Monogamy and Trade-Off Relations for Correlated Quantum Coherence

Monogamy and trade-oﬀ relations for correlated quantum coherence

Marcos L. W. Basso1, ∗ 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 One of the fundamental diﬀerences between classical and quantum mechanics is in the ways correlations can be distributed among the many parties that compose a system. While classical correlations can be shared among many subsystems, in general quantum correlations cannot be freely shared. This unique property is known as monogamy of quantum correlations. In this work, we study the monogamy properties of the correlated coherence for the l1-norm and relative entropy measures of coherence. For the l1-norm the correlated coherence is monogamous for a particular class of quantum states. For the relative entropy of coherence, and using maximally mixed state as the reference incoherent state, we show that the correlated coherence is monogamous for tripartite pure quantum systems. Keywords: Trade-oﬀ relations; Monogamy relations; Measures of quantum coherence; Correlated coherence

I. INTRODUCTION

The wave-particle duality is one of the most fascinating aspects of quantum mechanics, and captures the role of quantum coherence in quantum systems [1]. Another intriguing aspect of quantum mechanics is quantum entanglement [2]. One of the reasons is that quantum entanglement, and, in general, quantum correlations, cannot be shared freely among many parties. This property is known as monogamy of quantum correlations [3]. More recently, it became known that quantum coherence in a composite system can be contained either locally or in the correlations between the subsystems. The portion of quantum coherence contained within correlations can be viewed as a kind of quantum correlation, called correlated coherence [4]. Thus, a natural question that arises, in the context of monogamy relations for quantum correlations, is whether correlated coherence also have this property. This is the question we try to answer in this article. In quantum mechanics, the state of a physical system can be described as a vector in a Hilbert space H. One of the features of such vector space is that any linear combination of vectors also belongs to H, thus allowing superposition of states [5]. This superposition of states is crucial to explain interference patterns in multiple-slit experiments, that otherwise can’t be explained by classical physics. One special kind of quantum superposition is quantum coherence, which became an important physical resource in quantum information and quantum computation [6]. More recently, it was shown that quantum coherence is a natural generalization of visibility for quantifying the wave aspect of a quanton in multiple-slit experiments [7–12]. It also has an important role in several research fields, such as quantum biology [13, 14] and quantum metrology [15]. An important step towards the quantification of quantum coherence was given by Baumgratz et al. [16]. They established reasonable conditions that a measure of coherence must satisfy to be considered a bona fide measure: Nonnegativity, monotonicity under incoherent completely positive and trace preserving maps (ICPTP), monotonicity under selective incoherent operations on average, and convexity under mixing of states. In the same work, they showed that the l1-norm and the relative entropy of coherence are bone fide measures of coherence, meanwhile the Hilbert-Schmidt (or l2-norm) coherence is not a coherence monotone, i.e., it is not monotone under ICPTP. Later, an equivalent and rigorous framework for quantifying coherence was given in [17]. Another fundamental difference between classical and quantum mechanics is in the ways correlations can be shared arXiv:2006.14037v2 [quant-ph] 3 Sep 2020 among the many parts that compose a system. While classical correlations can be freely shared among many subsystems, in general quantum correlations cannot be freely shared. For instance, if we consider a pair of perfectly correlated classical bits A and B, it’s possible to maximally correlate the bit A also with a third bit C. Mathemat- 1 ically, if e.g. ρAB = 2 (|0, 0ih0, 0| + |1, 1ih1, 1|) describes the maximally classically-correlated state of subsystems A 1 and B, it’s possible to construct a joint quantum state of three subsystems ρABC = 2 (|0, 0, 0ih0, 0, 0| + |1, 1, 1ih1, 1, 1|) such that A is also maximally classically-correlated with C, once the reduced states [18] are equal ρAC = ρAB. In contrast, for two perfectly quantum-correlated (maximally entangled) qubits A and B, it is not possible for A to be maximally quantum-correlated with a third qubit C. Mathematically, if e.g. |Ψi = √1 (|0, 0i + |1, 1i ) A,B 2 A,B A,B

