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A multi-party correlation measure based on cumulant

We propose a genuine multi-party correlation measure for a multi-party quantum system as the trace norm of the cumulant of the state. The legitimacy of our multi-party correlation measure is explicitly demonstrated by proving it satisfies the five basic conditions required for a correlation measure. As an application we construct an efficient algorithm for the calculation of our measures for all stabilizer states.


INTRODUCTION
Although a composite system can contain several parties, its physical properties do not necessarily equal to the sum of its parts. Correlations among different parts in a composite system are usually invoked to describe the difference between the physical properties of a composite system and the sum of its parts. Roughly speaking, the physical properties for a composite system are reflected both in the sum of the physical properties of its parts and the correlations (among different parts) of the composite system. What makes a composite system more interesting in some sense can be largely attributed to the existence of correlations among the constituents of the system.
The rapid development of quantum information science in recent years has called for serious efforts to characterize correlations in a composite quantum system. Depending on whether quantum nonlocal resources are needed in the preparation of a quantum state, the correlations among different parties in a composite system can be further classified into classical and quantum correlation [1].
In the language of quantum information science, quantum correlation is often called quantum entanglement to emphasize the inseparability of the quantum state of a composite system into those of its parties. Quantum entanglement is widely believed to be a useful resource in implementing quantum computation and information tasks [2].
The aim of this article is to provide a genuine multiparty correlation measure for arbitrary quantum states. Our choice for the genuine multi-party correlation measure to be shown and justified below is intimately related to the cumulant (or the Ursell expansion) of a multi-party density matrix [7,8,9,10]. The research on the cumulant of a multi-party state has had a long history. Cumulants were first introduced by T. N. Thiele in 1899, who called them half-invariants [11]. The name of cumulant is given by R. Fisher and J. Wishart in 1931 [12]. It was first introduced by Ursell into classical physics in 1927 [13], and Kahn and Ulenbeck gave the quantum mechanical treatment in 1938 [14].
The cumulant of a multi-party density matrix was known to be related to multi-party correlation, and was even called the correlation operator [15]. However, it does not provide a legitimate measure to compare the correlations in different quantum states.
This article is arranged as follows. In Sec. II, we outline three preliminaries: First, we will introduce the trace distance between two Hermitian operators of the same trace. Second, the cumulant of a multi-party density matrix is introduced in an instructive way. Third, the conditions for a legitimate multi-party correlation measure are discussed. In Sec. III, we define our multi-party correlation measure as the trace norm of the cumulant of the multi-party density matrix. We further prove that it is a legitimate multi-party correlation measure by explicitly demonstrate that it satisfies the five basic conditions for a multi-party correlation measure. We then apply our correlation measure to study three cases: simple two-party states, three-party states, and more general multi-party stabilizer states [16,17]. Finally, we summarize our results with a discussion.

PRELIMINARIES
This section contains the three preliminaries for a convenient introduction of our correlation measure. First, we generalize the trace distance between two quantum states to the trace distance between two Hermitian operators with the same trace. An important property of this distance is briefly reviewed, which constitutes the key element in proving the legitimacy of the proposed correlation measure. Second, we briefly review the concept of the cumulant for a multi-party density matrix in an instructive way. Third, we discuss the basic requirements for a legitimate multi-party correlation measure.

Trace distance
The trace distance between two Hermitian operators ρ and σ with the same trace is defined by This definition is a simple extension of the trace distance between two quantum states [2]. Note that there is an alternative useful expression for the trace distance: where the maximum is taken over all projectors P . It is easy to check that it satisfies the three basic requirements for a distance, i.e., 1. D(ρ, σ) = 0 ⇔ ρ = σ, 2. D(ρ, σ) = D(σ, ρ), 3. D(ρ, τ ) ≤ D(ρ, σ) + D(σ, τ ). The trace distance has the following important property, which is listed here as a theorem.
Theorem 1: (Trace-preserving quantum operations are contractive) Suppose E is a trace-preserving quantum operation. Let ρ and σ be Hermitian operators with the same trace. Then This theorem is a simple generalization of the wellknown theorem where ρ and σ are density operators [2]. It plays an important role in proving our correlation measure is legitimate. The proof of this theorem is omitted here because its proof is almost the same to that of the original theorem, which can be found in several reference books.

