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Distilling perfect GHZ states from two copies of non-GHZ-diagonal mixed states

MARK

⁎

Xin-Wen Wanga,b,c, , Shi-Qing Tanga, Ji-Bing Yuana,d, Deng-Yu Zhanga a

College of Physics and Electronic Engineering, Hengyang Normal University, Hengyang, 421002 China Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Hunan Normal University, Changsha, 410081 China c Synergetic Innovation Center for Quantum Eﬀects and Applications, Hunan Normal University, Changsha, 410081 China d CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Beijing, 100190 China b

A R T I C L E I N F O

A BS T RAC T

Keywords: Entanglement puriﬁcation Non-GHZ-diagonal states Non-asymptotic way Perfect output states

It has been shown that a nearly pure Greenberger–Horne–Zeilinger (GHZ) state could be distilled from a large (even inﬁnite) number of GHZ-diagonal states that can be obtained by depolarizing general multipartite mixed states (non-GHZ-diagonal states) through sequences of (probabilistic) local operations and classical communications. We here demonstrate that perfect GHZ states can be extracted, with certain probabilities, from two copies of non-GHZ-diagonal mixed states when some conditions are satisﬁed. This result implies that it is not necessary to depolarize these entangled mixed states to the GHZ-diagonal type, and that they are better than GHZ-diagonal states for distillation of pure GHZ states. We ﬁnd a wide class of multipartite entangled mixed states that fulﬁll the requirements. Moreover, we display that the obtained result can be applied to practical noisy environments, e.g., amplitude-damping channels. Our ﬁndings provide an important complementarity to conventional GHZ-state distillation protocols (designed for GHZ-diagonal states) in theory, as well as having practical applications.

1. Introduction As we know, quantum entanglement is the key resource in quantum information science. That is, implementation of many quantum information processing (QIP) tasks relies on the existence of entanglement between quantum systems (or diﬀerent types of degree of freedom of a system) [1,2]. Generally, perfect achievements (e.g., with unit success probability) of most QIP tasks require the use of certain bipartite or multipartite maximally entangled pure states. In reality, however, those states will not be available with unit ﬁdelity, due to unavoidable quantum decoherence resulting from entangled particles interacting with environments or being sent through noisy quantum channels [3]. So, it is a central problem to establish methods to increase the ﬁdelity of entangled mixed states (relative to a ceratin maximally entangled pure state) by some means. Ideally, the ability of enhancing the ﬁdelity of entangled states to unit is desired. Entanglement distillation or puriﬁcation, at least in principle, provides a possible method to accomplish this task [4–9]. Entanglement distillation protocols (EDPs) function as distilling a few high-ﬁdelity entangled states from many low-ﬁdelity entangled states by using local operations and classical communication (LOCC). ⁎

Using eﬃcient EDPs, one could obtain an entangled state with near unit ﬁdelity from a large (even inﬁnite) number of copies of general entangled mixed states [8]. Extracting an entangled pure state from two or ﬁnite copies of especial entangled mixed states is also possible [10,11]. Generally, puriﬁcation of multipartite entangled mixed states is far more diﬃcult than that of bipartite entangled mixed states. This is due to the fact that there are lots of LOCC-inequivalent multipartite entangled states and puriﬁcation schemes for diﬀerent classes of noisy entangled states are usually not the same [12–15]. Greenberger–Horne–Zeilinger (GHZ) state [16] is a peculiar type of multipartite entangled state, and has widespread applications in quantum information ﬁeld (see e.g., [17–23]). Distillation of GHZ states has attracted considerable interest [24–32]. It has been shown that a general multipartite mixed state can be depolarized to a state which is diagonal in the GHZ-state basis (i.e., GHZ-diagonal state), by a sequence of probabilistic LOCC operations [26,27]. With the GHZdiagonal states as input states, two kinds of puriﬁcation protocols have been proposed [24–28]. One is the so-called recurrence protocol, in which the ﬁdelity of GHZ-diagonal states (relative to a pure GHZ state) can be improved step by step as long as the ﬁdelity of the initial input state is over a threshold. In each step, i.e., each round of distillation

Corresponding author at: College of Physics and Electronic Engineering, Hengyang Normal University, Hengyang 421002, China. E-mail addresses: xww[email protected] (X.-W. Wang), [email protected] (D.-Y. Zhang).

http://dx.doi.org/10.1016/j.optcom.2016.12.002 Received 6 November 2016; Received in revised form 27 November 2016; Accepted 1 December 2016 0030-4018/ © 2016 Published by Elsevier B.V.

