Average fidelity and fidelity deviation in noisy quantum teleportation
aa r X i v : . [ qu a n t - ph ] F e b Average fidelity and fidelity deviation in noisy quantumteleportation
Wooyeong Song ∗ Department of Physics, Hanyang University, Seoul 04763, Korea
Junghee Ryu ∗ Division of National Supercomputing,Korea Institute of Science and Technology Information, Daejeon 34141, Korea
Kyunghyun Baek and Jeongho Bang † School of Computational Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea (Received February 11, 2021)
Abstract
We analyze the average fidelity (say, F ) and the fidelity deviation (say, D ) in noisy-channelquantum teleportation. Here, F represents how well teleportation is performed on average and D quantifies whether the teleportation is performed impartially on the given inputs, that is, thecondition of universality. Our analysis results prove that the achievable maximum average fidelityensures zero fidelity deviation, that is, perfect universality. This structural trait of teleporta-tion is distinct from those of other limited-fidelity probabilistic quantum operations, for instance,universal-NOT or quantum cloning. This feature is confirmed again based on a tighter relationshipbetween F and D in the qubit case. We then consider another realistic noise model where F decreases and D increases due to imperfect control. To alleviate such deterioration, we proposea machine-learning-based algorithm. We demonstrate by means of numerical simulations that theproposed algorithm can stabilize the system. Notably, the recovery process consists solely of themaximization of F , which reduces the control time, thus leading to a faster cure cycle. Keywords: Quantum teleportation; Quantum machine learning ∗ Wooyeong Song and Junghee Ryu contributed equally to this work † Electronic address: [email protected] . INTRODUCTION Quantum teleportation makes possible the deterministic transmission of unknown quan-tum states from one location to another [1]. It has been acknowledged as a fundamentalscheme of state transfer. A shared quantum channel between a sender and a receiver isone of the essential ingredients for quantum teleportation, and quantum entanglement inthe channel is necessary to ensure that the fidelity is superior to that of classical commu-nication protocols. Moreover, quantum teleportation provides a useful framework to studyquantum nonlocality [2–5] and is one of the basic steps in constructing element gates, for ex-ample, single-qubit and CNOT gates, which are used in continuous-variable (CV) quantumcomputation [6, 7].Fidelity f is used to measure the closeness between the input and the teleported states.One of the methods to quantify the performance of teleportation based on f without anydependence on the input states involves averaging f over all the possible inputs. The av-erage fidelity F , defined as a uniform average of f , has been widely used as a relevantinput-independent measure of teleportation performance. It is well-known that F can reach1 in quantum teleportation, whereas the maximally attainable F is limited in any classicalstate-transfer scheme without quantum entanglement [1]. However, F does not consideruniversality, which indicates whether teleportation is performed equally for all of the inputstates. For example, a non universal teleportation protocol would be successful only fora specific set of inputs. This limitation can hinder the use of a teleportation protocol toimplement element gates in CV quantum computation [6, 7]. To quantify the universal-ity condition, a fidelity deviation D defined in terms of the standard deviation of f wasintroduced [8, 9].In this study, we analyze the two measures F and D in the context of noisy qudit—a d -level quantum system—teleportation. We show that perfect universality, D = 0, is at-tainable without any dependence on the quantum channel condition, while the maximumaverage fidelity F max is a function of the degree of entanglement in the quantum channel.We prove that the condition of F max is adequate to guarantee perfect universality. For thecase of a qubit, that is, d = 2, we demonstrate a more general and tighter relationshipbetween F and D that addresses the aforementioned behaviors in a clearer manner. Then,we consider a realistic situation in which operational noises can deteriorate the teleporta-1ion performance, as represented by a decrease in F and an increase in D . We propose asignificant machine-learning-based method to alleviate such deterioration. We numericallydemonstrate that the proposed machine-learning-based method is effective within a certainrate of noise occurrence. Note that the recovery of F and D can be implemented solely byusing the process of F → F max . This is because F max itself suffices the condition of zerodeviation. This feature allows us to reduce the control and/or learning time, which increasesthe recovery rate. II. AVERAGE FIDELITY AND FIDELITY DEVIATION
