In a N-party Bell-type experiment, measurements on each spatially-separated particle are assumed to be performed with two distinct results (d distinct results for d-level Bell-type experiments) from two different observables. In each run of the experiment, each party chooses one observable for a simultaneous measurement on the particle in parallel. Let us denote the local measurement setting by M = (V1, V2, ..., VN), where Vi represents the observable chosen by the ith party. After measurements, a set of results, (v1, v2, ..., vN), where vi ∈ {0, 1} (vi ∈ {0, 1, ..., d − 1} for d-level Bell-type experiments), is acquired. If sufficient runs of such measurements have been made under the chosen local measurement setting, the correlation between experimental outcomes can be revealed through analytical analyses of experimental records. In analogy, experiments for bipartite multilevel systems work in the same way as mentioned above.
For quantum mechanical representation, we introduce an operator of the form
Vˆi = X1 vi=0
(−1)viˆvi, (2.1)
where ˆvi = |viiViVihvi| and {|viiVi} is a complete set of orthonormal basis vectors for the observable ˆVi. Each N-product operator of the form ˆV± = ±NN
i=1Vˆi can be represented explicitly by
Vˆ±= ˆC0±+ ˆC1±, (2.2)
where
= denotes equality modulus two. Expectation values of ˆV± for some physical states, denoted by h ˆV±i, are typically called N-point correlation functions. Here we will give a new insight into h ˆV±i via their elements ˆC0± and ˆC1±. Determining the expectation values of ˆC0± and ˆC1± can provide information about correlation between the subsystems composed of the first m qubits and the rest of the system.
Theorem 1. If measured outcomes show that expectation values of operators satisfy h ˆC0±i > 0 and h ˆC1±i > 0, or, h ˆC0±i < 0 and h ˆC1±i < 0, the outcomes of measurements performed on the subsystem of the first m qubits are correlated with the ones performed on the subsystem of the last N − m qubits [111].
Proof. If the subsystems are independent, we have the following relations
h ˆC0+i = (hˆ0mi − hˆ1mi)hˆ0m¯i, h ˆC1+i = (hˆ1mi − hˆ0mi)hˆ1m¯i, (2.6) and
h ˆC0−i = (hˆ0mi − hˆ1mi)hˆ1m¯i, h ˆC1−i = (hˆ1mi − hˆ0mi)hˆ0m¯i. (2.7)
Since hˆ0mi + hˆ1mi = 1, hˆ0m¯i ≥ 0, and hˆ1m¯i ≥ 0 for any physical systems, it turns out that h ˆC0+ih ˆC1+i ≤ 0 and h ˆC0−ih ˆC1−i ≤ 0. Thus a contradiction reveals the dependency of one subsystem on another one.
Then h ˆV±i is not just a N-point correlation function but a general one composed of h ˆC0,1±i that gives nc conditions of dependence for correlations between any two subsystems with m qubits and N − m ones, where
nc =
⌊N/2⌋X
k=1
f (N, k) N!
k!(N − k)!, (2.8)
f (N, k) = 2−δ[k,⌊N/2⌋] for even N, δ[·] denotes Kronecker delta symbol, and f(N, k) = 1 for odd one. Take N = 3 for example. A correlation function h ˆV1Vˆ2Vˆ3i involves three conditions, i.e., nc = 3, to describe correlations between subsystems including the fol-lowing classifications, {[1, 2, 3]}: [1|2, 3], [2|1, 3], and [3|1, 2], where [i|j, k] denotes the correlation between the ith qubit and the subsystem composed of the j th and kth ones.
For N qubits, we use the denotation {[1, 2, ..., N]} or {[m, ¯m]} to represent nc differ-ent kinds of partitions for correlation, and we sometimes use the notations ˆC0[m, ¯± m] and Cˆ1[m, ¯± m] emphasizing the correlations between two specific subsystems denoted by m and
¯
mrespectively.