Optics Communications 392 (2017) 185–189

X.-W. Wang et al.

Considering two copies of ρ, i.e., ρ⊗2 . All parties apply local controlled-NOT (CNOT) operations to their particles, with the particle belonging to one copy as control, the other as target. Making use of the deﬁnition of the CNOT gate |i〉|j〉 = |i〉|i ⊕ j〉 with ⊕ denoting addition modulo 2, one can readily check that the action of such a multilateral CNOT (MXOR) operation is given by

procedure, two or more copies of a GHZ-diagonal state will be sacriﬁced for increasing the ﬁdelity of one copy of the GHZ-diagonal state. In principle, one can ﬁnally get a nearly pure GHZ state with arbitrarily high ﬁdelity in an asymptotic way by sacriﬁcing even inﬁnite number of copies of the initial GHZ-diagonal state. The other kind of puriﬁcation protocol is the hashing (or breeding) protocol that is capable to produce a nearly pure GHZ state as an output state with nonzero yield by collectively manipulating a large number of GHZdiagonal states, provided the initial ﬁdelity is suﬃciently high. In this paper, we show that it may be better to directly distill a GHZ state from general multipartite entangled mixed states (non-GHZ-diagonal states) when certain conditions are satisﬁed. That is to say, it is not necessary to transform non-GHZ-diagonal states to the GHZ-diagonal type by formidable probabilistic LOCC before carrying out the GHZ-state distillation protocol in some cases. Particularly, we demonstrate that a perfect GHZ state can be simply obtained, with a certain probability, from two copies of a non-GHZ-diagonal state with certain characteristics. The paper is organized as follows. We ﬁrst give in Section 2.1 the suﬃcient conditions that the multipartite mixed state should satisfy such that a perfect GHZ state can be extracted from two copies of it by a simple EDP, and then present in Section 2.2 a recycle scheme for improving the distillation eﬃciency. Subsequently, we construct a wide class of quantum states that fulﬁll these requirements in Section 3.1, and take an example of a physical model that matches the theoretical result in Section 3.2. Finally, we summarize the paper in Section 4.

|Φp , i〉|Φp r , i 〉 = |Φp ⊕ p r , i〉|Φp r , i ⊕ i 〉, r r ′ ′ ′ ′ ′

where i = i1i2…in , i′ = i′1i′2 …i′n , and i ⊕ i′ = i1 ⊕ i′1 , i2 ⊕ i′2 , …, in ⊕ i′n denotes bitwise addition modulo 2 (for convenience, the shorthand denotations will be used in the following context). Now, the target particles are measured in the computation basis {|0〉, |1〉}. If all the parties get the outcome “1”, the control particles will collapse into the following state (non-normalized):

p ′r , i, p ′l , j

ξ p(1)r , i; p l , j = ′ ′

ρ1 =

∑ λp,i1… in|Φp,i1… in〉〈Φp,i1… in|,

∑ ( − 1) pr ⊕p ′r ⊕pl ⊕p ′l λ pr ,i; pl ,jλ pr ⊕p ′r ,i; pl ⊕p ′l ,j. ξ p(1)r , i; p l ≠ p r , j ′ ′ ′

= 0 (Appendix). Then ρ1 reduces to

ξp(1), i; p , j|Φp , i〉〈Φp , j|. ′

′

′

(7)

′

(8)

According to Eqs. (7) and (8), we obtain that ρ1 is exactly the GHZ state |Φ0,0⋯0〉 if the following conditions are satisﬁed (Appendix):

λ 0, 0;0, 0λ1, 0;1, 0 = λ 0, 0;1, 0λ1, 0;0, 0 ,

λp, i; p, j = ± λ p , i; p, jor ± λp, i; p , j,

(9)