We first review the two measures, namely, average fidelity F and fidelity deviation D .Consider a map (or a general quantum operation) T : | φ i → ˆ ρ out , where | φ i is an input andˆ ρ out an output. The fidelity f is defined as [10] f = Tr (cid:20)(cid:16)p ˆ ρ t ˆ ρ out p ˆ ρ t (cid:17) / (cid:21) , (1)where ˆ ρ t is a target state that is assumed to be a pure state | φ t i . Then, formula (1) can berewritten as f = h φ t | ˆ ρ out | φ t i . (2)In a well-designed T , almost all of the input states | φ i are correctly transferred to theircorresponding targets | φ t i . The average fidelity represents how well a state transfer is per-formed. It is defined by averaging f over all possible inputs | φ i as F = Z d φ f, (3)where d φ is the Haar measure that satisfies R d φ = 1. In general, f varies with respect tothe input states for a given channel T . If f is uniform for a task, the task is said to beuniversal. To devise a measure of universality, we employ the fidelity deviation D in termsof the fluctuation of f [8, 9]: D = (cid:18)Z d φ f − F (cid:19) . (4)One can show perfect universality, that is, D = 0 when f = F . Otherwise, it is strictlypositive. Additionally, D ≤ Z d φ f − F = F (1 − F ) ≤ , (5)2here the last equality holds when F = 1 /
2. Thus, D is bounded as 0 ≤ D ≤ / F = 1 holds iff f = 1 for all possible inputs, andtherefore, F = 1 implies perfect universality. However, when the attainable maximum of F is less than one ( F max < F max in general. Forexample, if F max <
1, then 0 ≤ D ≤ p F (1 − F ) [see Eq. (5)]. This is true for almost allprobabilistic tasks, for instance, universal-NOT [11, 12] and quantum cloning [13]. Therefore,it is natural to consider the minimization of D independently of the maximization of F . III. NOISY QUANTUM TELEPORTATION
In this section, we describe noisy quantum teleportation [14]. First, a sender, say Alice,has a d -dimensional pure state ˆ ρ φ = | φ ih φ | that is to be transferred to a receiver, say Bob(in general, Alice has no information about the input state ˆ ρ φ ). A maximally entangledstate | Ψ i = √ d P d − j =0 | j i ⊗ | j i is shared by Alice and Bob. Second, Alice performs a jointmeasurement by using a set of maximally entangled bases {| Ψ α i} ( α = 0 , , . . . , d −
1) onthe composite system of the unknown state and one of the entangled pairs. The entangledbasis is obtained as | Ψ α i = ( ˆ U α ⊗ ˆ d ) | Ψ i , (6)where ˆ d is the identity of the d -dimensional Hilbert space and ˆ U α is a unitary conditionedby completeness, that is, P d − α =0 | Ψ α i h Ψ α | = ˆ d [14, 15]. The outcome of Alice’s jointmeasurement is communicated to Bob through a classical channel. Lastly, Bob applies alocal operation ˆ V α based on the measurement outcomes received from Alice. Then, the state | φ i is reproduced on Bob’s side (see Fig. 1).Sharing the entanglement between Alice and Bob is the crucial step in ensuring that thebetter performance is superior to that of any classical protocol [16]. However, under realisticconditions, quantum entanglement often becomes noisy [17, 18], and thus, we need to resolvethe noisy case. Herein, we consider the following ( d × d )-dimensional noisy entanglement asthe quantum channel [19]:ˆ ρ Ψ = γ | Ψ i h Ψ | + 1 − γd ˆ d , (7)which is a statistical mixture of the maximally entangled state | Ψ i and white (symmetric)noise, and γ is a fraction of | Ψ i . 3 lice Bob Noisy Entangled source
Unknown quantum state
Joint measurements Local operation
Classical channel { ˆ U α } { ˆ V α } ˆ ρ Ψ Teleported state
Unitaries are controlled by control parameter p p ( u ) p ( v ) FIG. 1: Schematic of d -dimensional quantum state teleportation. The quantum channel, i.e.,shared entangled state ˆ ρ Ψ , can be noisy [as in Eq. (7)]. The errors can also arise in ˆ U α and ˆ V α due to the imperfections of control. It can be cured in our scheme (see Sect. V for details).. IV. ANALYSIS OF F AND D IN NOISY QUANTUM TELEPORTATION
In this section, we discuss the relationship between the two measures, namely, the averagefidelity F and fidelity deviation D , according to the noise parameter γ and dimension d . A. Qudit-teleportation
By definition, the value of F ranges between 0 and 1. However, for a given set of γ and d , one can restrict the range by using the following relationship: Result 1.
