Monogamy relations for nonclassical correlations of multi-qubit states

# Monogamy relations for nonclassical correlations of multi-qubit states

## Abstract

Nonclassical correlations have been shown to be useful resources in many quantum information processing tasks, and various measures have been proposed to quantify these correlations. In this work, we first show for two-qubit states that a particular nonlocal effect, called measurement-induced nonlocality (MIN), is upper bounded by the Horodecki parameter which quantifies the maximal violation of Bell inequalities. We then derive a tight monogamy relation for MIN for pure three-qubit states. Furthermore, following from this monogamy relation, we provide an alternative way to derive monogamy relations for other correlation measures, including squared negativity, quantum discord, and geometric quantum discord. Finally, we generalize all of these monogamy relations to mixed three-qubit states and multi-qubit states.

###### Keywords:
Monogamy relation Measurement-induced nonlocality Quantum discord Entanglement

## 1 Introduction

Quantum correlations play an indispensable role in quantum information and quantum computation (1). Their peculiar properties, which have significant departures from the classical regime, attract strong theoretical and applied research interest (2); (3); (4). One of the most fundamental properties is the existence of quantum nonlocality, especially Bell nonlocality (4), corresponding to a fundamental incompatibility between quantum mechanics and local realism (5). Indeed, Bell nonlocality can be revealed even in the very simple scenario of two-qubit systems, shared by distant observers, where each observer chooses one of two dichotomic measurements on each qubit (6).

Unlike Bell nonlocality, MIN has been introduced to characterize the global effects caused by the local measurements on one side  (7). Restricting to qubit systems, we explore the potential relationships between these two kinds of nonlocality. They are also compared to entanglement, and to nonclassical correlations beyond entanglement, such as quantum discord (8); (9) and geometric quantum discord (10); (11).

In particular, such quantum correlations cannot be freely shared between members of multi-party systems. This phenomenon is termed monogamy, and for Bell nonlocality ensures the security of quantum cryptographic protocols (12). Monogamy relations are already known for concurrence (13); (14), negativity (15), Bell nonlocality (16); (17); (18), quantum steering (19); (20); (21); (22), and quantum discord (23); (24). Along this line, we are interested in whether MIN obeys a similar monogamy relation. We give an affirmative answer for qubit systems, and thus disprove claims in Ref. (25) that MIN does not satisfy such a monogamy relation.

This work is structured as follows. In Sec. 2, we introduce the basic definitions required in the two-qubit scenario and show MIN is no larger than the Horodecki parameter (26), which quantifies the maximal violation of a Bell inequality. We then derive a tight monogamy relation for MIN in Sec. 3, and also recover known monogamy relations for negativity, quantum discord, and geometric quantum discord as byproducts. In Sec. 4 we obtain monogamy relations for more general cases, including mixed three-qubit states and multi-qubit states. Finally, we conclude with some discussion in Sec. 5.

## 2 Measurement-induced nonlocality v.s. Bell nonlocality

Any two-qubit state can be written as

 ρAB=14(\mathbbm1A⊗\mathbbm1B+a⋅σ⊗\mathbbm1B+\mathbbm1A⊗b⋅σ+3∑j,k=1Tjkσj⊗σk), (1)

where refers to the vector of Pauli spin operators, and are identity operators, and correspond to the Bloch vectors of Alice’s and Bob’s reduced states, and is the spin correlation matrix, with .

Complementary to the locally accessible information and , the spin correlation matrix is of great importance in encoding the global information, and the strength of the quantum correlations of the qubits (2); (3); (4). Here, we are interested in the nonlocal correlations, corresponding to measurement-induced nonlocality and Bell nonlocality.

First, to quantify the maximal nonlocality caused by local measurements, Luo and Fu introduced measurement-induced nonlocality (MIN) via (7)

 DA→BM:=maxΠA||ρAB−ΠA(ρAB)||2. (2)

Here, the maximum is taken over all von Neumann measurements that preserve Alice’s local state, that is, , and denotes the Hilbert-Schmidt norm. In this work, we use the notation to specify Alice as the measuring party. In general, MIN is asymmetric with .

Fortunately, for two-qubit states, it can be computed analytically via (7) {numcases} D^A→B_M = 14(Tr[ T T^⊤ ]-1a2a^⊤T T^⊤a ) &
14(Tr[ T T^⊤ ]-s_3) & Here are the three eigenvalues of the symmetric matrix , and .