(i , j) ≠ (0 , 0),

(10)

where 0 = 00⋯0 (means i1 = i2 = ⋯ = in = 0 ) and (i , j) ≠ (0 , 0) denotes that i and j do not simultaneously equal 0 . The states that can be utilized to produce perfect GHZ states will be referred to as GHZ-source states (or GHZ-SSs for short). If a mixed state ρ is a GHZ-SS, its a local-unitary (LU) equivalent state ρ′ is also a GHZ-SS, because ρ′ can be transformed to ρ by LU operations prior to being subjected to distillation protocols. Thus, Eqs. (9) and (10) are suﬃcient conditions for an entangled mixed state or its LU equivalent states being source states of the GHZ state |Φ0,0⋯0〉. These conditions display that the non-diagonal elements in the state ρ given in Eq. (2) are not simultaneously equal to zero if it is a GHZ-SS. It tells us that it may be not necessary to depolarize a general entangled mixed state to a GHZ-diagonal type by formidable probabilistic LOCC before carrying out GHZ-state distillation protocols, and that directly distilling GHZ states from some non-GHZ-diagonal states may gain better output states or higher yields.

(1)

(2)

where λ pr , i1… in; pl , j … j ( pr , pl ∈ {0, 1}) satisfy the normalization condition 1 n and the conditions such that ρ is Hermitian and semi-positive. In principle, one can depolarize a general mixed state ρ to a GHZdiagonal state ρdiag without changing the diagonal elements by applying probabilistic LOCC operations [26,27], i.e.,

ρ → ρdiag =

∑ p ′, i, j

where p and the i's are zero or one, and a bar over a bit value indicates its logical negation. These 2n+1 orthonormal GHZ states {|Φp, i1… in〉} construct a complete basis in the Hilbert space of n + 1 qubits. Then, any (n + 1)-qubit state ρ can be expressed by the GHZ-state basis, i.e.,

∑ λ pr ,i1… in; pl ,j1 … jn|Φpr ,i1… in〉〈Φpl ,j1 … jn|,

(6)

pr , pl

It can be proved that

We deﬁne a set of (n + 1)-qubit (n > 0 ) GHZ states

ρ=

ξ p(1)r , i; p l , j|Φp r , i〉〈Φp l , j|, ′ ′ ′ ′

where

2.1. Protocol and conditions

1 (|0i1i2…in〉 + ( − 1) p |1i1i2…in〉), 2

∑

ρ1 =

2. Distillation of a perfect GHZ sate from two copies of a mixed state

|Φp, i1… in〉 =

(5)

(3) 2.2. Recycle scheme and eﬃciency

with λp, i1⋯ in ≡ λp, i1… in; p, i1… in . Mathematically, one can obtain ρdiag by throwing away all the non-diagonal elements in ρ. After that, one can distill a nearly pure GHZ state with a near-to-unit ﬁdelity from almost inﬁnite copies of ρdiag with the multipartite entanglement distillation protocols presented in Refs. [24–28], provided that the ﬁdelity of the state ρdiag relative to a GHZ state is larger than a threshold. In the following, we shall show that it is not necessary to transform the general mixed state ρ to the GHZ-diagonal type ρdiag in some cases. That is to say, it can be better to directly distill GHZ states from general entangled mixed states when certain conditions are satisﬁed. Particularly, we will demonstrate that a perfect GHZ state can be extracted from two copies of a non-GHZ-diagonal state with certain characteristics. Without loss of generality, we set the GHZ state

|Φ0,0…0〉 =

1 (|00⋯0〉 + |11⋯1〉) 2

We now analyze the eﬃciency of the aforementioned GHZ-state distillation. Distillation eﬃciency (also known as yield in literature) is conventionally deﬁned as the ratio of the number of ﬁnal output states to that of initial input states. In this section, we present a recycle scheme for increasing the eﬃciency, in which some discarded states can be recycled and each recycle process will contribute to the ﬁnal eﬃciency. We shall formulate the eﬃciency as a function of the parameters in Eqs. (9) and (10). As mentioned before, the parties will share a perfect (n + 1)-qubit GHZ state if all of them get the measurement outcome |1〉. The success probability is given by