For a given set of noise parameter γ and dimension d , F has a relation such that d − γd ( d + 1) ≤ F ≤ γ + (1 − γ ) d . (8)4his result is derived from the following extensive analysis of F . First, we write thetransmitted state ˆ ρ out asˆ ρ out = d − X α =0 (cid:16) ˆ d ⊗ ˆ V α (cid:17) (cid:10) Ψ A α (cid:12)(cid:12) ˆ ρ tot (cid:12)(cid:12) Ψ A α (cid:11) (cid:16) ˆ d ⊗ ˆ V † α (cid:17) , (9)where (cid:12)(cid:12) Ψ A α (cid:11) = | Ψ α i ⊗ ˆ d denotes Alice’s joint measurement, ˆ d ⊗ ˆ V α represents Bob’s localoperation, and ˆ ρ tot = ˆ ρ φ ⊗ ˆ ρ Ψ is the total initial state of ˆ ρ φ that is to be transferred, whereˆ ρ Ψ is the shared entangled state. We can rewrite Eq. (9) asˆ ρ out = γd d − X α =0 ˆ X α ˆ ρ φ ˆ X † α + 1 − γd ˆ d , (10)where ˆ X α = ˆ V α ˆ U † α . The fidelity f for a given ˆ ρ φ = | φ i h φ | is f = γd d − X α =0 ξ α + 1 − γd , (11)where ξ α is the fidelity between ˆ ρ φ and ˆ X α ˆ ρ φ ˆ X † α , and it is expressed as ξ α = Tr( ˆ X α ˆ ρ φ ˆ X † α ˆ ρ φ ) = (cid:12)(cid:12)(cid:12) h φ | ˆ X α | φ i (cid:12)(cid:12)(cid:12) . (12)By using the above descriptions, we evaluate F and D . First, we consider the averagefidelity F . From Eqs. (3) and (11), we have F = Z d φ f = γd d − X α =0 Z d φ ξ α + 1 − γd . (13)The integral R d φ ξ α can be calculated by using a lemma of identity, called Schur’s lemma [14,20]: Z G d g (cid:16) ˆ U † g ⊗ ˆ U † g (cid:17) ˆ X (cid:16) ˆ U g ⊗ ˆ U g (cid:17) = a ˆ d + b ˆ P , (14)where a = d Tr( ˆ X ) − d Tr( ˆ X ˆ P ) d ( d − ,b = d Tr( ˆ X ˆ P ) − d Tr( ˆ X ) d ( d − , (15)for any operator ˆ X in the d × d -dimensional Hilbert space. Here, d g denotes the (normalized)Haar measure on the unitary group G = U ( d ), satisfying R G d g = 1; ˆ U g is an irreducible5epresentation of g ∈ G ; and ˆ P denotes the swap operator defined by ˆ P | ij i = | ji i . Byapplying Schur’s lemma, we can compute the integral R d φ ξ α such that Z d φ ξ α = h | Z d φ (cid:16) ˆ U † φ ⊗ ˆ U † φ (cid:17) (cid:16) ˆ X α ⊗ ˆ X † α (cid:17) (cid:16) ˆ U φ ⊗ ˆ U φ (cid:17) | i = 1 d ( d + 1) (cid:18)(cid:12)(cid:12)(cid:12) Tr( ˆ X α ) (cid:12)(cid:12)(cid:12) + d (cid:19) , (16)where | φ i = ˆ U φ | i . Then, we arrive at the final form of F as F = F max − γd + 1 d − d d − X α =0 (cid:12)(cid:12)(cid:12) Tr( ˆ X α ) (cid:12)(cid:12)(cid:12) ! , (17)where F max = γ + (1 − γ ) d , which is the maximum value of F for a given γ .We rewrite Eq. (17) in a more useful form as follows: F = dE + 1 d + 1 , (18)where E is defined as E = 1 d d − X α =0 h Ψ α | ˆ ρ Ψ | Ψ α i = γd d − X α =0 (cid:12)(cid:12)(cid:12) Tr( ˆ X α ) (cid:12)(cid:12)(cid:12) + 1 − γd . (19)We find that E is maximized when Tr( ˆ X α ) = d (or equivalently, ˆ X α = ˆ d ) for all α andminimized when Tr( ˆ X α ) = 0 for all α . Then, the value of E is bounded by γ such that(1 − γ ) d ≤ E ≤ γ + (1 − γ ) d , (20)where the upper bound γ + (1 − γ ) d is called the (fullest) entanglement fraction of the chan-nel [20]. In this manner, by using Eq. (18) we can finally prove Result 1. Notably, this resultis consistent with the results obtained in previous studies [16, 20].Next, we consider the fidelity deviation D . Moreover, because the range of F is limited,we derive a condition of D for a given γ and d as follows. Result 2.