Second, Bell nonlocality is one of the most fundamental nonclassical features of quantum mechanics. In the simplest scenario where each side involves two dichotomic measurements on two qubits, Bell nonlocality can be witnessed by violating the Bell-Clauser, Horne, Shimony, and Holt (CHSH) inequality (6)

 ⟨B⟩2=(Tr[Bρ])2≤4, (3)

with the Bell operator . The Horodecki parameter is defined by (26)

 M=14max⟨B⟩2=s1+s2=Tr[TT⊤]−s3, (4)

where the maximum is over all measurement directions . Hence, quantifies the maximal violation of Bell-CHSH inequality.

It follows immediately from Eqs. (2) and (4) that for . For , note that for any unit vector . Hence, choosing , it follows via Eq. (2) that

 DA→BM≤14M (5)

for 2-qubit states generally. This implies that the Horodecki parameter provides an upper bound for MIN. Since is symmetric under the interchange of Alice and Bob, one similarly has , corresponding to the choice .

It is of interest to also consider geometric quantum discord, defined by (10)

 DA→BG:=minΠA||ρAB−ΠA(ρAB)||2. (6)

Here, in contrast to Eq. (2), the minimum is taken over all von Neumann measurements. It is obvious that . For any two-qubit state, both the computable entanglement measure  (27) and quantum discord  (8); (9) are upper bounded by  (28); (29), although this ordering does not hold for higher dimensional systems generally (30).

It follows from the above that

 N2,(DA→B)2≤2DA→BG≤2DA→BM≤12M, (7)

for two-qubit states. This ordering of nonclassical correlation measures becomes useful in exploring the monogamy phenomenon for multi-party systems. For example, if there exist monogamy relations for correlation measures in the right side of above ordering (7), such as MIN or Bell nonlocality, it is highly possible that these correlation measures in the left side also obey a similar monogamy relation. We will show it is indeed the case in the next section.

## 3 Monogamy relations for pure three-qubit states

Consider a pure three-qubit state , shared by Alice, Bob, and Charlie. The reduced bipartite states and can be expressed similarly to Eq. (1). Denote as the partition between Alice and the other parties. We are interested in the problem whether for a given measure , there is a monogamy relation

 EA:BC≥EA:B+EA:C. (8)

It is well known that the state of a pure quantum system is equivalent to a pure two-qubit state, up to local unitary operations. As a consequence, we can quantify the nonclassical correlations shared by Alice and the rest of the parties easily, because for the pure three-qubit state , one has (7); (10); (26); (27)

 (NA:BC)2=2DA→BCG=2DA→BCM=MA:BC−1=1−a2, (9)

while the quantum discord coincides with entropy of entanglement and is given (28)

 DA→BC=h(a):=−1−a2log1−a2−1+a2log1+a2. (10)

### 3.1 Pure states with a=0

In this case, Alice’s qubit is in a maximally mixed state. From Eq. (9), this immediately yields

 4DA→BCM=MA:BC=2. (11)

Furthermore, it has been proven in (18) that there is a tight monogamy relation for Bell nonlocality,

 MA:B+MA:C≤2=MA:BC, (12)

where the final equality derives from Eq. (11). We note that, in contrast to the monogamy relation obtained in (17), the above relation allows Alice to make different measurements in each case. With Eq. (5), this immediately leads to

 DA→BM+DA→CM≤14=DA→BCM. (13)

Hence, we obtain analytically a monogamy relation for MIN when . This relation is tight. For example, the state saturates both the monogamy relation (12) for Bell nonlocality and the monogamy relation (13) for MIN. We note that the monogamy relation (13) contradicts the claim and numerical results in (25).

### 3.2 General pure three-qubit states

When , the equality in Eq. (12) does not hold any more and hence the monogamy relation could be violated generally. For example, given the state with , we have .

However, for MIN, we can show that there still exists a strong monogamy relation:

 DA→BM+DA→CM≤12(1−a2)=DA→BCM. (14)

The proof is as follows. First, note that any pure three-qubit state can be locally transformed into the standard form (31); (32)

 a0|000⟩+a1eiϕ|100⟩+a2|101⟩+a3|110⟩+a4|111⟩, (15)

with and . Second, from the equalities and for pure states, we obtain two useful relations: (21)

 Tr[TABT⊤AB]=1+2c2−a2−b2, Tr[TACT⊤AC]=1+2b2−a2−c2.

Combining these results with Eqs. (2) and (9), we are able to calculate

 DA→BM+DA→CM−DA→BCM = 14[2+b2+c2−2a2−1a2(a⊤TABT⊤ABa+a⊤TACT⊤ACa)]−12(1−a2) = −2a20a2[4a21a22a23sin2ϕ+(a4(2a22+2a23+2a24−1)+2a1a2a3cosϕ)2] ≤ 0, (16)

as claimed in Eq. (14). Hence, MIN obeys this monogamy relation for any pure three-qubit state.