P1(1) = Tr[(|1 t 〈1| ⊗ Ic )⊗(n +1)MXORρ⊗2 MXOR] = (4)

1 (1) ξ0, 0;0, 0, 2

(11)

where I denotes the identity operator in two-dimensional Hilbert space and the subscript ‘ c’ (‘ t’) stands for the ‘control’ (‘target’) qubit that the

as the output state (or aim state). 186

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X.-W. Wang et al.

operator acts on. When all the parties get the same measurement outcome |0〉, the control qubits will be still in a genuine multipartite entangled mixed state and thus can be utilized again. Particularly, the resulting state of control qubits corresponding to the measurement outcome |0 ⊗(n +1) of target qubits is

ρ0 =

1 η1

λ p(1)r , i; p l , j|Φp r , i〉〈Φp l , j|, ′ ′ ′ ′

∑ p ′r , i, p ′l , j

λ 0, 0;0, 0 = Aγ , λ1, 0;1, 0 = Bγλ 0, 0;1, 0 = λ1, 0;0, 0 = γ AB , λ 0, 0;0, i = − λ 0, 0;1, i

BCi γ 1−γ , λ 0, i;0, i = − λ 0, i;1, i = Ci + Di , λ1, i;1, i 2 2 2 γ 1−γ = − λ1, i;0, i = Ci + Di , λ 0, i;0, j ≠ i = − λ 0, i;1, j ≠ i 2 2 γ γ = CiCj ≠ i , λ1, i;1, j ≠ i = − λ1, i;0, j ≠ i = CiCj ≠ i , i , j ≠ 0 . (21) 2 2 = − λ1, i;1, 0 = γ

(12)

where

λ p(1)r , i; p l , j = ′ ′

Evidently, the coeﬃcients in Eq. (21) meet the relations in Eqs. (9) and (10).

∑ λ pr ,i; pl ,jλ pr ⊕p ′r ,i; pl ⊕p ′l ,j,

(13)

pr , pl

3.2. Amplitude-damped states

and η1 is the normalization coeﬃcient given by

η1 =

∑ λp(1)′,i; p ′,i .

We now show that the decohered state of a generalized GHZ state in local amplitude-damping (AD) environments belongs to the abovementioned GHZ-SSs. AD model is applicable to many practical qubit systems, including vacuum-single-photon qubit with photon loss, photon-polarization qubit traveling through a polarizing optical ﬁber or a set of glass plates oriented at the Brewster angle, atomic qubit with spontaneous decay, and superconducting qubit with zero-temperature energy relaxation. The action of the AD noise on a qubit can be described by two Krauss operators [3]

(14)

p ′, i

The probability of this event is

1 η. 2 1

P0(1) =

(15)

The parties can repeat the above distillation procedure with two copies of ρ0 and attain a perfect (n + 1)-qubit GHZ state of control qubits when they get again the measurement outcome |1 ⊗(n +1) of target qubits. The success probability of the second round distillation reads

P1(2)

1 (2) = ξ0, 0;0, 0, 2η12

M0 = |0〉〈0| +

⎡

⎤2

∑ ( − 1) pr ⊕pl ⎢⎣λ p(1)r ,0; pl ,0⎥⎦

. (17)

pr , pl

By the same token, they can carry on with the third, the fourth, and up to the hth round of distillation. Considering the fact that the target copy is sacriﬁced in each round, the ﬁnal eﬃciency (total yield) of the recycle scheme is given by

E = Y1 + Y2 + ⋯ + Yh, Y1 = =

ξ0,(h0);0, 0 2h

2 η1η2…ηh −1

d |1〉〈1|, M1 =

d |0〉〈1|,

(22)

where d stands for the damping rate satisfying 0 ≤ d < 1 and d = 1 − d . The AD channel is trace preserving, that is, ∑l =0,1 Ml†Ml = I . Note that d=0 denotes the noise-free case, and it will not be considered in the following context. The issue on entangled particles interacting with local noises is well related to many practical scenarios, e.g., a sender preparing an entangled state in his/her lab and sending each receiver one qubit via a quantum channel modeled by amplitude damping. Assume n + 1 qubits are initially in a generalized GHZ state