∗Electronic address: [email protected] †Electronic address: [email protected] 2

describes the maximally quantum-correlated state of A and B, there is no tripartite quantum state |ΨiA,B,C such that ρAB = ρAC . Hence, if a pair of qubits A and B are maximally entangled, then the system A (or B) cannot be maximally entangled to a third system C [3, 19]. Actually, the more a qubit is entangled with another qubit, the less it can be entangled√ with a third one. This fact can be exempliﬁed with the state 2 |ΨiA,B,C = x(|0, 0iA,B + |1, 1iA,B) |jiC + 1 − x (|0, 0iA,C + |1, 1iA,C ) |jiB, where j = 0, 1 and x ∈ [0, 1]. One can see that for x = 1 the subsystems A and B are perfectly correlated and A and C are uncorrelated, whereas for x = 0 the parties A and C are perfectly correlated but A and B are uncorrelated. This indicates that there is a limitation in the distribution of entanglement [20]. This unique property, known as entanglement monogamy, has received a lot of attention by researchers [21–25]. Mathematically, for a tripartite quantum system described by the density matrix ρA,B,C , the monogamy of an arbitrary quantum correlation measure Q is expressed by:

Q(ρA|BC ) ≥ Q(ρAB) + Q(ρAC ), (1) where Q(ρA|BC ) denotes the quantum correlation Q between A and BC taken as a unit [26]. For instance, in the example of the classical bits given above, if the correlation measure is the classical mutual information [6], we have I(A : B) + I(A : C) = 2 > I(A : BC) = 1. In contrast, it was shown in Ref. [3] that the inequality in Eq. (1) is always satisfied for the squared concurrence, an quantum entanglement measure, applied to three qubit states. However, it is known that entanglement is not the only quantum correlation existing in multipartite quantum systems [27–31]. For example, quantum discord is a type of quantum correlation that describes the incapacity of a local observer to obtain information about a subsystem without perturbing it [32]. In [22], the authors showed that quantum discord and entanglement of formation obey the same monogamy relationship for tripartite pure cases. But, it’s known that, in general, quantum discord is not monogamous, except when the quantum states satisfy certain properties [23]. Meanwhile, the monogamy of coherence was first studied in [33], and later addressed in [34]. In addition, Yu [35] studied its polygamy. More recently, it became known that quantum coherence in a composite system can be contained either locally or in the correlations between the subsystems [31]. The portion of quantum coherence contained within correlations can be viewed as a kind of quantum correlation, called correlated coherence [4]. Thus, a natural question that arises, in the context of monogamy relations for quantum correlation, is whether correlated coherence fulfills a monogamy relation like (1). In this article we show that, for the l1-norm, the correlated coherence is monogamous for a given class of quantum states, and we conjecture that is monogamous, at least, for tripartite pure quantum states, what is noteworthy once the l1-norm correlated coherence doesn’t vanish for uncorrelated separable states. Also, for the relative entropy of coherence, and using a maximally mixed state as the reference incoherent state, we show that the correlated coherence is monogamous for tripartite pure quantum systems. Another interesting finding is that the relative entropy of correlated coherence is equal to the quantum mutual information when one uses maximally mixed state as the reference incoherent state. Finally, we also establish some trade-off relations between tripartite and bipartite quantum systems using correlated coherence. It’s worth mentioning that trade-off relations between coherence and entanglement were studied in [36], and between coherence and correlation were addressed in [37].

We organized the remainder of this article in the following manner. In Sec.II, we give the deﬁnition of correlated coherence and discuss some of its properties. The main results of this article are reported in Sec.III, where we show that, for the l1-norm, the correlated coherence is monogamous for a given class of quantum states. Meanwhile, for the relative entropy of coherence, and using maximally mixed state as the reference incoherent state, we show that the correlated coherence is monogamous for tripartite pure quantum systems. Furthermore, for a bipartition of a quantum system, the relative entropy of correlated coherence is equal to the mutual quantum information. We also present some trade-oﬀ relations between tripartite and bipartite quantum systems using correlated coherence. Lastly, in Sec.IV we give our conclusions.

