Genuine entanglement, distillability and quantum information masking under noise
GGenuine entanglement, distillability and quantum information masking under noise
Mengyao Hu ∗ and Lin Chen
1, 2, † School of Mathematical Sciences, Beihang University, Beijing 100191, China International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, China (Dated: February 5, 2021)Genuineness and distillability of entanglement play a key role in quantum information tasks, andthey are easily disturbed by the noise. We construct a family of multipartite states without genuineentanglement and distillability sudden death across every bipartition, respectively. They are realizedby establishing the noise as the multipartite high dimensional Pauli channels. Further, we constructa locally unitary channel as another noise such that the multipartite Greenberger-Horne-Zeilingerstate becomes the D¨ur’s multipartite state. We also show that the quantum information maskingstill works under the noise we constructed, and thus show a novel quantum secret sharing schemeunder noise. The evolution of a family of three-qutrit genuinely entangled states distillable acrossevery bipartition under noise is also investigated.
I. INTRODUCTION
Bipartite pure entangled states are genuinely entangled (GE) states [1]. They play a key rolein quantum computing, secret sharing and superdense coding [2–7]. An explicit protocol has beenproposed for faithfully teleporting an arbitrary two-qubit state by using a four-qubit GE state [8].A (2 n + 1)-qubit GE state has been constructed to perform controlled teleportation of an arbitrary n -qubit state [9]. However, Genuine entanglement may be influenced by the unavoidable noise inenvironment. Genuine entanglement sudden death (GESD) describes that GE states evolve intonon-GE states under noise. So it is indispensable to sustain the genuine entanglement of quantumstates under noise. We refer to the
GESD-free states as multipartite quantum states without GESD.We construct such states, and it is the first motivation of this paper.A bipartite quantum state is entangled or separable [10, 11]. The phenomenon of a finite-timedisappearance of entanglement under noise is known as entanglement sudden death (ESD). Forexample, ESD is exactly GESD for bipartite states. Bipartite entangled states can be divided intofree and bound types [12, 13]. Free entangled states can be asymptotically distilled into pure-stateentanglement under local operations and classical communications (LOCC), while bound entangledstates cannot. Only free entangled states are essential ingredients in quantum-information tasks.The notion of distillability sudden death (DSD) describes that free entangled states can evolve intonondistillable (bound entangled or separable) states in finite time under local noise [14, 15]. It is thusdesirable to construct DSD-free states, i.e., states without DSD under noise. Actually, two-qutritDSD-free states have been proposed in [14]. We extend the notion of DSD-free states from bipartitestates to multipartite states in any dimension. We refer to the later as multipartite DSD-free states .As far as we know, little is known about multipartite DSD-free states. We investigate a family ofthese states which go through the noise characterized by the multipartite high dimensional Paulichannel. This is the second motivation of this paper.The family of Pauli channels represents a wide class of noise process [16]. The experimental realiza-tion of the optimal estimation protocol for a Pauli noise channel has been constructed [17]. Thoughsome noise channels are not Pauli channels, they can be well approximated by Pauli channels without ∗ [email protected] † [email protected] (corresponding author) a r X i v : . [ qu a n t - ph ] F e b introducing new errors [18, 19]. Quantum noise is connected to quantum measurement and ampli-fication [20, 21]. Thus it is crucial to characterize quantum noise channels reliably and efficiently.Recently, a protocol is described and implemented experimentally on a 14-qubit superconductingquantum architecture in terms of an estimation of the effective noise over qubit systems [22]. Itthus applies to well-known multiqubit states such as the Greenberger-Horne-Zeilinger (GHZ) state[23]. It is widely used to split quantum information in secret sharing [24] and works as a channelin quantum teleportation [8, 25], as well as quantum information masking in bipartite systems [26].Moreover, quantum information masking plays a key role in quantum secret sharing [27–29]. Thisconcept has been extended to m -uniform quantum information masking in multipartite systems [30].This kind of masking allows quantum secret sharing from a ”boss” to his ”subordinates” since the m subordiantes cannot retrieve the information even if they collaborate. However, the existing schemesof quantum information masking work for pure states only, and they should be investigated undernoise from a practical point of view.In this paper, we construct a family of GESD-free states and multipartite DSD-free states fromGHZ states. We begin with reviewing the definition of GE state, and introduce the multipartite DSD-free states in Definition 1. We present the realignment criterion in Lemma 2 to detect entanglement.We review the fact from Ref. [31] about GE states coupled with white noise in Lemma 3. It showsthat the state η := p | GHZ d,n (cid:105)(cid:104)
GHZ d,n | + (1 − p ) d n I d n is GE if p > d n − +3 . In Theorem 5 we constructthe n -partite d -dimensonal Pauli channel such that the d -level n -partite GHZ state becomes the state η . We show that η is GESD-free if p ∈ ( d n − +3 , η is distillable acrossevery bipartition, that is, multipartite DSD-free, if and only if p ∈ ( d n − ,