(16)

where

ξ0,(2)0;0, 0 =

ACi , λ1, 0;0, i = − λ1, 0;1, i = λ 0, i;1, 0 2

= λ 0, i;0, 0 = − λ1, i;0, 0 = γ

|Ψ 〉 = α|0 n +1〉 + β|1n +1 〉,

(23) n +1

⊗(n +1)

(l=0,1) has been used. where the shorthand denotation |l 〉 = |l Then each one of the last n qubits undergoes an AD decoherence of strength d, and the state |Ψ 〉 degenerates to a mixed sate given by

ξ0,(2)0;0, 0 P (1) P (2) 1 (1) 1 (1) P1 = ξ0, 0;0, 0, Y2 = 0 · 1 = 4 , Yh 2 4 2 2 2 η1

n

n

ϱd = α 2|0 n +1〉〈0 n +1| + β 2d |1n +1 〉〈1n +1 | + αβ d (|0 n +1〉〈1n +1 | + |1n +1 〉〈0 n +1|)

, (18)

n

+ β 2|1〉〈1| ⊗

where

∑ d kd n −k[|0 k1n −k 〉〈0 k1n −k | + ⋯],

(24)

k =1

ξ0,(h0);0, 0 =

⎡

⎤

∑ ( − 1) pr ⊕pl ⎢⎣λ p(rh,−1) 0; pl , 0 ⎥ ⎦

2

, ηh −1 =

pr , pl

=

∑ p ′r , p ′l

where “⋯” denotes all permutations of the ﬁrst term in the square bracket. The state ϱd can be rewritten as the form of Eq. (20) with

(h −1) ∑ λp(h,i−1) ; p, i , λ pr , i; pl , i p, i

λ p(hr−2) λ (h −2) , h ≥ 2, λ p(0), i; p , i ≡ λ p , i; p , i . r l r l ′ , i; p ′l , i p ′r ⊕ pr , i; p ′l ⊕ pl , i

n 2

A=

(19)

n 2

n−| i|

(α + β d ) (α − β d ) d |i|d ,B= , Ci = 0, Di = n , 2 2 n 2 2 n 1−d 2(α + β d ) 2(α + β d ) n

|i| = k ≥ 1, γ = α 2 + β 2d , 3. Examples of GHZ-SS

where |i| denotes the number of one in the multi-indices i1i2…in , e.g., |10⋯0| = 1. Using the aforementioned recycle scheme, the eﬃciency of distilling the perfect GHZ state |Φ0,0⋯0〉 from the noisy state ρd is given by

3.1. A wide class of GHZ-SSs We here give a wide class of GHZ-SSs which can be used to produce perfect GHZ states. This class of entangled mixed states can be uniformly expressed as

∼ |ψ 〉 = ϱ = γ |ψ 〉〈ψ | + (1 − γ )ϱ,

A |Φ0, 0〉 +

B |Φ1, 0〉 +

∑ i≠0

|Φ1, i〉), ∼ ϱ=

∑ i≠0

Di (|Φ0, i〉 − |Φ1, i〉)(〈Φ0, i| − 〈Φ1, i|), 2

(25)

h −2

Y12

n

Ed = Y1 + Y2 + ⋯ + Yh, Y1 = α 2β 2d , Yh =

h

n

Yh −1

n

2α 2 + 2 ∑k =0 ( k )(β 2d kd

Ci (|Φ0, i〉 − 2

,

h ≥ 2.

h −1 n−k 2

)

(26)

As an example, we take n=2, h=10, and α = β = 1/ 2 and plot Ed as a function of d in Fig. 1. Fig. 1 displays clearly that one can obtain a perfect 3-qubit GHZ state from two copies of the 3-qubit amplitudedamped state in Eq. (24) with a non-zero probability, in contrast to the conventional protocols [24–28] where the noisy state ρd should be

(20)

where A + B + ∑i ≠ 0 Ci = ∑i ≠ 0 Di = 1 and A ≠ B . It can be rewritten as the form of Eq. (2) with 187

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X.-W. Wang et al.