II. PRELIMINARIES

In this section, we give the definition of correlated coherence and explore some of its properties. Definition 1. For some coherence measure C, the correlated coherence of a multipartite quantum system described by the density operator ρA1,...,An in the composite Hilbert space HA1 ⊗ ... ⊗ HAn is defined as

n c X C (ρA1...An ) := C(ρA1...An ) − C(ρAi ), (2) i=1

where ρAi is the reduced density matrix of the subsystem Ai, with i = 1, ..., n. 3

c Nn In this article, we won’t deﬁne the correlated coherence as C (ρA ...A ) = C(ρA ...A ) − C( ρA ), because not Nn Pn1 n 1 n i=1 i every coherence measure satisﬁes the property C( i=1 ρAi ) = i=1 C(ρAi ), as we’ll show in this section. Now, let d −1 {|i i } Am be an orthonormal local basis of the Hilbert space H , with m = 1, ..., n. Then m Am im=0 Am X X ρ = ρ |i , ..., i i hj , ..., j | , (3) A1,...,An i1...in,j1...jn 1 n A1,...,An 1 n i1,...,in j1,...,jn and the state of the subsystem Am, which is obtained by tracing over the other subsystems, is given by X X X ρ = ρAm |i i hj | = ρ |i i hj | , (4) Am im,jm m Am m i1...,im,...,in,i1...,jm,...,in m Am m im,jm im,jm iα,∀α6=m where P means summation for all i such that α 6= m for a given m. Throughout this article, we’ll use the iα,∀α6=m α following coherence measures:

Definition 2. The l1-norm measure of quantum coherence is defined as X Cl (ρ) := min ||ρ − ι||l = |ρjk|, (5) 1 ι∈I 1 j6=k where kAk = P |A |, for A ∈ n×n, and I is the set of all incoherent states. Meanwhile, the relative entropy of l1 j,k jk C quantum coherence is defined as

Cre(ρ) := min S(ρ||ι) = S(ρdiag) − S(ρ), (6) ι∈I where S(ρ||ι) = Tr(ρ ln(ρ − ι)) is the relative entropy. In both cases, the closest incoherent state is given by ι = P ρdiag = j ρjj |jihj|. Also, we already omitted the upper limits of the summations for convenience. The following theorem holds for any multipartite quantum system: d −1 Theorem 1. Let ρ be a multipartite quantum state and let {|i i } Am be an orthonormal local basis for A1,...,An m Am im=0 the Hilbert space H , with m = 1, ..., n. Then Cc (ρ ) ≥ 0, where Cc is the correlated coherence using the Am l1 A1...An l1 l1-norm as coherence measure. Proof. By deﬁnition n X Cc (ρ ) := C (ρ ) − C (ρ ) (7) l1 A1...An l1 A1...An l1 Am m=1

n X X X X := |ρi1...in,j1...jn | − ρi1...,im,...,in,i1...,jm,...,in , (8)

(i1,...,in)6=(j1,...,jn) m=1 im6=jm iα,∀α6=m where X X X X X X X ≡ + +... + + +... + +... + . (9)

(i1,...,in)6=(j1,...,jn) i16=j1 i1=j1 i1=j1 i16=j1 i16=j1 i16=j1 i2=j2 i26=j2 i2=j2 i26=j2 i2=j2 i26=j2 ...... in=jn in=jn in6=jn in=jn in6=jn in6=jn P P Using the fact that | i zi| ≤ i |zi| ∀ zi ∈ C, we have n X X X X Cc (ρ ) ≥ |ρ | − |ρ | (10) l1 A1...An i1...in,j1...jn i1...,im,...,in,i1...,jm,...,in (i1,...,in)6=(j1,...,jn) m=1 im6=jm iα,∀α6=m X X X = +... + +... + |ρi1...in,j1...jn | (11)

i16=j1 i16=j1 i16=j1 i26=j2 i2=j2 i26=j2 ...... in=jn in6=jn in6=jn ≥ 0, (12) once we have the sum of non negative real numbers. 4

It’s worth pointing out that the theorem stated above was ﬁrst proved in [4] for the bipartite case, and implicitly proved in [38] for the three-qubit case, since they proved the following result: Cl1 (ρABC ) ≥ Cl1 (ρA)+Cl1 (ρB)+Cl1 (ρC ). Now, let’s restrict ourselves to a bipartite quantum system. Then, for a separable uncorrelated quantum state P A B ρA,B = ρA ⊗ ρB = i,j,k,l ρikρjl |i, jiA hk, l|, the correlated coherence is not necessarily zero: X X X X Cc (ρ ⊗ ρ ) = ρA ρB − ρA − ρA = ρA ρB = C (ρ )C (ρ ) ≥ 0, (13) l1 A B ik jl ik jl ik jl l1 A l1 B (i,j)6=(k,l) i6=k j6=l i6=k j6=l