1] in Lemma 6. Nextin Theorem 7, we show that the state η is not at the same time GE and positive partial transpose(PPT) across any given bipartition. In Theorem 8, we construct the locally unitary channel suchthat the N -partite GHZ state becomes the D¨ur’s multipartite state [32–34]. The condition for thebiseparability of such a state is given in Theorem 9. We also study the quantum information maskingunder noise in terms of the state η and D¨ur’s multipartite state. Furthermore, we investigate theevolution of a three-qutrit GE state which is distillable across every bipartition under global (i.e.,local and collective) noise. The numerical results show that the state will undergo DSD and becomePPT.The noise characterized by the channels proposed in this paper can be investigated more. Thesechannels are operational in experiment [18, 35]. Many noisy quantum channels in nature are aconvex combination of unitary channels [36]. It is important to characterize these channels reliablyand efficiently with high precision since it is the central obstacle in quantum computation and secretsharing. The quantum information masking under noise we proposed is capable of being applied insecret sharing and future quantum communication protocols.The rest of this paper is organized as follows. In Sec. II, we introduce the preliminary knowledgeused in this paper. In Sec. III, we construct a family of GESD-free states from GHZ states by usingthe high dimensional Pauli channels. In Sec. IV, we investigate the multipartite DSD-free statesconstructed from GHZ states. In Sec. V, we investigate quantum information masking under noise.In Sec. VI, we give an example of a GE state distillable across every bipartition under local andcollective noise. Finally, we conclude in Sec. VII. II. PRELIMINARIES
In this section, we introduce the fundamental knowledge used throughout the paper. Let C d bethe d -dimensional Hilbert space and H A be the Hilbert space of system A. Given a bipartite state ρ on H AB , the partial trace with regard to system A is defined as ρ B := Tr A ρ := (cid:80) i ( (cid:104) a i | ⊗ I B ) ρ ( | a i (cid:105) ⊗ I B ) := (cid:80) i (cid:104) a i | A ρ | a i (cid:105) A , where | a i (cid:105) is an arbitrary orthonormal basis in H A . We call ρ B the reduceddensity matrix of system B. Next, we review the GE states [31, 37]. Let | ψ n (cid:105) be an n -partite purestate in the Hilbert space H A A ...A n = H A ⊗ H A ⊗ · · · ⊗ H A n = C d ⊗ C d ⊗ · · · ⊗ C d n . Then | ψ n (cid:105) is an n -partite GE pure state if | ψ j (cid:105) (cid:54) = | a, b (cid:105) ∈ H A j ...jk ⊗ H A jk +1 ...jn , where { j , . . . , j n } = { , . . . , n } .Further, we say that ρ ∈ B ( H A A ...A n ) is a mixed n -partite GE state if ρ = r (cid:88) j =1 p j | ψ j (cid:105)(cid:104) ψ j | , (1)where | ψ j (cid:105) is a pure n -partite GE state in every decomposition, (cid:80) rj =1 p j = 1 , p j ≥
0. Genuineentanglement sudden death (GESD) describes that GE states evolve into biseparable states, that is,it cannot be written as the form in Eq. (1). In Sec. III, we shall construct GESD-free states byusing GHZ states.Next, we present the definition of multipartite DSD-free states.
Definition 1
Multipartite DSD-free states are n -partite quantum states without DSD in every bipar-tition. They are distillable and non-positive partial transpose (NPT) [54] across every bipartition. Such multipartite DSD-free states will be constructed in Sec. IV. DSD-free states always haveNPT. It has been proven that a bipartite state has PPT is non-distillable under LOCC [12, 14]. SoPPT entangled states must be bound entangled [38, 39]. However, there are evidences showing thatbound entangled states with NPT exist [40, 41]. It will be investigated in Sec. VI.In the following, we introduce the realignment criterion proposed in [42]. It provides a necessarycriterion for separable states [42, 43] and a method of detecting BE states [13]. We only considerbound entangled states with PPT. Realignment criterion provides a computable necessary criterionto detect separability [42]. It will be used to detect entanglement in Sec. VI.
Lemma 2
If an m × n bipartite density matrix ρ is separable, then for the m × n matrix ( ρ ) Rij,kl = ρ ij,kl , one can obtain that (cid:107) ρ R (cid:107) ≤ . III. GESD-FREE STATES
In this section, we construct a family of GESD-free states in terms of the well-known GHZ statewhich goes through the multipartite high dimensional Pauli channels. The first main result of thissection is Theorem 5. This is supported by Lemma 3. We present the keys of the three-qubit statein Example 4. This is a more implementable case due to the practically realizable n -qubit states upto ten [44–46].One can show that the pure GE state is NPT across every bipartition. An example is the d -level n -partite GHZ state | GHZ d,n (cid:105) := 1 √ d d − (cid:88) j =0 | j, j, ..., j (cid:105) . (2)In practice, the GHZ state will be coupled with white noise. We need to know whether the resultingstate is GE. The following fact is from Example 4 in [31]. Lemma 3
Consider the GHZ state with additional isotropic (white) noise, η := p | GHZ d,n (cid:105)(cid:104)
GHZ d,n | + (1 − p ) 1 d n I d n . (3) Then η is GE for p > d n − +3 . We construct a locally unitary channel ∆ such that | GHZ d,n (cid:105) becomes the state η in Eq. (3) when d = 2 and n = 3. Example 4
Let d = 2 and n = 3 , that is, | GHZ , (cid:105) := √ (cid:80) j =0 | j, j, j (cid:105) . We denote the Paulimatrices as follows, σ x = (cid:20) (cid:21) , σ y = (cid:20) − ii (cid:21) , σ z = (cid:20) − (cid:21) . (4) Let the Kraus operators be K = I ⊗ I ⊗ I , K = I ⊗ I ⊗ σ z ,K = I ⊗ I ⊗ σ x , K = I ⊗ σ z ⊗ σ x ,K = I ⊗ σ x ⊗ I , K = I ⊗ σ x ⊗ σ z ,K = σ x ⊗ I ⊗ I , K = σ x ⊗ I ⊗ σ z . Let the locally unitary channel ∆( · ) = pK ( · ) K † + − p (cid:80) i =1 K i ( · ) K † i . We refer to it as the tripartitePauli channel. One can verify that ∆( | GHZ , (cid:105)(cid:104) GHZ , | )= pK | GHZ , (cid:105)(cid:104) GHZ , | K † + 1 − p (cid:88) i =1 K i | GHZ , (cid:105)(cid:104) GHZ , | K † i (5) is exactly the state in (3) with d = 2 and n = 3 . From Lemma 3, we know that η is GE if p ∈ ( , . (cid:117)(cid:116) We extend the above channel ∆ to a channel of any d and n . We refer to it as the n -partite d -dimensional Pauli channel. In this position, we present the main result of this section. Theorem 5
There exists an n -partite d -dimensional Pauli channel such that | GHZ d,n (cid:105) becomes thestate η in (3) . It is GESD-free when p ∈ ( d n − +3 , . Proof.