0.35

conventional schemes (designed for GHZ-diagonal states) as long as the initial mixed states are appropriate non-GHZ-diagonal states. We also constructed a wide class of quantum states that meet the requirements. Moreover, we found that noisy generalized GHZ states in the AD environment can act as the GHZ-SSs. It has been mentioned before that AD decoherence would occur in many physical systems, e.g., a polarized photon traveling through a polarizing optical ﬁber or a set of glass plates oriented at the Brewster angle [33]. Thus, our ﬁndings have practical meaning and could be demonstrated in lab with current technology. Finally, we should realize that it is very diﬃcult to derive the necessary conditions for distilling a normal GHZ state from two or more copies of a multipartite entangled mixed state. On the one hand, it is formidable to ﬁnd all the states that are LU equivalent to a GHZ state, due to the fact that the single-qubit reduced matrix of a GHZ state is the identity matrix [34]. As a consequence, when a multipartite entangled pure state can be extracted from two or more copies of an entangled mixed state, one can not necessarily justify whether it is equivalent to a GHZ state under LU operations. One the other hand, the requirements that the source states should fulﬁll may rely on the distillation protocols. However, such a problem deserves to be paid more attention, and is expected to be fully solved in the future.

0.3 0.25 Ed

0.2 0.15 0.1 0.05 0

0

0.2

0.4

0.6

0.8

1

d Fig. 1. The eﬃciency (Ed) of distilling 3-qubit perfect GHZ states from 3-qubit amplitude-damped GHZ states with diﬀerent damping rate d.

transformed into a GHZ-diagonal state in advance and the yield of getting a (nearly) pure GHZ state tends to zero when d does not approach zero. Moreover, the above scheme can work for all values of d, but previous protocols cannot. 4. Summary

Acknowledgments

In summary, we have shown the possibility of distilling a pure GHZ state from ﬁnite number of entangled mixed states. We have given the suﬃcient conditions for extracting a perfect N-qubit GHZ state from two copies of an N-qubit entangled mixed state. It was found that the GHZ-SSs are not diagonal in the GHZ-state basis. The results indicate that the proposed GHZ-state distillation scheme will do better than

This work was supported by the NSFC (Gant No. 11547258), the HNNSF (Grant No. 2015JJ3029 and No. 2016JJ2009 ), the Scientiﬁc Research Fund of Hunan Provincial Education Department (Grant Nos. 15A028 and 16B036), and Opening fund of Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education (Grant No. QSQC1207).

Appendix A. Appendix When p′l ≠ p′r (i.e., p′l = p ′r ) in Eq. (7), we have

ξ p(1)r , i; p r , j = ′ ′

⎞

⎛

∑ ( − 1) pr ⊕pl λ pr ,i; pl ,jλ pr ⊕p ′r ,i; pl ⊕p ′r ,j = ∑ ⎜⎝λ pr ,i; pr ,jλ pr ⊕p ′r ,i; pr ⊕p ′r ,j−λ pr ,i; pr ,jλ pr ⊕p ′r ,i; pr ⊕p ′r ,j⎟⎠,

pr , pl

pr

(27)

where the identity pr ⊕ p ′r = pr ⊕ p′r has been used. It can be directly veriﬁed that ξ p(1)r , i; p r , j = 0 for both p′r = 0 and p′r = 1. ′ ′ When p′l = p′r = p′, Eq. (7) reduces to

ξp(1), i; p , j = ′ ′

∑ ( − 1) pr ⊕pl λ pr ,i; pl ,jλ pr ⊕p ′,i; pl ⊕p ′,j.

(28)

pr , pl

For p′ = 0 , we have

ξ0,(1)i;0, j = λ 0,2 i;0, j + λ1,2i;1, j − λ 0,2 i;1, j − λ1,2i;0, j.

(29)

As to the case p′ = 1, we get

ξ1,(1)i;1, j = 2(λ 0, i;0, jλ1, i;1, j − λ 0, i;1, jλ1, i;0, j).

(30)

Substituting Eqs. (9) and (10) into Eqs. (29) and (30), one can directly obtain that

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