which implies that Cl1 (ρA ⊗ ρB) 6= Cl1 (ρA) + Cl1 (ρB), as mentioned before (for a related discussion, see [39]). P A B P P A B Analogously, for a separable state ρAB = α pαρα ⊗ ρα = α i,j,k,l pαρα,ikρα,jl |i, jiA hk, l|, we have X X Cc (ρ ) = p ρA ρB ≥ 0. (14) l1 AB α α,ik α,jl i6=k α j6=l N However, using the relative entropy of coherence as the measure of correlated coherence [40], since S( ρA ) = P N P m m m S(ρAm ), we have Cre( m ρAm ) = m Cre(ρAm ), and therefore the relative entropy of correlated coherence c N satisﬁes Cre(ρA1,...,An ) = 0 for ρA1,...,An = m ρAm . More generally, it is worth mentioning that Rényi’s entropy is also additive [41, 42], hence correlated coherence measures based on Rényi’s entropy must satisfy such relation.

III. MONOGAMY AND TRADE-OFF RELATIONS FOR CORRELATED COHERENCE

A. l1-norm correlated coherence

In [38], the authors proved that the conjecture C(ρABC ) ≥ C(ρAB) + C(ρAC ), made by Yao et al. [31], for the l1- norm is invalid. They considered a counter example using the following the quantum state: |Ψi = a000 |0, 0, 0iA,B,C + a100 |1, 0, 0iA,B,C . However, it is interesting to note that for the correlated coherence this type of trade-oﬀ relation c c c holds for the quantum state mentioned above. Since Cl (ρABC ) ≥ Cl (ρAB)+Cl (ρAC ) implies Cl1 (ρABC )+Cl1 (ρA) ≥ 1 1 1 ∗ Cl1 (ρAB) + Cl1 (ρAC ), and Cl1 (ρABC ) = Cl1 (ρA) = Cl1 (ρAB) = Cl1 (ρAC ) = 2|a000a100|. More generally, we have the following theorem:

dA−1 dB −1 dC −1 Theorem 2. Let ρABC be a tripartite quantum state and let {|iiA}i=0 ,{|jiB}j=0 ,{|kiC }k=0 be a orthonormal local basis of HA, HB, and HC , respectively. If the reduced quantum system ρA = TrB,C ρA,B,C satisﬁes

X X X X Cl1 (ρA) = ρijk,ljk = |ρijk,ljk|, (15)

i6=l j,k i6=l j,k then Cc (ρ ) ≥ Cc (ρ ) + Cc (ρ ). (16) l1 ABC l1 AB l1 AC Proof. Cc (ρ ) − Cc (ρ ) − Cc (ρ ) = C (ρ ) + C (ρ ) − C (ρ ) − C (ρ ) (17) l1 ABC l1 AB l1 AC l1 ABC l1 A l1 AB l1 AC

X X X X X X X = |ρijk,lmn| + ρijk,ljk − ρijk,lmk − ρijk,ljn (18)

(i,j,k)6=(l,m,n) i6=l j,k (i,j)6=(l,m) k (i,k)6=(l,n) j

X X X X X ≥ + − |ρijk,lmn| + ρijk,ljk (19)

i6=l i=l i6=l i6=l j,k j6=m j6=m j=m k6=n k6=n k=n X X = + |ρijk,lmn| (20) i6=l i=l j6=m j6=m k6=n k6=n ≥ 0, (21) P P Above we used the fact that | i zi| ≤ i |zi|, ∀ zi ∈ C. This completes the proof. 5

Now, following the same reasoning:

Theorem 3. If the reduced quantum systems ρA, ρB, ρC satisfy

X X X X Cl1 (ρA) = ρijk,ljk = |ρijk,ljk|, (22)

i6=l j,k i6=l j,k

X X X X Cl1 (ρB) = ρijk,imk = |ρijk,imk|, (23)

j6=m j,k j6=m i,k

X X X X Cl1 (ρC ) = ρijk,ijn = |ρijk,ijn|, (24)

k6=n i,j k6=n i,j (25) then

Cc (ρ ) ≥ Cc (ρ ) + Cc (ρ ) + Cc (ρ ). (26) l1 ABC l1 AB l1 AC l1 BC Proof. X X Cc (ρ ) − Cc (ρ ) − Cc (ρ ) − Cc (ρ ) = C (ρ ) + C (ρ ) − C (ρ ) (27) l1 ABC l1 AB l1 AC l1 BC l1 ABC l1 α l1 αβ α=A,B,C α<β=A,B,C