For a given set of noise parameter γ and dimension d , the fidelity deviation D has a condition such that ≤ D ≤ γ ∆ , (21) where ∆ = d P d − α =0 ∆ α , and ∆ α = Z d φ ξ α − (cid:18)Z d φ ξ α (cid:19) , (22) which can be regarded as the fidelity deviation of ξ α .
6o prove this result, we express D as follows by using Eqs. (11) and (13) D = (cid:18)Z d φ f − F (cid:19) = γd d − X α,β =0 C αβ ! , (23)where C αβ are elements of covariance matrix C given by C αβ = Z d φ ξ α ξ β − Z d φ ξ α Z d φ ξ β . (24)Note that C is symmetric, that is, C αβ = C βα , and its diagonal elements C αα are equal to∆ α in Eq. (22). Furthermore, each element of C is bounded as | C αβ | ≤ q ∆ α ∆ β , (25)which is known as the variance-covariance inequality [21]. Then, by using Eq. (25), weobtain Result 2. The perfect universality D = 0 can be achieved when ξ α is constant for all α . Based on the above results, we discuss the conditions of the sets { ˆ U α } and { ˆ V α } fromthe viewpoint of achieving the maximum average fidelity and perfect universality. First, weconsider the maximization of F . According to Eq. (17), F max is achieved whenˆ X α = ˆ d , or equivalently, ˆ U α = ˆ V α , ( ∀ α ) . (26)That implies that the maximization of E straightforwardly leads to F max . Therefore, ourmain result is as follows: Result 3.
The condition of Eq. (26) naturally suffices perfect universality, that is, ˆ X α = ˆ d → the perfect universality D = 0 . (27)The proof is simple. By using Eq. (12), we rewrite condition (26) as ξ α = 1 ( ∀ α ) . (28)Then, C becomes a zero (or null) matrix because C αβ = 0 for ξ α = const ( ∀ α ), thus leadingto D = 0, see Eq. (23). However, the opposite is not always true, that is, perfect universalitydoes not guarantee the maximum fidelity. We emphasize that Result 3 is a non-commontrait in limited maximum fidelity tasks, for instance, universal-NOT or cloning [8].7 . Qubit-teleportation In the case of a qubit, that is, d = 2, we can derive a tighter relationship between F and D . To this end, we first write the input state ˆ ρ φ in terms of the Bloch representation asˆ ρ φ = 12 (cid:0) ˆ + φ T σ (cid:1) , (29)where φ = ( φ x , φ y , φ z ) T is a Bloch vector of unit norm (i.e., | φ | = 1) in three-dimensional realvector space R , and σ = (ˆ σ x , ˆ σ y , ˆ σ z ) T is a vector operator whose components ˆ σ j ( j = x, y, z )are Pauli operators. Then, we can express ξ α in Eq. (12) as ξ α = 12 (cid:0) φ T R α φ (cid:1) , (30)where R α is a 3 × R , whose elements [ R α ] jk are given as[ R α ] jk = 12 Tr( ˆ X α ˆ σ j ˆ X † α ˆ σ k ) for j, k = x, y, z. (31)The rotation angles ϑ α and axes n α of R α are found in the general expression of a single-qubitunitary asˆ X α = e − i ϑα n T α σ = cos ϑ α − i sin ϑ α n T α σ . (32)Then, by using Eqs. (13) and (30), the F corresponding to d = 2 can be expressed as F = γ X α =0 (cid:20) Z d φ (cid:0) φ T R α φ (cid:1)(cid:21) + 1 − γ , (33)where d φ is the Haar measure over the surface of the Bloch sphere, and it is normalized as R d φ = 1. Here, we again employ Schur’s lemma to calculate the integral R d φ ( φ T R α φ ): Z G d g O g XO T g = 1 r Tr( X ) I r , (34)where I r is the identity matrix in R r , O g is an irreducible orthogonal representation of anelement g ∈ G , and d g is the measure, normalized as R g d g = 1. By applying this lemma to O (3) of three-dimensional rotations, we can calculate the integral in Eq. (33) as Z d φ ( φ T R α φ ) = 13 Tr( R α ) , (35)and we immediately obtain the following form of F : F = 12 + γ X α =0 Tr( R α ) . (36)8ext, we consider D . First, we express D as D = γ X α =0 ∆ α + X α = β C αβ ! . (37)Subsequently, we (re)calculate the inequality of Eq. (25) for d = 2, and a tighter lower boundcan be found as follows (for more details, see appendix B in Ref. [8]): −