Let | a qi ,i , ··· ,i n − (cid:105) = 1 √ d d − (cid:88) j =0 e πid jq | j, j ⊕ i , j ⊕ i , · · · , j ⊕ i n − (cid:105) , (6)where j ⊕ i = ( j + i ) mod d , and q, j, i , i , · · · , i n − = 0 , , · · · , d −
1. By using (6), the d n -dimensional identity matrix can be expressed as I d n = d − (cid:88) q,i ,i , ··· ,i n − =0 | a qi ,i , ··· ,i n − (cid:105)(cid:104) a qi ,i , ··· ,i n − | . (7)For d -level system, a set of local orthogonal observables (LOOs) [47] is a set of d observables A µ ( µ = 1 , , · · · , d ) satisfying orthogonal relations Tr( A µ A ν ) = δ µν , where µ, ν = 1 , , · · · , d . Forexample, a standard complete set of LOOs is defined as { A µ } = { A m = | m (cid:105)(cid:104) m | , A ± m,n } , where A + m,n = | m (cid:105)(cid:104) n | + | n (cid:105)(cid:104) m |√ m < n ) , (8) A − m,n = | m (cid:105)(cid:104) n | − | n (cid:105)(cid:104) m | i √ m < n ) , and m, n = 0 , , · · · , d −
1. For convenience, we denote A + m,n in Eq. (8) as A + j ⊕ i,j = √ A + j ⊕ i,j ( j ⊕ i < j ) ,A + j ⊕ i,j = √ A + j,j ⊕ i ( j ⊕ i > j ) ,A + j ⊕ i,j = I d , ( j ⊕ i = j ) , (9)Let the Fourier matrix W = √ d [ e πid jq ] j,q . Suppose the Kraus operators are K = I d ⊗ I d ⊗ · · · ⊗ I d ,K r = W ⊗ A + j ⊕ i ,j ⊗ · · · ⊗ A + j ⊕ i n − ,j . (10)where r = 1 , , · · · , d n . Then using (7) and (10), one can verify that (cid:80) d n r =1 K r | GHZ d,n (cid:105)(cid:104)
GHZ d,n | K † r = I d n . We define the n -partite d -dimensional Pauli channel U as U ( · ) := pK ( · ) K † + 1 − pd n d n (cid:88) r =1 K r ( · ) K † r . (11)We have U ( | GHZ d,n (cid:105)(cid:104)
GHZ d,n | )= pK | GHZ d,n (cid:105)(cid:104)
GHZ d,n | K † + 1 − pd n d n (cid:88) r =1 K r | GHZ d,n (cid:105)(cid:104)
GHZ d,n | K † r . (12)One can verify that the state is exactly the state η in Eq. (3). From Lemma 3, we know that η isGE if p ∈ ( d n − +3 , p ∈ ( d n − +3 , (cid:117)(cid:116) FIG. 1: The state | GHZ d,n (cid:105) goes through the n -partite d -dimensional Pauli channel U in (12), and becomes thestate η in (3). We illustrate Theorem 5 by FIG. 1. By selecting p ∈ ( d n − +3 , η in experiment. When d or n is large enough, then d n − +3 →
0. Under these circumstances,the state η is almost always GE. On the other hand, there has been substantial progress towards therealization of GHZ states and noise in experiment. Recently, the ten-photon GHZ state in experi-ment has been demonstrated [46]. Further, a three-particle GHZ state entangled in three levels forevery particle has been realized in 2018 [7]. By choosing local basis rotations properly, a complete basis of GHZ states can be constructed. The four qubit GHZ states have been demonstrated exper-imentally by entangling two photons in polarization and orbital angular momentum [48]. Moreover,the multipartite high dimensional Pauli channel in this paper is constructed from LOOs. One cansee that they are constructed by the computational basis in C d . Based on LOOs, Ref [47] alsoproposed a family of entanglement witness and corresponding positive maps that are not completelypositive. Thus, we provide an operational way to construct a family of GE states in (12) by the d -level n -partite GHZ state through locally unitary channels in experiment. IV. MULTIPARTITE DSD-FREE STATES
In this section, we construct the multipartite DSD-free states under white noise in Lemma 6.We also investigate their genuine entanglement and distillability in Theorem 7. It is supported byLemmas 3 and 6. Next in Theorem 8, we construct the locally unitary channel such that the N -partite GHZ state becomes the D¨ur’s multipartite state. Further, the condition for the biseparabilityof the D¨ur’s multipartite state is given in Theorem 9.First, we begin with the distillability of the state η in (3). It will be used in the proof of Theorem7. Lemma 6
The state η is distillable across every bipartition if and only if p ∈ ( d n − , . Proof.