X X X X X X X X X X ≥ − − − |ρijk,lmn| + ρijk,ljk + ρijk,imk + ρijk,ijn (28)

i6=l i6=l i=l i=l i6=l j,k j6=m i,k l6=n i,j j6=m j=m j6=m j=m k6=n k=n k=n k6=n X = |ρijk,lmn| (29) i6=l j6=m k6=n ≥ 0. (30)

This kind of trade-off relation is equivalent to the monogamy relation expressed by the Eq. (1) for the correlated coherence. Theorem 4. A tripartite quantum system that satisfies the trade-off relation

c c c c C (ρABC ) ≥ C (ρAB) + C (ρAC ) + C (ρBC ) (31) also satisﬁes the monogamy relation

c c c C (ρA|BC ) ≥ C (ρAB) + C (ρAC ), (32)

c c for any coherence measure, where C (ρA|BC ) := C (ρABC )−C(ρA)−C(ρBC ) denotes the correlated coherence between A and BC. Proof. The proof follows directly from the deﬁnition

c c c X X C (ρA|BC ) − C (ρAB) − C (ρAC ) = C(ρABC ) + C(ρα) − C(ραβ) (33) α=A,B,C α<β=A,B,C c c c c = C (ρABC ) − C (ρAB) − C (ρAC ) − C (ρBC ) (34) ≥ 0. (35) 6

Figure 1: The function M := Cc (ρ ) − Cc (ρ ) − Cc (ρ ) as function of p and for the state |Φ(p, )i. l1 A|BC l1 AB l1 AC

For instance, the pure quantum state |G, H, Z, W iA,B,C = λ1 |0, 0, 0iA,B,C + λ2 |0, 0, 1iA,B,C + λ3 |0, 1, 0iA,B,C + 2 2 2 2 2 λ4 |1, 0, 0iA,B,C + λ5 |1, 1, 1iA,B,C , with |λ1| + |λ2| + |λ3| + |λ4| + |λ5| = 1, satisﬁes the monogamy relation for the correlated coherence. Also, for the state |ΦiA,B,C = a000 |0, 0, 0iA,B,C + a101 |1, 0, 1iA,B,C + a |1, 1, 0i + a |1, 1, 1i such that |a |2 + |a |2 + |a |2 + |a |2 = 1, we have Cc (ρ ) − 1,1,0 A,B,C 111 A,B,C 000 101 110 111 l1 A|BC c c ∗ Cl (ρAB) − Cl (ρAC ) = 2|a000a111| ≥ 0. This state was considered by Giorgi [22] in the form |Φ(p, )i = √ 1 1 p p p |0, 0, 0iA,B,C + p(1 − ) |1, 1, 1iA,B,C + (1 − p)/2(|1, 1, 0iA,B,C + |1, 0, 1iA,B,C ), with p, ∈ [0, 1]. Hence, for the correlated coherence, |Φ(p, )i is monogamous for any value of p, ∈ [0, 1]. In Fig.1, we plotted M := Cc (ρ ) − Cc (ρ ) − Cc (ρ ) as function of p and . l1 A|BC l1 AB l1 AC Also, let’s consider the following state |ΨiA,B,C = λ1 |0, 0, 0iA,B,C + λ2 |0, 1, 1iA,B,C + λ3 |1, 0, 0iA,B,C + 2 2 2 2 ∗ ∗ ∗ ∗ λ4 |1, 1, 1iA,B,C , with |λ1| + |λ2| + |λ3| + |λ4| = 1 [43], such that Cl1 (ρA) = 2|λ1λ3 + λ2λ4|= 6 2(|λ1λ3| + |λ2λ4|). Then

Cc (ρ ) − Cc (ρ ) − Cc (ρ ) = 2(|λ λ∗| + |λ λ∗|) + 2(|λ λ∗ + λ λ∗| − |λ λ∗| − |λ λ∗|) (36) l1 A|BC l1 AB l1 AC 1 4 2 3 1 3 2 4 1 3 2 4 ∗ ∗ ∗ ∗ + 2(|λ1λ2| + |λ3λ4| − |λ1λ2 + λ3λ4|) ∗ ∗ ∗ ∗ ∗ ∗ ≥ 2(|λ1λ4| + |λ2λ3|) + 2(|λ1λ3 + λ2λ4| − |λ1λ3| − |λ2λ4|) (37) ≥ 0. (38)

