. Rev. A 85, 010301(R) (2012). [16] R. Schnabel, Squeezed states of light and their applications in laser interferometers, Phys. Rep. 684, 1 (2017). [17] D. Girolami, A. M. Souza, V. Giovannetti, T. Tufarelli, J. G. Filgueiras, R. S. Sarthour, D. O. Soares-Pinto, I. S. Oliveira, and G. Adesso, Quantum discord determines the interferometric power of quantum states, Phys. Rev. Lett. 112, 210401 (2014). [18] M. Howard, J. J. Wallman, V. Veitch, and J. Emerson, Contextuality supplies the magic for quantum computation, Nature 510, 351 (2014). [19] H.-L. Shi, S.-Y. Liu, X.-H. Wang, W.-L. Yang, Z.-Y. Yang, and H. Fan, Coherence depletion in the Grover quantum search algorithm, Phys. Rev. A 95, 032307 (2017). 16

[20] T. Theurer, N. Killoran, D. Egloff, and M. B. Plenio, Resource Theory of Superposition, Phys. Rev. Lett. 119, 230401 (2017). [21] J. Åberg, Catalytic Coherence, Phys. Rev. Lett. 113, 150402 (2014). [22] K. von Prillwitz, L. Rudnicki, and F. Mintert, Contrast in Multipath Interference and Quantum Coherence, Phys. Rev. A 92, 052114 (2015). [23] A. Streltsov, G. Adesso, and M. B. Plenio, Quantum coherence as a resource, Rev. Mod. Phys. 89, 041003 (2017). [24] E. Chitambar, X. Ma, and A. Streltsov, Preface: Quantum coherence, J. Phys. A: Math. Theor. 51, 410301 (2018). [25] A. Messiah, Quantum Mechanics (Dover, New York, 2014). [26] U. Fano, Description of States in Quantum Mechanics by Density Matrix and Operator Techniques, Rev. Mod. Phys. 29, 74 (1957). [27] M. Wilde, Quantum Information Theory (Cambridge University Press, New York, 2013). [28] J. Maziero, Hilbert-Schmidt quantum coherence in multi-qudit systems, Quantum Inf. Process. 16, 274 (2017). [29] T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying Coherence, Phys. Rev. Lett. 113, 140401 (2014). [30] C.-s. Yu, Quantum coherence via skew information and its polygamy, Phys. Rev. A 95, 042337 (2017). [31] Z. Puchała and ŁRudnicki and K. Chabuda and M. Paraniak and K. Życzkowski, Certainty relations, mutual entanglement and non-displacable manifolds, Phys. Rev. A 92, 032109 (2015). [32] K. Korzekwa, M. Lostaglio, D. Jennings, and T. Rudolph, Quantum and classical entropic uncertainty relations, Phys. Rev. A 89, 042122 (2014). [33] E. Bagan, J. A. Bergou, S. S. Cottrell, and M. Hillery, Relations between Coherence and Path Information, Phys. Rev. Lett. 116, 160406 (2016). [34] U. Singh, M. N. Bera, H. S. Dhar, and A. K. Pati, Maximally coherent mixed states: Complementarity between maximal coherence and mixedness, Phys. Rev. A 91, 052115 (2015). [35] S. Cheng and M. J. W. Hall, Complementarity relations for quantum coherence, Phys. Rev. A 92, 042101 (2015). [36] F. Liu, F. Li, J. Chen, and W. Xing, Uncertainty-like relations of the relative entropy of coherence, Quantum. Inf. Process. 15, 3459 (2016). [37] O. Pessoa Júnior, Conceitos de Física Quântica (Editora Livraria da Física, São Paulo, 2006). [38] T. Qureshi, Coherence, Interference and Visibility, Quanta, 8, 24 (2019). [39] S. Dürr, Quantitative wave-particle duality in multibeam interferometers, Phys. Rev. A, 64, 042113 (2001). [40] B.-G. Englert, D. Kaszlikowski, L. C. Kwek, and W. H. Chee, Wave-particle duality in multi-path interferometers: General concepts and three-path interferometers, Int. J. Quant. Inf. 6, 129 (2008). [41] S. Mishra, A. Venugopalan, and T. Qureshi, Decoherence and visibility enhancement in multi-path interference, Phys. Rev. A 100, 042122 (2019). [42] R. A. Bertlmann and P. Krammer, Bloch vectors for qudits, J. Phys. A: Math. Theor. 