For the state η A ··· A n = p | GHZ d,n (cid:105)(cid:104)
GHZ d,n | + (1 − p ) d n I d n , one can obtain that this k -partite reduced density matrices η A j ··· A jk of η A ··· A n for any given k ∈ { , · · · , n − } are the same,where j , · · · , j k ∈ { , · · · , n } . Thus it suffices to show that the state η A ··· A n is distillable across anyone of the bipartite cuts η A | A ··· A n , η A A | A ··· A n , · · · , η A ··· A (cid:98) n (cid:99) | A (cid:98) n (cid:99) +1 ··· A n .Let P = (cid:80) i =0 | i, · · · , i (cid:124) (cid:123)(cid:122) (cid:125) k (cid:105)(cid:104) i, · · · , i (cid:124) (cid:123)(cid:122) (cid:125) k | ⊗ (cid:80) j =0 | j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) n − k (cid:105)(cid:104) j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) n − k | , where k = 1 , , · · · , (cid:98) n (cid:99) . Then wecan obtain that P ( η A ··· A n ) P † = pd ( (cid:88) i =0 | i, · · · , i (cid:124) (cid:123)(cid:122) (cid:125) n (cid:105) )( (cid:88) j =0 (cid:104) j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) n | )+ 1 − pd n (cid:88) i =0 | i, · · · , i (cid:124) (cid:123)(cid:122) (cid:125) k (cid:105)(cid:104) i, · · · , i (cid:124) (cid:123)(cid:122) (cid:125) k | ⊗ (cid:88) j =0 | j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) n − k (cid:105)(cid:104) j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) n − k | . (13)It is a state on C ⊗ C n − . Using the fact that any 2 ⊗ N ( N ≥
2) states are distillable underLOCC if and only if they are NPT [49], it suffices to show that the state P ( η A ,...,A n ) P † in (13) isNPT across any one of the bipartite cuts η A | A ...A n , η A A | A ...A n , . . . , η A ...A (cid:98) n (cid:99) | A (cid:98) n (cid:99) +1 ...A n .First we prove the ”only if” part. Suppose that P ( η A ··· A n ) P † is NPT. The partial transpose withregard to system A · · · A k is( P ( η A ··· A n ) P † ) Γ = pd (cid:88) i =0 1 (cid:88) j =0 | i, · · · , i (cid:124) (cid:123)(cid:122) (cid:125) k , j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) n − k (cid:105)(cid:104) j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) k , i, · · · , i (cid:124) (cid:123)(cid:122) (cid:125) n − k | + 1 − pd n (cid:88) i =0 | i, · · · , i (cid:124) (cid:123)(cid:122) (cid:125) k (cid:105)(cid:104) i, · · · , i (cid:124) (cid:123)(cid:122) (cid:125) k | ⊗ (cid:88) j =0 | j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) n − k (cid:105)(cid:104) j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) n − k | . (14)It implies that if P ( η A ··· A n ) P † is NPT, then − pd + (1 − p ) d n <
0. So we have p > d n − . Second we prove the ”if” part. Suppose p > d n − . Then from (14), we can obtain that theminimum eigenvalue of ( P ( η A ··· A n ) P † ) Γ is negative.Therefore, the state p | GHZ d,n (cid:105)(cid:104)
GHZ d,n | + (1 − p ) d n I d n is distillable across every bipartition if andonly if p ∈ ( d n − , (cid:117)(cid:116) Hence, when the state | GHZ d,n (cid:105) goes through the n -partite d -dimensional Pauli channel U in (12),the output state η is multipartite DSD-free if and only if p ∈ ( d n − , Theorem 7
The state η is not at the same time GE and PPT across any given bipartition. Proof.
From Lemma 6, we know that the state η = p | GHZ d,n (cid:105)(cid:104)
GHZ d,n | + (1 − p ) d n I d n is NPT ifand only if it is distillable, and it is distillable across every bipartition if and only if p ∈ ( d n − , p ≤ d n − . From Lemma 3, we know that η is GE if p > d n − +3 . Notethat d n − +3 is larger than d n − . Thus η cannot at the same time be GE and PPT across any givenbipartition. This completes the proof. (cid:117)(cid:116) Lemma 6 and Theorem 7 imply that η will always be GE and distillable across every bipartition.It provides an operational way in experiments to construct a family of this kind of states [7, 46, 48].We describe Theorem 5 and Lemma 6 in FIG. 2. FIG. 2: When the state | GHZ d,n (cid:105) goes through the locally unitary channel U = pK ( · ) K † + − pd n (cid:80) d n r =1 K r ( · ) K † r , theoutput state η in (3) is GESD-free if p ∈ ( d n − +3 ,
1] and multipartite DSD-free if p ∈ ( d n − , So far, we have investigated the GHZ states under white noise. Actually, they may undergo othernoise and become other state. For example, the D¨ur’s multipartite state [32–34] ρ N ( x ) = x | Ψ G (cid:105)(cid:104) Ψ G | + 1 − x N N (cid:88) k =1 ( P k + P k ) , (15)where | Ψ G (cid:105) = √ ( | ⊗ N (cid:105) + e iα N | ⊗ N (cid:105) ) is the N -partite GHZ state, P k is the projector onto theproduct state | φ k (cid:105) = | (cid:105) A | (cid:105) A · · · | (cid:105) A k · · · | (cid:105) A N , and P k is the projector onto the product state | ϕ k (cid:105) = | (cid:105) A | (cid:105) A · · · | (cid:105) A k · · · | (cid:105) A N . We may take e iα N = 1 in | Ψ G (cid:105) since it can be eliminated bylocal unitary transformations. In the following we construct the locally unitary channel such thatthe multipartite GHZ state becomes D¨ur’s multipartite state ρ N ( x ). Theorem 8
There exists the channel Λ such that the N -partite GHZ state | Ψ G (cid:105) becomes the state ρ N ( x ) in (15) . The state ρ N ( x ) is bound entangled if ≤ x ≤ N +1 and free entangled if N +1 < x ≤ . Proof.