∗ ∗ ∗ ∗ ∗ ∗ To prove the last passage, let’s suppose, by contradiction, that |λ1λ3| + |λ2λ4| ≥ |λ1λ4| + |λ2λ3| + |λ1λ3 + λ2λ4|. Squaring this expression, we obtain

∗ 2 ∗ 2 ∗ ∗ ∗ ∗ ∗ ∗ 0 ≥ |λ1λ4| + |λ2λ3| + 2(|λ1λ3| + |λ2λ4|)|λ1λ3 + λ2λ4| + 2Re(λ1λ3λ2λ4) (39) ∗ ∗ 2 ∗ ∗ ∗ ∗ = |λ1λ3 + λ2λ4| + 2(|λ1λ3| + |λ2λ4|)|λ1λ3 + λ2λ4|, (40) which is an absurd because the right-hand side is the sum of positive real numbers. On the other hand, if the 4 coefficients {λi} are real and positive, the monogamy is obviously satisfied. We can check this considering the state i=1 √ p p in the form |Ψ(p, )i = p |0, 0, 0iA,B,C + p(1 − ) |1, 1, 1iA,B,C + (1 − p)/2(|1, 0, 0iA,B,C + |0, 1, 1iA,B,C ), with p, ∈ [0, 1], where, in Fig.2, we plotted M := Cc (ρ ) − Cc (ρ ) − Cc (ρ ) as function of p and . Therefore, l1 A|BC l1 AB l1 AC the conditions expressed by the equations (22), (23), and (24) are sufficient but not necessary, and seems reasonably to conjecture that the monogamy relation Cc (ρ ) ≥ Cc (ρ ) + Cc (ρ ) holds for any tripartite pure quantum l1 A|BC l1 AB l1 AC state. Finally, it is possible to establish a weaker trade-off relation for an arbitrary tripartite quantum state, i.e.,

c 1 c c c C (ρABC ) ≥ C (ρAB) + C (ρAC ) + C (ρBC ) , (41) l1 2 l1 l1 l1 7

Figure 2: The function M := Cc (ρ ) − Cc (ρ ) − Cc (ρ ) as function of p and for the state |Ψ(p, )i. l1 A|BC l1 AB l1 AC once

c 1 c c c 1 C (ρABC ) − C (ρAB) + C (ρAC ) + C (ρBC ) = Cl (ρABC ) − Cl (ρAB) + Cl (ρAC ) + Cl (ρBC ) (42) l1 2 l1 l1 l1 1 2 1 1 1 X 1 X X X ≥ + ( + + ) |ρ | (43) 2 ijk,lmn i6=l i=l i6=l i6=l j6=m j6=m j=m j6=m k6=n k6=n k6=n k=n ≥ 0. (44)

B. Relative entropy of correlated coherence

First, we’ll recall an upper bound for the correlated coherence of a bipartite quantum system ρAB [47]:

c Cre(ρAB) = Cre(ρAB) − Cre(ρA) − Cre(ρB) (45)

= S(ρABdiag ) − S(ρAB) − S(ρAdiag ) + S(ρA) − S(ρBdiag ) + S(ρB) (46)

≤ S(ρA) + S(ρB) − S(ρAB), (47)

since S(ρABdiag ) − S(ρAdiag ) − S(ρBdiag ) ≤ 0, by the subadditivity of von Neumann’s entropy [6]. In addition, Xi et al. [29] proved that Cre(ρAB) ≥ Cre(ρA) + Cre(ρB), by using the fact that local operations never increase the relative c entropy, which implies Cre(ρAB) ≥ 0. Thus c 0 ≤ Cre(ρAB) ≤ S(ρA) + S(ρB) − S(ρAB) := IA,B, (48) where IA,B is the quantum mutual information, which is the total amount of correlations in any bipartite quantum state [44, 45]. Also, it’s known that the quantification of quantum coherence depends on a particular reference basis, i.e., these quantum coherence measures are basis-dependent [46]. Following Ma et al. [47], we’ll define the basis-independent relative entropy of coherence, or intrinsic relative entropy of quantum coherence (IREQC). Since the maximally mixed state, I/d, is the only basis-independent incoherent state, the IREQC is defined by taken the maximally mixed state as the reference incoherent state, i.e.,

I Cre(ρ) := S(ρ||I/d) = log d − S(ρ), (49) where S(ρ||I/d) = Tr(ρ ln ρ − ρ ln I/d) is the relative entropy. Now, deﬁning the intrinsic relative entropy of correlated coherence (IRECC) for a bipartite quantum system as