12 ∆ α ∆ β ≤ C αβ ≤ ∆ α ∆ β ( α = β ) , (38)where we obtain the lower bound when the two rotation axes n α and n β are orthogonalto each other, that is, n T α n β = 0, and the upper bound when n α and n β are parallel orantiparallel, that is, n T α n β = ±
1. Then, the upper bound of D is [from Eq. (38)] D ≤ γ vuut X α =0 ∆ α + X α = β ∆ α ∆ β = γ ∆ d =2 , (39)where ∆ d =2 = P α =0 ∆ α . Here, ∆ α are given in the Bloch form, such that∆ α = 12 (cid:20)Z d φ (cid:0) φ T R α φ (cid:1) −
19 Tr( R α ) (cid:21) . (40)Now, we calculate R d φ ( φ T R α φ ) by using the product form of Schur’s lemma: Z d g (cid:0) O T g ⊗ O T g (cid:1) X ( O g ⊗ O g ) = a I d + b D + c P , (41)where a = ( r + 1)Tr( X ) − Tr( XD ) − Tr( XP ) r ( r − r + 2) ,b = − Tr( X ) + ( r + 1)Tr( XD ) − Tr( XP ) r ( r − r + 2) ,c = − Tr( X ) − Tr( XD ) + ( r + 1)Tr( XP ) r ( r − r + 2) . Here, P is a swap matrix P ( x i ⊗ x j ) = x j ⊗ x i , or equivalently, P = P r − i,j =0 ( x j ⊗ x i ) ( x i ⊗ x j ) T , and D = (cid:0)P r − i =0 x i ⊗ x i (cid:1) (cid:16)P r − j =0 x j ⊗ x j (cid:17) T where { x i } is an or-thonormal basis set in R r . Then, by using this lemma, we can rewrite R d φ ( φ T R α φ ) as x T00 Z d φ (cid:16) O T φ ⊗ O T φ (cid:17) ( R α ⊗ R α ) (cid:16) O φ ⊗ O φ (cid:17) x , (42)9here x = x ⊗ x and φ = O φ x . Eq. (40) is then calculated and we have∆ α = 12 √ (cid:18) −
13 Tr( R α ) (cid:19) , (43)where we used the following properties:Tr( R α ⊗ R α ) = Tr( R α ) , Tr( R α ⊗ R α D ) = Tr( R α R T α ) = Tr( I ) = 3 , Tr( R α ⊗ R α P ) = Tr( R α ) . (44)Consequently, we can derive a tighter relationship between F and D as follows: D ≤ √ F max − F ) , (45)where F max = γ . This confirms Result 3, that is, D = 0 iff F = F max . Here, we can provethat the maximum value of D , that is, the worst case of universality, implies the minimumaverage fidelity, F min = − γ . V. MACHINE-LEARNING-BASED STABILIZATION OF CONTROL IN TELE-PORTATION
Result 3 implies the following: [
T.1 ] The maximization of F naturally includes the min-imization of D in noisy teleportation. This is indeed a structural trait of teleportation.We conjecture that quantum teleportation degraded by operational noises can be effec-tively cured and further provide a benefit of utilizing the aforementioned trait [
T.1 ] can beachieved. To investigate this, we assume the operational noises in the control of Alice’s jointmeasurements (i.e., { ˆ U α } ) and Bob’s local operations (i.e., { ˆ V α } ), which deteriorate F and D . We propose a machine-learning-based algorithm to stabilize the teleportation systemagainst noise. For qubit teleportation (i.e., d = 2), we demonstrate by means of numericalsimulations that our machine-learning-based algorithm can cure the deterioration of both F and D solely by maximizing F (see Fig. 1).10 . Effects of operational noise In general, a unitary operation in the d -dimensional Hilbert space can be parameterizedas follows:ˆ U ( p ) = exp ( − i p T G ) , (46)where p = ( p , p , . . . , p d − ) T is a ( d − G =(ˆ g , ˆ g , . . . , ˆ g d − ) T is a vector operator whose components are SU( d ) group generators ˆ g j ( j = 1 , , . . . d −
1) [22–24]. Here, the components of p correspond to a set of control pa-rameters in a real experiment, such as multiport beam splitters and phase shifters in linearoptical systems [25] or the pulse sequences of solid system qubits [26]. The operationalnoises of { ˆ U α } and { ˆ V α } can be formulated as follows: the control parameters of ˆ U α and ˆ V α fluctuates such that p → p + η ǫ , (47)where ǫ = ( ǫ , ǫ , . . . , ǫ d − ) T is a vector of the stochastic errors ǫ j ∈ [ − π, π ]. The factor η ∈ [0 ,