Let the Kraus operators be K = I ⊗ · · · ⊗ I ,K i = ( | (cid:105) A | (cid:105) A · · · | (cid:105) A k · · · | (cid:105) A N )( (cid:104) | A (cid:104) | A · · · (cid:104) | A k · · · (cid:104) | A N ) , i = 1 , · · · , N,K j = ( | (cid:105) A | (cid:105) A · · · | (cid:105) A k · · · | (cid:105) A N )( (cid:104) | A (cid:104) | A · · · (cid:104) | A k · · · (cid:104) | A N ) , j = N + 1 , · · · , N,K N +1 = √ N · K − N (cid:88) i =1 K i , (16)where k = 1 , · · · , N . Suppose the locally unitary channelΛ( · ) := xK ( · ) K † + 1 − x N N +1 (cid:88) i =1 K i ( · ) K † i , (17)where K i , i = 0 , · · · , N + 1 are the Kraus operators in (16). One can verify that xK † K + − x N (cid:80) N +1 i =1 K † i K i = I N . Using the channel Λ, we can obtain that Λ( | Ψ G (cid:105)(cid:104) Ψ G | ) = x | Ψ G (cid:105)(cid:104) Ψ G | + − x N (cid:80) Nk =1 ( P k + P k ) is exactly the state ρ N ( x ) in (15). The second claim has been proved in [34]. Wehave proven the assertion. (cid:117)(cid:116) This theorem implies that if we change the channel Λ by x from x > N +1 to x ≤ N +1 , then thestate ρ N ( x ) will undergo DSD. We can also obtain that when the N -partite GHZ state | Ψ G (cid:105) goesthrough the channel Λ, the output state ρ N ( x ) is multipartite DSD-free if and only if x ∈ ( N +1 , N is large enough, then N +1 →
0. Under this circumstance, the state ρ N ( x ) will alwaysbe free entangled. Next, we give the condition for the biseparability of the D¨ur’s multipartite state ρ N ( x ). Theorem 9
The state ρ N ( x ) is biseparable if x ∈ [0 , ] . Proof.
Let the state ρ A ··· A N ( x ) := ρ N ( x ) = x | Ψ G (cid:105)(cid:104) Ψ G | + 1 − x N N (cid:88) k =1 ( P k + P k ) , (18)and α ( k ) A ··· A N = xN | Ψ G (cid:105)(cid:104) Ψ G | + − x N ( P k + P k ), N ≥
4. For simplicity, we do not normalize α ( k ) A ··· A N .Further, we have ρ A ··· A N ( x ) = (cid:80) Nk =1 α ( k ) A ··· A N .For the state α ( k ) A ··· A N , we bond systems A , · · · , A k − , A k +1 , · · · , A N and denote it as A k . De-fine a bijection from {| , · · · , (cid:105) , | , · · · , (cid:105)} in ( C ) ⊗ ( N − to the basis {| (cid:105) , | (cid:105)} in C as follows: | , · · · , (cid:105) → | (cid:105) , | , · · · , (cid:105) → | (cid:105) . Then we rewrite the state α ( k ) A ··· A N on C ⊗ C as α ( k ) A k A k = x N ( | (cid:105) + | (cid:105) )( (cid:104) | + (cid:104) | ) + 1 − x N ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) . The partial transpose of α ( k ) A k A k is( α ( k ) A k A k ) Γ = x N ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) + 1 − x N ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) . (19) It is known that α ( k ) A k A k is separable if and only if α ( k ) A k A k is PPT [10, 50]. One can verify that thefour eigenvalues of ( α ( k ) A k A k ) Γ in (19) are N , − x N , x N , x N . So if α ( k ) A k A k is separable, we have x ∈ [0 , ].That is, α ( k ) A ··· A N is biseparable if x ∈ [0 , ]. Since ρ N ( x ) is the convex sum of α ( k ) A ··· A N , we obtainthat ρ N ( x ) is biseparable if x ∈ [0 , ]. This completes the proof. (cid:117)(cid:116) Thus the D¨ur’s multipartite state ρ N ( x ) is not GE if x ∈ [0 , ]. Further, the value of x is in-dependent from N . So the genuine entanglement of | Ψ G (cid:105) is not pretty robust against the noise − x N (cid:80) Nk =1 ( P k + P k ). V. QUANTUM INFORMATION MASKING UNDER NOISE
For bipartite systems, an operation S is defined as quantum information masker if it maps states {| a k (cid:105) A ∈ H A } to states {| ψ k (cid:105) ∈ H A ⊗ H A } such that all the reductions to one party of | ψ k (cid:105) areidentical. For multipartite systems, an operation S is an m -uniform quantum information maskerif it maps states {| a k (cid:105) A ∈ H A } to states {| ψ k (cid:105) ∈ ⊗ n(cid:96) =1 H A (cid:96) } such that all the reductions to m parties of | ψ j (cid:105) are identical. Moreover, if m = (cid:98) n (cid:99) , then this m -uniform masking is strong quantuminformation masking [30]. The action of the masker is a physical process. It can be modeledby a unitary operator U S on the system A plus some ancillary systems { A , · · · , A n } , given by S : U S | a k (cid:105) A ⊗ | b (cid:105) A ··· A n = | ψ k (cid:105) . As far as we know, quantum information masking under noise islittle studied. In this section, we investigate the quantum information masking under the n -partite d -dimensional Pauli channel U in (11) and the locally unitary channel Λ in (17), respectively. FIG. 3: Quantum information masking under noise. The state | k (cid:105) is to be encoded in the m -uniform quantuminformation masking process modelled by U S and the n -partite d -dimensional Pauli channel U , where | b i (cid:105) , ≤ i ≤ n are the initial states of the ancillary systems. The reduced density matrices ( | ψ k (cid:105)(cid:104) ψ k | ) A of any m subsystems are thesame, where A = { A (cid:96) , · · · , A (cid:96) m } . Then the state | ψ k (cid:105)(cid:104) ψ k | goes through the channel U , the reduced density matrices( p | ψ k (cid:105)(cid:104) ψ k | + − pd n I d n ) A of the resulting state p | ψ k (cid:105)(cid:104) ψ k | + − pd n I d n are still independent of the encoded state | k (cid:105) . Thusthis m -uniform masking under noise still works. First, as shown in FIG. 3, we consider the quantum information masking and the n -partite d -dimensional Pauli channel U . A masker S : U S | k (cid:105) A ⊗ | b (cid:105) A ··· A n = | ψ k (cid:105) masks quantum in-formation contained in {| (cid:105) , | (cid:105) , · · · , | d − (cid:105)} ∈ C d into {| ψ (cid:105) , | ψ (cid:105) , · · · , | ψ d − (cid:105)} ∈ ( C d ) ⊗ n , where | ψ k (cid:105) = √ d (cid:80) d − j =0 e πid jk | j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) n (cid:105) , k = 0 , , · · · , d −
1, and the state | b (cid:105) A ··· A n = | b (cid:105) ⊗ · · · ⊗ | b n (cid:105) .One can verify that for all reduced density matrices ( | ψ j (cid:105)(cid:104) ψ j | ) A (cid:96) ··· A (cid:96)m of ( | ψ j (cid:105)(cid:104) ψ j | ) A ··· A n for any given m ∈ { , · · · , n − } are d (cid:80) d − j =0 | j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) m (cid:105)(cid:104) j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) m | , where (cid:96) , · · · , (cid:96) m ∈ { , · · · , n } . They arethe same. Thus the messages of | (cid:105) , · · · , | d − (cid:105) are strongly masked. Moreover, the masker S is m -uniform. Next, the state | ψ j (cid:105) goes through the n -partite d -dimensional Pauli channel U , we have U ( | ψ k (cid:105)(cid:104) ψ k | )= pK | ψ k (cid:105)(cid:104) ψ k | K † + 1 − pd n d n (cid:88) r =1 K r | ψ k (cid:105)(cid:104) ψ k | K † r , = p | ψ k (cid:105)(cid:104) ψ k | + 1 − pd n I d n . (20)The m -partite reduced density matrices ( p | ψ k (cid:105)(cid:104) ψ k | + − pd n I d n ) A (cid:96) ··· A (cid:96)m of the state in (20) for any given m ∈ { , · · · , n − } are d (cid:80) d − j =0 | j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) m (cid:105)(cid:104) j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) m | + I d m . Thus the resulting state is still stronglyquantum information masked. Hence, the state in {| (cid:105) , | (cid:105) , · · · , | d − (cid:105)} ∈ C d can be strongly maskedby the masker S and the n -partite d -dimensional Pauli channel U .Next, for D¨ur’s multipartite state ρ N ( x ), we study the quantum information masking andthe locally unitary channel Λ in (17). A masker S : U S | i (cid:105) A ⊗ | b (cid:105) A , ··· ,A N = | Ψ G i (cid:105) masks quantum information contained in {| (cid:105) , | (cid:105)} ∈ C into {| Ψ G (cid:105) , | Ψ G (cid:105)} ∈ ( C ) ⊗ N , where | Ψ G i (cid:105) = √ (cid:80) j =0 ( − ij | j, · · · , j (cid:105) , i = 0 ,
1. The reduced density matrices ( | Ψ G i (cid:105)(cid:104) Ψ G i | ) A (cid:96) ··· A (cid:96)m of( | Ψ G i (cid:105)(cid:104) Ψ G i | ) A ··· A n for any given m ∈ { , · · · , n − } are identical, that is, (cid:80) j =0 | j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) m (cid:105)(cid:104) j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) m | ,where (cid:96) , · · · , (cid:96) m ∈ { , · · · , n } . Hence we have no information about the value of i and it is a strongquantum information masking. Let | Ψ G i (cid:105) go through the locally unitary channel Λ in (17). Thestate | Ψ G i (cid:105) will become Λ( | Ψ G i (cid:105)(cid:104) Ψ G i | )= x | Ψ G i (cid:105)(cid:104) Ψ G i | + 1 − x N N (cid:88) k =1 ( P k + P k ) . (21)The ( N − N − x + 12 N (cid:88) j =0 | j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) N − (cid:105)(cid:104) j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) N − | + 1 − x N N − (cid:88) k =1 ( P (cid:63)k + P (cid:63)k ) , where P (cid:63)k is the projector onto the product state | φ k (cid:105) = | (cid:105) A | (cid:105) A · · · | (cid:105) A k · · · | (cid:105) A N − , and P (cid:63)k isthe projector onto the product state | ϕ k (cid:105) = | (cid:105) A | (cid:105) A · · · | (cid:105) A k · · · | (cid:105) A N − . So the resulting state is1-uniformly masked.The above two examples show that quantum information masking under noise (the channels weconstructed) still works. The traditional definition of quantum information masking asks for a mapfrom a pure state to another pure state. Our channels indirectly maps a pure state to a mixed stateand the masking effect still works. It is known that noise is not avoidable in nature. Thus quantuminformation masking under noise plays a key role in quantum secret sharing in experiment. VI. DSD OF TRIPARTITE GE STATES
A bipartite system under dephasing will evolve into the dynamics that DSD must precede ESD[15]. In this section, we consider the three-qutrit GE state distillable across every bipartition [51]. We investigate the evolution of it under global (i.e., local and collective) dephasing in (24). It turnsout that the state will undergo DSD and become PPT. We study the existence of PPT entanglementusing realignment criterion and indecomposable positive map.Assume that | η i (cid:105) = | (cid:105) + ( − i | (cid:105) , | ξ j (cid:105) = | (cid:105) + ( − j | (cid:105) , where i, j = 0 ,
1. Let B = {| ψ ( i, j ) (cid:105) = | (cid:105)| η i (cid:105)| ξ j (cid:105) , | ψ ( i, j ) (cid:105) = | η i (cid:105)| (cid:105)| ξ j (cid:105) , | ψ ( i, j ) (cid:105) = | (cid:105)| ξ j (cid:105)| η i (cid:105) , | ψ ( i, j ) (cid:105) = | η i (cid:105)| ξ j (cid:105)| (cid:105) , | ψ ( i, j ) (cid:105) = | ξ j (cid:105)| (cid:105)| η i (cid:105) , | ψ ( i, j ) (cid:105) = | ξ j (cid:105)| η i (cid:105)| (cid:105) , | ψ (0 , (cid:105) − | ψ (0 , (cid:105) , | ψ (0 , (cid:105) − | ψ (0 , (cid:105) , | ψ (0 , (cid:105) − | ψ (0 , (cid:105) , ( | (cid:105) + | (cid:105) + | (cid:105) )( | (cid:105) + | (cid:105) + | (cid:105) )( | (cid:105) + | (cid:105) + | (cid:105) ) , i, j = 0 , } . We review the unextendible biseparable base [15], B = B \ { (cid:83) (cid:96) =1 | ψ (0 , (cid:105) (cid:96) } and ρ (0) := 15 I ⊗ I ⊗ I − (cid:88) | ψ (cid:105)∈B | (cid:101) ψ (cid:105)(cid:104) (cid:101) ψ | , (22)where | (cid:101) ψ (cid:105) is a normalized state of | ψ (cid:105) . Then ρ (0) is a three-qutrit rank-five GE state, and it isdistillable across every bipartite cut [15]. One can prove that the three bipartite reduced densitymatrices ρ β (0) := Tr α [ ρ (0)] with β ∈ { BC, CA, AB } and α ∈ { A, B, C } , respectively, are identical.Hence it suffices to check whether the state ρ (0) will undergo ESD and DSD across the cut ρ A | BC (0). FIG. 4: Two systems A and BC are collectively interacting with the stochastic magnetic field B ( t ) and separatelyinteracting with the stochastic magnetic field b A ( t ) and b BC ( t ). As shown in FIG. 4, the initial state ρ (0) interacting with global dephasing only correlates withthe three stochastic magnetic fields b A ( t ) , b BC ( t ) and B ( t ) [14, 52]. The time-dependent densitymatrix for the system is ρ ( t ) = ε ( ρ (0)) = (cid:80) µ K † µ ( t ) ρ (0) K µ ( t ), where the Kraus operators K µ ( t )of the channel ε representing the influence of statistical noise are completely positive and tracepreserving, that is, (cid:80) µ K † µ K µ = I . By bonding systems B and C , We suppose that systems A , BC only correlate with the local magnetic fields in FIG. 4. That is, ρ (0) interacts with the local andcollective dephasing noise. It can be expressed as ρ g ( t ) = (cid:88) i =1 9 (cid:88) j =1 ( E Ai ) † ( F Bj ) † ρ (0) E Ai F Bj , (23)where E Ai and F Bj describe the interaction with the local magnetic fields b A ( t ) and b BC ( t ) respectively. The operators E Ai and F Bj are E ( t ) = diag(1 , γ A ( t ) , γ A ( t )) ⊗ I ,E ( t ) = diag(0 , ω A ( t ) , ⊗ I ,E ( t ) = diag(0 , , ω A ( t )) ⊗ I ,F ( t ) = I ⊗ diag(1 , γ B ( t ) , γ B ( t ) , . . . , γ B ( t )) ,F ( t ) = I ⊗ diag(0 , ω B ( t ) , , . . . , ,. . .F ( t ) = I ⊗ diag(0 , , . . . , , ω B ( t )) . (24)The time-dependent parameters are γ A ( t ) = γ B ( t ) = e − Γ t/ , ω A ( t ) = (cid:112) − γ A ( t ), ω B ( t ) = (cid:112) − γ B ( t ), and Γ is the dephasing rate. FIG. 5: The three-dimensional plots of the minimum eigenvalue λ min of ρ Γ g ( t ) in different viewing angles with thelocal asymptotic dephasing rate Γ ∈ [0 ,