Cumulants
We introduce an instructive way to understand the cumulant for an N -party density matrix ρ (12...N ) . Suppose we know all the (N − 1)-party reduced density matrices, we can construct a pseudo-N -party density matrix ρ (12···N ) such that all of its reduced density matrices are correct. Then the cumulant of ρ (12...N ) is defined by The pseudo-N -party density matrixρ (12···N ) can be directly constructed by the following method. First, we find all the partitions of T N ≡ {12 · · · N }. Any partition can be denoted by where a {Si} are constants dependent on the specific partition {S i }. Third, all the constants can be determined by the conditions Tr iρ (12···N ) = Tr i ρ (12···N ) , or To determine the constants a {Si} in Eq. (7), we will use the cumulants to expand the density matrix, which is the so-called the Ursell expansion of the density matrix [9]. Let us denote ρ (1) = C (1) and C(ρ (12···N ) ) = C (12···N ) . From Eqs. (6) and (7), the density matrix where the sum is taken over all the partitions {S i } of S with S ⊆ T N . Using the condition (8) and the mathematical induction method, we can prove that the constants b {Si} = 1. Using Eq. (9), we obtain the unique solution of the constants in Eq. (7) satisfying the condition (8): [7,8,9]. The cumulant of an N -party density matrix has the following important property.

Conditions of a genuine multi-party correlation measure
From a general physical consideration, a genuine Nparty correlation measure M C (ρ (12···N ) ) should satisfy the following five conditions [5].
1. Negative correlation has no physical interpretation.
2. Any product state implies no genuine N -party correlation. If an N -party density matrix 4. The correlation measure is invariant when the system is augmented by locally non-correlated auxiliary sub-systems.
Here we emphasized that condition 2 is stronger than the following condition N -product version, As we have mentioned in the introduction, this condition can be used to define the total correlation, which includes different types of correlations in the state.
We include the extra condition 4 as a general requirement for a legitimate correlation measure because correlations in a system should not depend on the rest of the world or ancillary systems (σ (12···N ) L ) if they are independent and uncorrelated (σ ). An optional requirement for a legitimate correlation measure is related to the so-called additivity, i.e., requiring 12···N ) ) for an absolute correlation scale. This additivity requirement is clearly stronger than our proposed condition 4. We feel such a strong condition is not needed as argued previously in the basic requirements of twoparty entanglement measure [4].

A GENUINE MULTI-PARTY CORRELATION MEASURE
In this section, we present our central result of a genuine multi-party correlation measure. It is first proposed and further proved to satisfy the aforementioned five basic requirements for a multi-party correlation measure, thus it constitutes a legitimate multi-party correlation measure. We end this section by demonstrating the applications of our proposed measures to several important class of examples.

General formalism
Definition: An N -party correlation measure of the state ρ (12···N ) is proposed as which constitutes a legitimate genuine multi-party correlation measure because of the following theorem, which is the main result of our paper. Main Theorem: M C (ρ (12···N ) ) is a legitimate Nparty correlation measure, i.e., it satisfies the five basic conditions for an N -party correlation measure.
Proof : Let us prove the five conditions respectively.
Note that we have used Theorem 2 in the first step of this proof.
3. Under the action of U L , where U Si = j∈Si U j . Using the expression (7) for the psedo-density matrix, we obtaiñ Thus, the cumulant Therefore, 4. It is easy to prove that 5. Under the action of E L , where E Si = j∈Si E (j) . Using the expression (7) for the psedo-density matrix, we obtaiñ 12···N ) ). Therefore, Note that Theorem 1 was used for the proof of the last inequality.

Two-party correlation
The two-party correlation measure is defined as The physical meaning of this measure is the distance between the state ρ (12) and its reduced product state ρ (1) ρ (2) [2]. Let us apply the two-party correlation measure to the following two typical states. The first state is the maximally classical correlated two-qubit state we find the two-party correlation according to our measure is given by M C (ρ (12) c ) = 1/2. The second state is the maximally entangled Bell state for which our two-party correlation measure gives M C (|B (12) (12) B|) = 3/4. The unique difference between the Bell state (16) and the maximally classical correlated two-qubit state (15) is the very existence of quantum coherence in |B (12) . Our result shows that quantum coherence will increase the two-party correlation [18].

