Detecting entangled qudits with two local measurement settings

Two-qudit Bell state

It is an important and interesting question whether one can construct an entanglement witness operator to detect a truly many-qudit entanglement without using much experi-mental effort. Instead of investigating this subject, we will show one can detect multi-level entanglement for states in the proximity of a d-level Bell state with two local measurement settings, which is preliminary to the previous question.

We use the correlators introduced in the fourth specification of |Ψi to construct an entanglement witness operator with a highly robustness, and with the fact that

Xd−1

the basic idea relies on the strategy introduced in examples (a), (b), and related discussions in Sec. 2.2. The kernel of our witness is of the form:

CˆΨ4 = ˆC + ˆCF, (2.93)

Note that the representation of the projector ˆk’s in ˆC is different from the one in ˆCF.

Then our witness operator is

W^Ψ4 = αΨ41− ˆCΨ4, (2.96)

where αΨ4 = γdand 1 denotes an identify operator with d² dimensions. If a state ρ shows Tr (W^Ψ4ρ) < 0, ρ is identified as an entanglement close to the state |Ψi. Especially, the witness W^Ψ4 is very robust. The robustness of W^Ψ4 is determined by the noise tolerance:

pnoise < δnoise, is such that

ρ = p_noise1/d²+ (1 − pnoise) |Ψi hΨ| (2.97)

is identified as an entanglement. The witness W^Ψ4 tolerates noise if pnoise < 1/2, inde-pendent of the number of levels.

Next, we will show that the operator W^Ψ4 is a witness. In order to achieve this aim, we compare WΨ4 with a project-based witness operator of the form:

WΨ^p = α^p_Ψ1− |Ψi hΨ| , (2.98)

where α^p_Ψ = 1/d [70]. If measured outcomes show that Tr(WΨ^pρ) < 0, the state ρ is identified as an entanglement close to |Ψi. Then one has to show if ρ satisfies Tr(WΨ4ρ) <

0, it also satisfies Tr(WΨ^pρ) < 0, i.e., W^Ψ4 − γ^Ψ4WΨ^p ≥ 0 where γ^Ψ4 is some positive constant. Let us consider the operator W = W^Ψ4 − dγ^d/(d − 1)WΨ^p. To diagonalize W , we propose a complete basis {|Ψ^kvi}, where

|Ψ^kvi = 1

√d Xd−1 v^′=0

exp(i2πkv^′/d) |v^′i ⊗ |v^′+ vi , (2.99)

for k, v = 0, 1, ..., d − 1, where the addition of v^′ and v is modulo d. Since both of WΨ4

and WΨ^p are diagonal in this basis, W is also diagonal. The diagonal elements of W in

this basis can be calculated analytically, and then we have

hΨ^kv| W |Ψ^kvi = d

d − 1γd, (2.100)

for k ≥ 1 and v ≥ 1, and hΨ^kv| W |Ψ^kvi = 0 otherwise. This proves our claim.

Entangled state comprised of subsystems with different dimensions

We proceed to give another witness to detect an entangled sate composed of a qutrit and a ququat (quantum four-level system) of the state vector:

|ǫi = 1

√3(|0i ⊗ |0i + |1i ⊗ |1i + |2i ⊗ |3i). (2.101)

where the kets on the left-hand side of tensor products denote single qutrits and form an orthonormal basis: {|0i , |1i |2i}, and the kets on the right-hand side of tensor products are ququants described by an orthonormal basis {|0i , |1i , |2i , |3i}. When |ǫi is in the above representation, one can easily derive two set of correlator operators to describe correlations from the knowledge of the state vector:

Cˆ₀⁽¹⁾ = (ˆ0 − ˆ1)ˆ0, ˆC₁⁽¹⁾ = (ˆ1 − ˆ2)ˆ1, ˆC₂⁽¹⁾ = (ˆ2 − ˆ0)ˆ3,

Cˆ₀⁽²⁾ = (ˆ0 − ˆ2)ˆ0, ˆC₁⁽²⁾ = (ˆ1 − ˆ0)ˆ1, ˆC₂⁽²⁾ = (ˆ2 − ˆ1)ˆ3. (2.102)

Each correlator proposed above is h ˆC_k^(q)i = 1/3. Similarly, by the knowledge of the state vector given by

|ǫi = 1

√3(|0iF ⊗ |0^′iF + |1iF ⊗ |1^′iF + |2iF ⊗ |2^′iF), (2.103)

where

|0^′iF = 1 2√

3(3 |0iF + |1iF − |2iF + |3iF),

|1^′iF = 1

6[(−3 +√

3) |1iF + 2√

3 |2iF + (3 +√

3) |3iF],

|2^′iF = 1

6[(3 +√

3) |1iF + 2√

3 |2iF + (−3 +√

3) |3iF], (2.104)

and |viF are defined by Eq. (2.50), we give the second type of correlator operators:

Cˆ_0F⁽¹⁾ = (ˆ0 − ˆ1)ˆ0^′, ˆC_1F⁽¹⁾ = (ˆ1 − ˆ2)ˆ1^′, ˆC_2F⁽¹⁾ = (ˆ2 − ˆ0)ˆ2^′,

Cˆ_0F⁽²⁾ = (ˆ0 − ˆ2)ˆ0^′, ˆC_1F⁽²⁾ = (ˆ1 − ˆ0)ˆ1^′, ˆC_2F⁽²⁾ = (ˆ2 − ˆ1)ˆ2^′. (2.105)

Each correlator is h ˆC_kF^(q)i = 1/3. Therefore, our witness consists of all of the correlator operators introduced above is

W^ǫ = αǫ1− ˆCǫ, (2.106)

where αǫ = 2 and ˆCǫ =P

q,kCˆ_k^(q)+ ˆC_kF^(q).

We proceed to prove the operator W^ǫ is a witness. To attain this aim, we have to compare W^ǫ with the following projector-based witness

Wǫ^p= α^p_ǫ1− |ǫi hǫ| , (2.107)

where α^p_ǫ = 1/3 [70]. With the whole knowledge of |ǫi, the witness Wǫ^p can be used to identify a state ρ as the one close to |ǫi if Tr(Wǫ^pρ) < 0. We find that W^ǫ− 3Wǫ^p≥ 0 and from which we deduce that if Tr(W^ǫρ) < 0 then Tr(Wǫ^pρ) < 0 also applies to ρ. Thus W^ǫ is an entanglement witness operator. In addition, W^ǫ is very robust against noise. When a pure state |ǫi is suffered from white noise, the mixed sate is identified as entanglement if the noise fraction is less than 0.5.

Four-qubit GHZ state shared by two parties

In the previous cases of entanglement detections for qubits, each party of a system has access to perform measurements on only one qubit. It is natural to ask how to design a strategy of detections of genuine multipartite entanglement if each party has access to measure more than one qubits in a Bell-type experiment. For instance, how to construct an entanglement witness operator for detecting a four-qubit GHZ state which is shared by two parties? Since more information about nonlocal properties of the GHZ state can be acquired via measurements, it is interesting to investigate the difference between the new witness and the previous one.

We will present a witness which requires only two local measurements to attain the aim mentioned above. First, let us assume two individual pairs of qubits of a four-qubit GHZ state are shared by two parties respectively, and then each party can perform two-qubit measurements on the qubits. For the first measurement setting, we use the correlator operators introduced in the third specification for the generalized GHZ state: ˆC0Z[m, ¯m]

and ˆC1Z[m, ¯m], i.e.,

Cˆ_{0Z[m, ¯}^m_]= (ˆ0ˆ0 − ˆ1ˆ1)ˆ0ˆ0, ˆC_{1Z[m, ¯}^m_]= (ˆ1ˆ1 − ˆ0ˆ0)ˆ1ˆ1. (2.108)

A four-qubit GHZ state of the form

|Φi = 1

√2(|00i ⊗ |00i + |11i ⊗ |11i),

gives h ˆC0Z[m, ¯m]i = h ˆC1Z[m, ¯m]i = 1/2. For the second measurement setting, we propose the operators

Cˆ_{0F [m, ¯}^m_]= (0^m_F − 1^mF)0m¯F, ˆC_{1F [m, ¯}^m_]= (1^m_F − 0^mF)1m¯F. (2.109)

where

Since a comparison between W^Φ2 and the projector-based witness WΦ^p satisfies W^Φ2− 2WΦ^p ≥ 0, WΦ2is a witness for detecting truly four-qubit entanglement for states close to

|Φi.

The witness W^Φ2is very robust against noise. The noise tolerance of W^Φ2is δΦ2= 1/2, and then W^Φ2is more robust than the witness W^Φ, Eq. (2.82), with δΦ = 4/11 ≃ 0.3636.

In other word, W^Φ2 based on two-qubit measurements for each party can detect more states in the proximity of |Φi and is finer than W^Φ.

Let us consider the above example in another way. We define each pair of qubits as a single ququat, and then the four-qubit GHZ state can be represented by

|Φi = 1

√2(|0i ⊗ |0i + |3i ⊗ |3i),

where |2v + v^′i = |vv^′i for v, v^′ = 0, 1. Therefore constructions of correlator operators for a four-qubit GHZ state is equivalent to the ones for a two-ququat entangled sate of the above form, and then one can observe that each vector in the orthonormal basis {|vv^′iF} chosen in the second measurement setting is derived from a vector in the basis {|2v + v^′i}

which is transferred by a single-ququat Fourier transformation. In this situation, further

two questions arise. What are the constructions of correlator operators for a N-ququat entangled state of the form: |Φi = (|0i^⊗N + |3i^⊗N)/√

2? Are the witnesses based on correlators finer than W^Φ? The investigations on these questions are the future works.

We proceed to consider another situation where one party has three qubits and another party has the rest of a four-qubit GHZ state. First, we follow the method just discussed and substitute eight-level state vector |4v + 2v^′+ v^′′i for three-qubit one |vv^′v^′′i to express

|Φi as

|Φi = 1

√2(|0i ⊗ |0i + |1i ⊗ |7i),

where the kets on the left-hand side of the tensor products in the above equation denote single qubits and constitute an orthonormal basis {|0i , |1i}, whereas the right-hand ones are three-qubit elements of the orthonormal basis {|0i , |1i , ..., |7i}. Then we give the following correlator operators for states shown in this representation:

Cˆ0 = (ˆ0 − ˆ1)ˆ0, ˆC1 = (ˆ1 − ˆ0)ˆ7, (2.112)

easily derived from the knowledge of the state vector and are given by

Cˆ0F = (ˆ0 − ˆ1)ˆ0^′, ˆC1F = (ˆ1 − ˆ0)ˆ1^′. (2.114)

Then our witness is

W^Φ3= 1 − X1 k=0

Cˆk+ ˆCkF, (2.115)

and fulfills the condition WΦ3 − 2WΦ^p ≥ 0 for detecting truly four-qubit entanglement.

The noise tolerance of WΦ3 is δ_Φ3= 1/2 and is also superior to the one of WΦ.

2.5.4 Witnesses composed of the kernels of Bell inequalities for