41, 235303 (2008). [43] K. Kuttler, Elementary Linear Algebra (Saylor Foundation, 2012). [44] M. Avendaño, Descartes’ rule of signs is exact!, J. Algebra 324, 2884 (2010). [45] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cam- bridge, 2000). [46] M. Ledoux, The Concentration of Measure Phenomenon (American Mathematical Society, 2001). [47] J. Maziero, Random Sampling of Quantum States: a Survey of Methods, Braz. J. Phys. 45, 575 (2015). [48] J. Maziero, Fortran Code for Generating Random Probability Vectors, Unitaries, and Quantum States, Front. ICT 3, 4 (2016). [49] R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 2013). [50] A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50, 221 (1978). [51] T. Hiroshima and S. Ishizaka, Local and nonlocal properties of Werner states, Phys. Rev. A 62, 044302 (2000). [52] E. P. Wigner and M. M. Yanase, Information contents of distributions, Proc. Nat. Acad. Sci. USA 49, 910 (1963). [53] M. Hillery, Coherence as a resource in decision problems: The Deutsch-Jozsa algorithm and a variation, Phys. Rev. A 93, 012111 (2016). [54] P. Kammerlander and J. Anders, Coherence and measurement in quantum thermodynamics, Sci. Rep. 6, 22174 (2016). [55] A. Streltsov, E. Chitambar, S. Rana, M. N. Bera, A. Winter, and M. Lewenstein, Entanglement and Coherence in Quantum State Merging, Phys. Rev. Lett. 116, 240405 (2016). [56] C. Yu, S. Yang, and B. Guo, Total quantum coherence and its applications, Quantum Inf. Process. 15, 3773 (2016). [57] X. Yuan, K. Liu, Y. Xu, W. Wang, Y. Ma, F. Zhang, Z. Yan, R. Vijay, L. Sun, and X. Ma, Experimental Quantum Randomness Processing, Phys. Rev. Lett. 117, 010502 (2016). [58] P. Giorda and M. Allegra, Coherence in quantum estimation, J. Phys. A: Math. Theor. 51, 025302 (2018). [59] F.-L. Zhang and T. Wang, Intrinsic coherence in assisted sub-state discrimination, EPL 117, 10013 (2017). [60] M. B. Pozzobom and J. Maziero, Environment-induced quantum coherence spreading of a qubit, Ann. Phys. 377, 243 (2017). [61] D. N. S. Buruaga and C. Sabín, Quantum coherence in the dynamical Casimir effect, Phys. Rev. A 95, 022307 (2017). [62] L. Li, J. Zou, H. Li, J.-G. L, Y.-M. Wang, and B. Shao, Controlling energy flux into a spatially correlated environment via quantum coherence, Eur. Phys. J. D 71, 62 (2017). [63] K. Brandner, M. Bauer, and U. Seifert, Universal Coherence-Induced Power Losses of Quantum Heat Engines in Linear Response, Phys. Rev. Lett. 119, 170602 (2017). [64] G. D. Scholes, G. R. Fleming, L. X. Chen, A. Aspuru-Guzik, A. Buchleitner, D. F. Coker, G. S. Engel, R. van Grondelle, A. Ishizaki, D. M. Jonas, J. S. Lundeen, J. K. McCusker, S. Mukamel, J. P. Ogilvie, A. Olaya-Castro, M. A. Ratner, F. 17

C. Spano, K. B. Whaley, and X. Zhu, Using coherence to enhance function in chemical and biophysical systems, Nature 543, 647 (2017). [65] C. Bengtson and E. Sjöqvist, The role of quantum coherence in dimer and trimer excitation energy transfer, New J. Phys. 19, 113015 (2017). [66] D. F. Pinto and J. Maziero, Entanglement production by the magnetic dipolar interaction dynamics, Quantum Inf. Proc. 17, 253 (2018). [67] K. Southwell, Quantum coherence, Nature 453, 1003 (2008). [68] F. g. Li, W.-S. Bao, S. Zhang, H. l. Huang, T. Li, X. Wang, and X. q. Fu, Role of coherence in adiabatic search algorithms, Phys. Lett. A 382, 2709 (2018). [69] M. S. Byrd and N. Khaneja, Characterization of the positivity of the density matrix in terms of the coherence vector representation, Phys. Rev. A 68, 062322 (2003). [70] G. Kimura, The Bloch vector for N-level systems, Phys. Lett. A 314, 339 (2003). [71] B.-G. Englert, Fringe visibility and which-way information: An inequality, Phys. Rev. Lett. 77, 2154 (1996). [72] F. G. S. L. Brandão and G. Gour, The general structure of quantum resource theories, Phys. Rev. Lett. 115, 070503 (2015). [73] E. Chitambar and G. Gour, Quantum Resource Theories, Rev. Mod. Phys. 91, 025001 (2019). [74] M. Huber and M. Perarnau-Llobet and K. V. Hovhannisyan and P. Skrzypczyk and C. Klöckl and N. Brunner and A. Acín, Thermodynamic cost of creating correlations, New J. Phys. 17, 065008 (2015). [75] J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, The role of quantum information in thermodynamics–a topical review, J. Phys. A: Math. Theor. 49, 143001 (2016). [76] J. Anders and M. Esposito, Focus on quantum thermodynamics, New J. Phys. 19, 010201 (2017). [77] M. Lostaglio, An introductory review of the resource theory approach to thermodynamics, Rep. Prog. Phys. 82 114001 (2019).