Three-party correlation
When dealing with three-party systems, our correlation measure becomes We again study several types of typical three-qubit states. Analogously, the first state is the maximally classical correlated three-qubit state for which our three-party correlation measure vanishes, i.e., M C (ρ (123) c ) = 0. The second state we consider is the GHZ state [19] for which our three-party correlation measure gives M C (|G (123) (123) G|) = 1/2. These two specific examples indicate that just like the case of two-parties, quantum coherence generally increases the three-party correlation. More specifically, we note that M C (ρ (123) c ) = 0, i.e., there exists no genuine three-party correlation in this state. Yet, even according to our definition of separate states included in the condition 2, this state is NOT really a non-correlated state. This example thus shows that even for a threequbit state, the first of the five conditions becomes a sufficient but not necessary condition, i.e., if the correlation measure M C (ρ (123) ) is zero, we do not know for sure if this state is non-correlated or not. Contrary to this for a mixed state, however, we find that for a three-qubit pure state, the following theorem is valid.
Theorem 3: If ρ (123) is a three-qubit pure state, and the correlation measure M C (ρ (123) ) = 0, then the state is non-correlated, i.e., it can be written as a direct product of two density matrixes of mutually independent parts. The proof of this theorem is attached as an appendix. Remarkably, this theorem implies the other typical three-qubit pure state, the so-called W state [20] possesses genuine three-qubit correlation.

Multi-party correlations in the N -qubit GHZ state
Let us study the correlations in the N -qubit GHZ state, which is defined by 1···N ) . (20) An analogous classically correlated state based on what we considered earlier is Since all reduced density matrices of the above two states are exactly the same, the difference of their respective cumulants simply consists of the off-diagonal terms Furthermore, the cumulant of the state ρ (N ) c is given by where ρ even = i ai∈even |a 1 · · · a N (1···N ) (1···N ) a 1 · · · a N |, and the coefficient Specifically, for any odd number N , c N = 0. Therefore for any odd number N , the N -party correlation measure gives which for |c N | ≥ 1/2, gives C M (|GHZ (N ) ) = C M (ρ (N ) c ). In Fig. 1, we show the N -dependence of the above correlation measures for the two states. Two interesting features are worthy of some attention. First, it is interesting to note that C M (|GHZ (N ) ) = C M (ρ (N ) c ) for N ≥ 8 because |c N | ≥ 1/2 for N ≥ 8. The physical meaning of this interesting equality is yet to be understood. Secondly, the exponentially increasing dependence of the total correlations on N in both cases reflects the exponentially increasing size of the Hilbert space.

Multi-party correlations in stabilizer states
The correlations (or entanglement) in stabilizer states have been discussed in Refs. [22,23,24]. We now apply our multi-party correlation measure (10) to characterize these correlations.
To compute our measure of Eq. (10) for a given state, we need firstly to obtain the corresponding reduced density matrices. This task is dramatically reduced if the given state is a stabilizer state. A stabilizer state is the unique simultaneous eigenstate of the maximal Abelian subgroup of the N -qubit Pauli group G N . The group G N consists of all 4 × 4 N local operators of the form M = α M M 1 ⊗ · · · ⊗ M N , where α M ∈ {±1, ±i} is an overall phase factor and M i is either the 2 × 2 identity matrix I i or one of the Pauli matrices X i , Y i , Z i .
A stabilizer S in the Pauli group G N is defined as an Abelian subgroup of G N which does not contain −I. A stabilizer consists of 2 k Hermitian Pauli operators (i.e. they must have real overall phase factors ±1), where k (≤ N ) is the number of the generators of S. As the operators in a stabilizer commute, they can be diagonalized simultaneously and, what is more, if k = N then there exists a unique state |ψ on N qubits such that M |ψ = |ψ for every M ∈ S. Such a state |ψ is called a stabilizer state and the group S = S(ψ) is called the stabilizer of |ψ . The expansion which describes a stabilizer state as a sum of all elements in its stabilizer, can be readily verified.
The support supp(M ) of an element M = α M M 1 ⊗ . . . ⊗ M n ∈ S(ψ) is the set of all i ∈ {1, . . . , n} such that M i differs from the identity I i . Let S be a subset of T N . Tracing out all qubits of |ψ outside S gives the reduced density matrix ρ (S) (ψ) associate with S, which is equal to This can easily be verified using the identity (29). This allows use to obtain the cumulant of the state by direct computation, with the result taking the following form where β M are real constants, and β M = 0 when M = N i=1 I i . The eigenvalues of the cumulant C(|ψ ψ|) can be obtained as follows. First we take a generator S G of the stabilizer group S, which includes n independent elements in S, denoted by G i (i = 1, 2 · · · , n). Therefore, our multi-party correlation measure for a stabilizer state becomes Without any complication, the above procedure we describe is easily extended for different partitions of the stabilizer states, or a reduced density matrix of the stabilizer states.
Let us demonstrate the above procedure with the three-qubit stabilizer state (N = 3), or the equivalent three-qubit GHZ state as considered earlier in Eq. (19). When viewed as a stabilizer state, its generators can be taken as follows.
The elements in the stabilizer group S can be written as where a 1 , a 2 , a 3 ∈ {0, 1}. So the three-party density matrix is The two-party density matrices are and the one-party density matrices are The cumulant thus becomes and our three-party correlation measure is given by the same result as obtained earlier in the line after Eq.
based on a direct calculation.