By the same idea of constructing ˆC0,1± for qubits, we introduce the following sets of operators for two-qudit correlations:
Cˆk(q) = [ˆk − T (ˆk)] ⊗ U(ˆk), (2.9)
for k = 0, 1, ..., d − 1 and q = 1, ..., γd, where T and U are injective maps such that T (ˆk) 7→ ˆk′, U(ˆk) 7→ ˆk′′, and k′ 6= k, and each set {T (ˆk)} composed of T (ˆk)’s is numbered by q. Take d = 3 for example, we have two sets of {T (ˆk)} and hence the sets of operators
{ ˆCk(q)} could be
{ ˆC0(1) = (ˆ0 − ˆ1)ˆ0, ˆC1(1) = (ˆ1 − ˆ2)ˆ1, ˆC2(1) = (ˆ2 − ˆ0)ˆ2}, { ˆC0(2) = (ˆ0 − ˆ2)ˆ0, ˆC1(2) = (ˆ1 − ˆ0)ˆ1, ˆC2(2) = (ˆ2 − ˆ1)ˆ2}.
where U(ˆk) = ˆk is used in this example. For d = 2, we get the sets of operators for qubits introduced above: ˆC0(1) = ˆC0+ and ˆC1(1) = ˆC1+ for U(ˆk) = ˆk, or ˆC0(1) = ˆC0− and ˆC1(1) = ˆC1− for U(ˆk) = ˆk′′ and k′′6= k. Then it is clear that the number of sets { ˆCk(q)} depends on the number of {T (ˆk)}. For general d, we have γdsets of { ˆCk(q)}, where γ2 = 1, γ3 = 2, γ4 = 9, γ5 = 44, γ6 = 285, and
γd= (d − 2)
d−1Y
v=3
v + (d − 1)γd−2, (2.10)
for d ≥ 7. The correlation between outcomes of measurements performed on two remote qudits can be revealed by the help of the following theorem.
Theorem 2. If measured outcomes show each expectation value h ˆCk(q)i in the qth set {h ˆCk(q)i} is positive or each one is negative, the outcomes of measurements performed on the first qudit are correlated with the ones performed on the second qudit [112].
Proof. If the subsystems are independent, one can recast h ˆCk(q)i as
h ˆCk(q)i = (hˆki − hT (ˆk)i)hU(ˆk)i, (2.11)
Since P
khˆki = P
khT (ˆk)i = 1 and hU(ˆk)i ≥ 0, h ˆCk(q)i > 0 for all k’s is impossible for independent subsystems. Then a contradiction indicates the dependency of the first qudit on the second one.
With the above two theorems, one can feature a many-qubit or two-qudit entangled state in sets of correlation conditions proposed under different local measurement set-tings. These conditions can be considered as necessary ones for the entangled state under
study. We call the expectation values h ˆCk(q)i and h ˆC0,1± i correlators due to their utilities for correlations. We give three concrete examples to illustrate how correlators work for ana-lyzations of the correlation structures of given states and the basic idea of entanglement detections based on correlators:
(a) A two-qubit pure entangled state in the following representation:
|φi = sin(ξ) |00i + cos(ξ) |11i , (2.12)
for 0 < ξ < π/4, where |v1v2i = |v1i ⊗ |v2i and |vii is the eigenstate of Pauli-operator σz with eigenvalue (−1)vi , can be described by correlators that correspond to the operators Cˆ0Z = ˆC0+ = (ˆ0 − ˆ1)ˆ0 and ˆC1Z = ˆC1+ = (ˆ1 − ˆ0)ˆ1. By a direct calculation, one obtains the correlators h ˆC0Zi = sin2(ξ) and h ˆC1Zi = cos2(ξ) for the state |φi, which reveals the correlation properties when observed in the local measurement setting Mz = (Z, Z) where Z = σz. The state |φi can also be shown in another representation, e.g.,
|φi = a(|00iX + |11iX) + b(|01iX + |10iX), (2.13)
where a = [cos(θ) + sin(θ)]/2, b = [cos(θ) − sin(θ)]/2, and |viiX is an eigenstate of Pauli-operator σx with an eigenvalue (−1)vi. This representation provides the information of probability distribution for {|v1v2iX} when measured in the setting Mx = (X, X) where X = σx. From which, one can construct correlators, and the characters of correlation can be described by h ˆC0Xi = h ˆC0Xi = sin(2ξ)/2 where ˆC0X = ˆC0+ = (ˆ0 − ˆ1)ˆ0 and Cˆ0X = ˆC1+ = (ˆ1 − ˆ0)ˆ1.