As a by-product, note that Eqs. (7), (9) and (14) imply corresponding monogamy relations for negativity and geometric quantum discord:

 (NA:B)2+(NA:C)2 ≤(NA:BC)2=1−a2, (17) DA→BG+DA→CG ≤DA→BCG=1−a22, (18)

while for quantum discord,

 (DA→B)2+(DA→C)2≤(DA→BC)2=h(0) (19)

Although these latter monogamy relations have been derived previously (15); (23); (24), we provide an alternative way to derive them, using the monogamy relation (14) for MIN. At the same time, from Eq. (18), we can also see the form of the residual for MIN, analogous to the 3-tangle for concurrence (13).

## 4 Monogamy relations for multi-qubit states

### 4.1 Mixed three-qubit states

Generally, it becomes difficult to obtain an analytical form such as Eq. (9) for the corresponding measures for the partition , and there are no corresponding monogamy relations of the form of (12) and (14). However, using the convexity of correlation measures, we have the weaker monogamy relation (18)

 MA:B+MA:C≤2, (20)

for mixed three-qubit states.

From the strength ordering (7), we can then obtain a chain of monogamy relations:

 (NA:B)2+(NA:C)2,  (DA→B)2+(DA→C)2 ≤ 2DA→BG+2DA→CG ≤ 2DA→BM+2DA→CM ≤ 12MA:B+12MA:C ≤ 1. (21)

Although we can also use the convex-roof construction (13) to derive tighter monogamy relations, the above trade-off relations require no optimization over all pure decompositions.

### 4.2 Multi-qubit states

For any n-qubit state , either pure or mixed, based on the monogamy relation (20), we can obtain

 MA:B+MA:C+MA:D+⋯≤2(n−12)(n−21)=n−1 (22)

for the Horodecki parameter. In particular, we can sum inequalities of the form of Eq. (20) with respect to Alice, with each individual parameter appearing in total times. This relation is tight because it can be saturated by the generalized GHZ-states .

We point out that the relation (22) is different from the tradeoff relation for Bell nonlocality in (33) where a tradeoff relation for all possible pairs is derived. Additionally, as we are constrained by quantum theory, the relation (22) is stronger than the one derived within the no-signaling theory (34); (35)

 √MA:B+√MA:C+√MA:D+⋯≤n−1. (23)

Consequently, the average of the maximal possible violation of a Bell-CHSH inequality, by Alice with one of the other parties, is bounded by

 ¯M:=MA:B+MA:C+MA:D+…n−1≤1. (24)

As pointed before, the bound is achieved by the generalized GHZ-states.

It follows from the inequalities (7) and (24) that the averages of other measures of nonclassical correlations satisfy

 ¯DM :=DA→BM+DA→CM+DA→DM+…n−1≤14. (25) ¯DG :=DA→BG+DA→CG+DA→DG+…n−1≤14, (26) ¯D :=DA→B+DA→C+DA→D+…n−1≤1√2, (27) ¯N :=NA:B+NA:C+NA:D+…n−1≤1√2. (28)

It is easy to see that the average amount of quantum correlations, shared by Alice and the other parties, is always upper bounded by half of the maximal amount for a bipartite system.

## 5 Conclusions

In summary, we have first investigated the strength ordering of different measures of quantum correlations and demonstrated that for two-qubit states, the nonlocal effect induced by measurements, MIN, is upper bounded by the maximal violation of the Bell-CHSH inequality as (5). Then, we have proven a tight monogamy relation of MIN (14) for all pure three-qubit states, which further gives an alternative derivation of known monogamy relations for negativity, quantum discord, and geometric quantum discord. Finally, we could obtain a chain of strong tradeoff relations of various measures (21) for general three-qubit states and the corresponding tradeoff relation (22) for any multi-qubit state.

It is the monogamy of quantum correlations that at least in parts underlies their usefulness as resources in quantum information processing, especially in cryptographic protocols (12). Our results may provide insight into exploiting this usefulness. It is still a rather complex problem to characterize and quantify nonclassical correlations in multi-party systems and thus it is also hoped that the further investigation of our work will help to expose the rich structures of quantum correlations. Finally, it would be of interest to study whether MIN based on relative entropy (36), trace-distance (37), and two-side form (38) satisfy corresponding monogamy relations or not.

###### Acknowledgements.
We thank Michael Hall for helpful comments. S. C. is supported by the ARC Centre of Excellence CE110001027, and Z. X. is supported by National Natural Science Foundation (NNSF) of China (Grant Nos. 61227902 and 61573343).

### Footnotes

1. email: shuming.cheng@griffithuni.edu.au
2. email: shuming.cheng@griffithuni.edu.au
3. email: shuming.cheng@griffithuni.edu.au

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