Ic I I I Cre (ρAB) := Cre(ρAB) − Cre(ρA) − Cre(ρA), (50) we see it is equal to the total amount of correlations in any bipartite quantum system: 8

Ic Theorem 5. The intrinsic relative entropy of correlated coherence of a bipartite quantum system, Cre (ρAB), is equal to the quantum mutual information IA,B = S(ρA) + S(ρB) − S(ρAB). Proof. The result follows directly from the deﬁnition:

Ic I I I Cre (ρAB) = Cre(ρAB) − Cre(ρA) − Cre(ρB) = S(ρA) + S(ρB) − S(ρAB) (51) = IA,B. (52)

Ic I Now, by considering a tripartite quantum system, the IRECC is given by Cre (ρABC ) = Cre(ρABC ) − P I α=A,B,C Cre(ρα). Hence, we have the following trade-oﬀ relation

Ic Ic Ic I I I Ic Cre (ρABC ) − Cre (ρAB) − Cre (ρAC ) = Cre(ρABC ) + Cre(ρA) − Cre(ρAB) − Cre (ρAB) (53) = log(dAdBdC ) − S(ρABC ) + log dA − S(ρA) − log(dAdB) + S(ρAB) − log(dAdC ) + S(ρAC ) (54)

= S(ρAB) + S(ρAC ) − S(ρABC ) − S(ρA) (55) ≥ 0, (56) since S(ρAB) + S(ρAC ) − S(ρABC ) − S(ρA) ≥ 0 (by the strong subadditivity of von Neumann’s entropy [6]).

Theorem 6. If a tripartite quantum system ρABC is pure, then ρABC satisﬁes the trade-oﬀ relation

Ic Ic Ic Ic Cre (ρABC ) = Cre (ρAB) + Cre (ρAC ) + Cre (ρBC ), (57) and consequently the monogamy relation

Ic Ic Ic Cre (ρA|BC ) = Cre (ρAB) + Cre (ρAC ). (58) Proof.

Ic X Ic I X I X I Cre (ρABC ) − Cre (ραβ) = Cre(ρABC ) + Cre(ρα) − Cre(ραβ) (59) α<β=A,B,C α=A,B,C α<β=A,B,C

= S(ρAB) + S(ρAC ) + S(ρBC ) − S(ρABC ) − S(ρA) − S(ρB) − S(ρC ) (60)

= TA,B,C, (61) since TA,B,C is the interaction information [48], and TA,B,C = 0 for tripartite pure states, since S(ρABC ) = 0, and S(ρAB) = S(ρC ), S(ρAC ) = S(ρB), S(ρBC ) = S(ρA) [49]. This completes the proof.

Ic Ic Ic Consequently, Cre (ρA|BC ) = Cre (ρAB) + Cre (ρAC ) is equivalent to IA|BC = IA,B + IA,B [50], once that Ic Ic Cre (ρA|BC ) = IA|BC , Cre (ρXY ) = IX,Y , ∀ X,Y = A, B, C such that X 6= Y .

IV. CONCLUSIONS

Monogamy relations are an important feature of quantum correlations, as they tells us that a specific quantum resource cannot be shared freely. Recently, the correlated coherence was used as a resource for remote state preparation and quantum teleportation [51]. Hence, it is important to know if the correlated coherence satisfies monogamy relations. In this paper, we have studied the monogamy properties of the correlated coherence for the l1-norm and relative entropy measures. For the l1-norm, the correlated coherence is monogamous for a given class of quantum states, and we conjectured that it is monogamous, at least, for tripartite pure quantum states. For the relative entropy of coherence, and using a maximally mixed state as the reference incoherent state, we showed that the correlated coherence is monogamous for tripartite pure quantum system. Another interesting finding is that the intrinsic relative entropy of correlated coherence (IRECC) is equal to quantum mutual information. Finally, we also established some trade-off relations between tripartite and bipartite quantum systems, and proved that the trade-off relation c c c c c c c C (ρABC ) ≥ C (ρAB)+C (ρAC )+C (ρBC ) is equivalent to the monogamy relation C (ρA|BC ) ≥ C (ρAB)+C (ρAC ). 9

Acknowledgments

This work was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), process 88882.427924/2019-01, and by the Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ), process 465469/2014-0.

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