(b) The probability distribution for |φi when measured with the setting Mz is the same as the one of the following mixture of product states:
ρφ= sin2(ξ) |00i h00| + cos2(ξ) |11i h11| . (2.14)
Then we have the correlators h ˆC0Ziρφ = sin2(ξ) and h ˆC1Ziρφ = cos2(ξ) and know outcomes
of measurements for these particles are dependent. When the state ρφ is represented in the basis {|v1v2iX}, the probability for observing an element in {|v1v2iXXhv1v2|} of ρφ is 1/4, which implies that these two particles are independent. This fact can be shown by the correlators h ˆC0Xi = h ˆC1Xi = 0.
From the above examples, one has P
l=X,Z which, it is worth noting that determining a sum of correlators associated with two dif-ferent local measurement settings can help us to distinguish the entangled state |φi from the separable state ρφ. This idea and approach can be applied to detections of truly many-qubit entanglement and bipartite entangled qudits. For any many-qubit system composed of two independent parts, outcomes of measurements should satisfy
for any measurement settings chosen. Whereas, for some specific entangled states, one can feature properties of entanglement to be created in |P
kh ˆCk[m, ¯± m]i| = 1 for several local measurement settings chosen and consider which as necessary conditions for the entangled state. Furthermore, we could give all conditions of correlations [m, ¯m] associated with any two subsystems of the many-qubit entangled state under study. Thus we can use these conditions of genuine many-qubit entanglement to rule out biseparable correlations. For two independent qudits observed under any measurement settings, a sum of correlators should follow the criteria con-sidered can be very useful to detect entangled qudit pairs. Using the idea introduced above can promote constructions of many-qubit and two-qudit entanglement witness op-erators that require only two local measurement settings. Even though the conditions
|P
kh ˆCk[m, ¯± m]i| = 1 and |P
kh ˆCk(q)i| = 1 cannot be satisfied by all entangled states con-sidered, the above approach still can be applied to entanglement detections if more local measurement settings are used. See the case discussed in the second subsection of Sec.
2.5.
(c) The state vector of a two-qubit singlet state is represented by
|ψi = 1
√2(|01i − |10i). (2.17)
If ˆV1 ∈ { ˆV1(1) = Z, ˆV1(2) = X} and ˆV2 ∈ { ˆV2(1) = −(Z + X)/√
2, ˆV2(2) = (Z − X)/√ 2}, we have four different local measurement settings M = ( ˆV1, ˆV2) to give four sets of correlators.
The operators of correlators are as follows: ˆC0(rt) = ˆC0+ = (ˆ0 − ˆ1)ˆ0, ˆC1(rt) = ˆC1+ = (ˆ1 − ˆ0)ˆ1 for (rt) ∈ {(11), (21), (22)} and ˆC0(12) = ˆC0− = (ˆ0 − ˆ1)ˆ1, ˆC1(12) = ˆC1− = (ˆ1 − ˆ0)ˆ0 , where the superscripts (rt) mean an observable ˆV1(r) and another one ˆV2(t) are chosen for measurements. The correlators can be easily calculated, and then we have h ˆC0(rt)i = h ˆC1(rt)i = 1/2√
2. When collecting all of the correlator operators proposed above, one gets
B =
Local-realistic theories predict that B ≤ 2, which is called the CHSH inequality [13], whereas the entangled state |ψi predicted by quantum mechanics provides a violation by P
rth ˆC(rt)i = 2√
2. It is remarkable that the kernel of the CHSH inequality [13] is com-posed of necessary conditions of the state |ψi in terms of the correlators (h ˆC0(rt)i, h ˆC1(rt)i).
In what follows, we will use correlators to analyze the most studied many-qubit and two-qudit entangled states: the N-qubit GHZ state [113] and the two-qudit Bell state.
The correlators proposed are necessary for states to be the entanglement under study and play important roles in identifying quantum correlations including ruling out biseparable correlations and ones predicted by local-realistic theories.