1] is casted to represent the degree of immaturity in control.We investigate the effects of operational noises on qubit teleportation. First, we assumethat the protocol is already set with the optimal condition as in Eq. (26), and in this case,operational noise occurs continuously in both { ˆ U α } and { ˆ V α } . To understand the effects ofoperational noise, we perform Monte-Carlo simulations with increasing η . The results areshown in Fig. 2, where each data point is created by averaging 10 simulations. F decreasesfrom 1 to F err , and D increases from 0 to D err . The fullest deterioration of F , that is, F err ≃ . F rand = , whereas D err increases to ≃ . D maxerr ≃ √ even in the worst case of F err ≃ . This implies thatthe deterioration of F is more conspicuous than that of D . Here, F rand is the average fidelityof the purely random protocol and D maxerr represents half (mean) of the maximum fidelitydeviation at F err , that is, D maxerr = √ ( F max − F err ) (see the dashed line in Fig. 2). B. Machine-learning-based algorithm for stabilization of control
To cure the unstable control, we propose a machine-learning-based algorithm built on theso-called differential evolution concept [8], where the control parameters of { ˆ U α } and { ˆ V α } η F η D FIG. 2: Deteriorations of F (left) and D (right) with respect to η . Each data point is created byaveraging 10 simulations. The error bar denotes the standard deviation. The teleportation systembreaks down due to operational noise, as indicated by F = 1 → F err < D = 0 → D err > are allowed to evolve during the process. The algorithm runs as follows: First, N pop setsof the control parameter vectors are prepared as the candidate solutions { p ( u ) α,i , p ( v ) α,i } , where p ( u ) α,i and p ( v ) α,i are respectively the control parameter vectors of the candidate operations ˆ U α,i and ˆ V α,i ( i = 1 , , . . . , N pop ). Thus, we have 2 d N pop parameter vectors. Then, the preparedsets of the candidate solutions are allowed to evolve through the following steps: (1) Wegenerate 2 N pop mutant vectors ν ( u,v ) α,i for ˆ U α,i and ˆ V α,i according to ν ( u,v ) α,i = p ( u,v ) α,a + W (cid:16) p ( u,v ) α,b − p ( u,v ) α,c (cid:17) , (48)where p ( u,v ) α,a , p ( u,v ) α,b , and p ( u,v ) α,c are randomly selected for a, b, c ∈ { , , . . . , N pop } . Thesevectors are selected to be different from each other. The free parameter W , also called adifferential weight, is a real and constant number. (2) Thereafter, all 2 d N pop parametervectors, p ( u ) α,i = p ( u )1 p ( u )2 ... p ( u ) d − , α,i , p ( v ) α,i = p ( v )1 p ( v )2 ... p ( v ) d − , α,i (49)12 IterationF p=1p BV p C IterationD p=1p BV p C FIG. 3: Remediation of fully broken teleportation. The graphs of F (left) and D (right) are plottedbased on the results of the numerical simulations. Here, we consider three cases: γ = γ C = , γ = γ BV = √ , and γ = 1. Each data point is created based on 10 repeating simulations. are reformed to trial vectors, τ ( u ) α,i = τ ( u )1 τ ( u )2 ... τ ( u ) d − , α,i , τ ( v ) α,i = τ ( v )1 τ ( v )2 ... τ ( v ) d − , α,i (50)by the rule: For each j = 1 , , . . . , d − τ ( u,v ) j ← p ( u,v ) j if R j > C r ,τ ( u,v ) j ← ν ( u,v ) j otherwise , (51)where R j ∈ [0 ,
1] is a randomly generated number, and the crossover rate C r is anotherfree parameter ranging between 0 and 1. Note that these free parameters W and C r areset to achieve the best learning efficiency. (3) Finally, the control parameter vectors areevaluated by using the fitness criteria, that is, how well do the given parameters fit to theprotocol. More specifically, { τ ( u ) α,i , τ ( v ) α,i } are taken if they yield a higher level of fitness; ifnot, { p ( u ) α,i , p ( v ) α,i } are retained. In our algorithm, we extract the best fitness among N pop andretain the corresponding parameters { p α, best , p α, best } . Steps (1)-(3) are then repeated.We investigate numerically whether even a fully broken teleportation system can becured using the proposed machine-learning-based algorithm. The numerical simulation isperformed for d = 2. Here, we take N pop = 100, and the free parameters of our algorithm areselected such that: W = 0 . C r = 0 .