DISCUSSION AND CONCLUSION
We proposed a multi-party correlation measure as the trace norm of the cumulant of the multi-party density matrix, as defined in Eq. (10). A natural question arises: Is this a unique multi-party correlation measure based on the cumulant? For example, by replacing the trace norm of Eq. (10) by a squared norm, can we obtain an alternative measure which then can be directly written as a sum of a complete set of squared correlation functions. We find that M ′ C (ρ (12···N ) ) satisfies the condition 1, 2, and 3 of a multi-party correlation measure. In particular, it is invariant under local unitary transformations, thus it is appropriate to call M ′ C (ρ (12···N ) ) a local unitary invariant multi-party correlation function (LUI-MCF). We do not know whether the LUI-MCF satisfies the condition 5, i.e., whether M ′ C is non-increasing under general local operations. Unfortunately, We find it does not satisfy the additional condition 4 proposed by us, thus the LUI-MCF is not a legitimate correlation measure.
As mentioned in the introduction, a total correlation was previously defined by Vedral to measure the total correlation in a multi-party quantum state [6]. Our work above also suggests an alternative total correlation measure as the distance between the quantum state and its reduced completely non-correlated state, i.e.
It is easy to check that it satisfies the conditions 1, 2 ′ , and 3 − 5, but does not satisfy the condition 2, which implies it is indeed a legitimate total correlation measure. We expect our correlation measure will find applications not only in quantum information science but also in many-body physics. This expectation is based on the observation that the usual correlation functions cannot characterize general correlations in a multi-party quantum state. On one hand, our correlation measure recovers to the usual correlation function when there is only a single nonzero correlation function. On the other hand, our correlation measure satisfies the basic general requirements for a legitimate correlation measure, which implies they will faithfully characterize the multi-party correlation in a quantum state. To our knowledge, this is for the first time a multi-party correlation measure capable of capturing genuine multi-party correlation of a multi-party quantum state is defined.
From a theoretical viewpoint, one open problem is how to extract the quantum part of the correlation from our correlation measure. This will then provide a measure of multi-party entanglement. In addition, as an open question for further investigation, it will be interesting to find out how our correlation measure is related to quantum entanglement measures [23,24,25] of interest in quantum information science. In many-body physics, we are especially interested in finding out what is really responsible for the quantum phase transition: Is it the quantum correlation, classical correlation, or the total correlation [26,27,28,29].
In summary, we have proposed a multi-party correlation measure based on the cumulant of multi-party density matrix, which is capable of characterizing genuine multi-party correlation. We proved that our correlation measure is a legitimate multi-party correlation because it satisfies the five basic requirements for a multi-party correlation measure. The fourth requirement is suggested by us based on a physical and operational considerations of multi-party correlation. As an application, we find an efficient algorithm to compute the multi-party correlations for all stabilizer states.
We acknowledge interesting discussions with Prof. C.P. Sun and Mr. B. Sun. This work is supported by NSF and CNSF.
We first consider the element According to the symmetry of the state, we obtain