1] and the time t ∈ [0 , ρ Γ g ( t ) is an NPT state for any t ∈ [0 ,
3] and Γ ∈ [0 , . t ∈ [0 , . Γ ∈ [0 , Let ρ Γ g ( t ) be the partial transpose of ρ g ( t ) in Eq. (23). Consider the local asymptotic dephasingrate Γ ∈ [0 ,
1] and the time t ∈ [0 , λ min ( ρ Γ g ( t )) of ρ Γ g ( t ).Then we find that ρ g ( t ) is NPT for any t ∈ [0 ,
3] and Γ ∈ [0 , . t ∈ [0 , . Γ ∈ [0 , ρ g ( t ) is entangled. For any t ∈ (1 . ,
3] andsome Γ ∈ (0 . , ρ g ( t ) is PPT. Thus the state ρ (0) will undergo multipartite DSD.In FIG. 5, we draw the three-dimensional plots of the minimum eigenvalue λ min ( ρ Γ g ( t )) in differentviewing angles. Next we use the realignment criterion in Lemma 2 to compute (cid:107) ρ Rg ( t ) (cid:107) − Γ ∈ [0 ,
1] and the time t ∈ [0 , (cid:107) ρ Rg ( t ) (cid:107) − t ∈ [0 , . Γ ∈ [0 , t ∈ [0 ,
3] and Γ ∈ [0 , . (cid:107) ρ Rg ( t ) (cid:107) − ρ Γ g ( t ) and the valueof (cid:107) ρ Rg ( t ) (cid:107) − ρ (0) is PPT entangled.In the rest of this section, we choose the local asymptotic dephasing rate Γ = 1. Then we can (cid:107) ρ Rg ( t ) (cid:107) − t ∈ [0 ,
3] and Γ ∈ [0 ,
1] in different viewing angles. Wecan obtain that (cid:107) ρ Rg ( t ) (cid:107) − t ∈ [0 , . Γ ∈ [0 , t ∈ [0 ,
3] and Γ ∈ [0 , . obtain that ρ g ( t ) will become a PPT state after t ≈ .
38. By computing (cid:107) ρ Rg ( t ) (cid:107) −
1, we obtain thatif t > .
18, then the value of (cid:107) ρ Rg ( t ) (cid:107) − ρ g ( t ) with Γ = 1 can not be distillableunder LOCC when t > .
38. Next, we use the indecomposable positive map proposed in [53] todetect whether ρ g ( t ) is PPT entangled. It can detect entanglement in a certain class of two-qutritPPT entangled states. The one parameter class of linear trace preserving maps Λ α isΛ α = 1 / ( α + 1 α ) α ( x + x ) − x − αx − x
21 1 α ( x + x ) − x − αx − x αx + α x , (25)where X = x x x x x x x x x is any 3 × α ∈ (0 , ρ g ( t ) in (23) is PPT andthe minimum eigenvalues of ( I ⊗ Λ α ) ρ g ( t ) is smaller than zero, then we can say that ρ g ( t ) is PPTentangled for some t and Γ . By choosing the local asymptotic dephasing rate Γ = 1, we computethe minimum eigenvalues λ min of ( I ⊗ Λ α ) ρ g ( t ) versus time t ∈ [0 ,
3] and α ∈ (0 , λ min of ( I ⊗ Λ α ) ρ g ( t ) in different viewingangles versus time t ∈ [0 ,
3] and α ∈ (0 , α ∈ (0 , t ∈ [0 , . λ min of ( I ⊗ Λ α ) ρ g ( t ) is negative. However, t = 0 . < .
38. The numericalresults do not show the PPT entanglement of ρ g ( t ) when Γ = 1. Hence, ρ g ( t ) with the dephasingrate Γ = 1 will become PPT, thus is not distillable under LOCC. That is, the state ρ (0) with localand collective noise in (24) when Γ = 1 will undergo multipartite DSD. λ min of ( I ⊗ Λ α ) ρ g ( t ) in different viewing anglesversus time t and α for the local asymptotic dephasing rate Γ = 1. For α ∈ (0 , λ min of( I ⊗ Λ α ) ρ g ( t ) is negative when t ∈ (0 , . VII. CONCLUSIONS
We constructed the GESD-free states and multipartite DSD-free states by establishing the mul-tipartite high dimensional Pauli channel. We presented the locally unitary channel such that the N -partite GHZ state becomes the D¨ur’s multipartite state. We also studied the quantum informa-tion masking under the channel we constructed. Further, we investigated the three-qutrit GE statedistillable across every bipartition under global noise. The numerical results showed that it mayundergo DSD and become PPT. Our future work is to give more constructions of GESD-free andmultipartite DSD-free states from other states such as Werner states as well as their relation toquantum information masking under more general noise. Acknowledgments
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