1. To use the structural trait [
T.1 ] of teleportation,13
IterationF
IterationD
FIG. 4: Real-time remediation of control fluctuations. The results of the numerical simulationsare given as graphs of F (left) and D (right). Each data point is created based on 10 simulations.It is assumed that fluctuation occurs after every 10 or 50 iterations of our algorithm. For moreconvincing analysis, we consider the worst case, namely, η = 1. It is observed that the teleportationcan be cured continuously. we define fitness in terms of F ; in other words, there is no minimization of D . Note thatin general, fitness should be defined as a function of F and D (for example, see Ref. [8]).Such a setting is indeed beneficial, as described later. In Fig. 3, we present the results inthe form of graphs of F and D for three cases: γ = γ C = , γ = γ BV = √ , and γ = 1.Here, γ C denotes the condition of the separability of the channel and γ BV is the criticalvalue that the entanglement of the channel allows the violation of CHSH inequality. Eachdata point is created based on 10 repeated simulations. The results indicates that thebroken teleportation system can be recovered; F approaches F max , and D decreases to zero.Specifically, we obtain ( F ≃ . D ≃ . γ = γ C , ( F ≃ . D ≃ . γ = γ BV , and ( F ≃ . D ≃ . γ = 1.We further investigate through numerical simulations whether the system can be stabi-lized, by assuming that the fluctuation occurs abruptly after intervals of some iterations ofour algorithm. Note that such a model is realistic [27, 28]. Here, we consider the scenarios inwhich the controls fluctuate after every 10 and 50 iterations. For more faithful and confidentanalysis, we consider the scenario with the worst deterioration, that is, η = 1. Fitness isdefined solely by F . In Fig. 4, we present our simulation results. F and D deteriorate totheir fullest extents, but the system is cured continuously.14 I. SUMMARY AND REMARKS
We analyzed the average fidelity and fidelity deviation for noisy teleportation. We provedthat teleportation can be zero fidelity deviations (or equivalently, the perfect universality)independently of the quantum channel condition, while the achievable maximum averagefidelity is limited by the fraction of entanglement in the channel. Based on these analyses,we derived Result 3: the maximum average fidelity ensures perfect universality in quan-tum teleportation. For the case of d = 2, we derived a tighter relationship between thetwo measures. Taking into account other realistic noises, namely, the fluctuations in sys-tem control, we proposed a machine-learning-based algorithm to stabilize teleportation. Wedemonstrated by means of numerical simulations that even the fullest deteriorations can becured. It is remarkable that the process of fidelity maximization guarantees the minimiza-tion of fidelity deviation without additional processes. The aforementioned trait (comingfrom Result 3) is indeed beneficial to reduce the algorithm time and realize faster system re-mediation; in fact, if the minimization of fidelity deviation was considered in the algorithm,we may not have obtained a cure cycle (i.e., sufficient time for iterations to cure abruptfluctuations). Such a gain is expected to be more conspicuous in large- d teleportation. Acknowledgments
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