Discriminating quantum correlations with networking quantum teleportation

Shih-Hsuan Chen, He Lu, Qi-Chao Sun, Qiang Zhang, Yu-Ao Chen, and Che-Ming Li1,3,5,6,∗ Department of Engineering Science, National Cheng Kung University, Tainan 70101, Taiwan School of Physics, Shandong University, Jinan 250100, China Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Center for Quantum Technology, Hsinchu 30013, Taiwan and Center for Quantum Frontiers of Research & Technology, National Cheng Kung University, Tainan 701, Taiwan

The Bell inequality, and its substantial experimental violation, offers a seminal paradigm for showing that the world is not in fact locally realistic. Here, going beyond the concept of Bell's inequality, we show that quantum teleportation can be used to quantitatively characterize quantum correlations using a generic physical model of genuinely classical processes. The validity of the proposed formalism is demonstrated by considering the problem of teleportation through a linear three-node quantum network. A hierarchy is derived between the Bell nonlocality, nonbilocality, steering and nonlocality-steering hybrid correlations based on a process fidelity constraint. The proposed formalism can be directly extended to reveal the nonlocality structure behind teleportation through any linear many-node quantum network. The formalism provides a faithful identification of quantum teleportation and demonstrates for the first time the use of quantum-information processing as a means of quantitatively discriminating quantum correlations.
Quantum teleportation [1] enables networking participants to move an unknown quantum state between the nodes of a quantum network [2]. Quantum teleportation experiments have been realized in laboratories [3][4][5], free-space [6,7], and even ground-to-satellite [8]. The ideal teleportation of a qubit with an unknown state ρ acts as an identity unitary transformation, χ I , on the transmitted state, i.e., χ I (ρ) = ρ, as illustrated in Figs. 1(a,b). However, if such a quantum process is attacked by an eavesdropper, or manipulated by untrusted networking participants, the performance of all the networking tasks underlying the quantum teleportation process becomes questionable [5]. Thus, the problem of identifying genuinely quantum teleportation through quantum networks, and ruling out any classical strategies of mimicry, poses an interesting but significant challenge to both quantum-information processing and practical implementation. In particular, while it is known that networking teleportation is fueled by entangled pairs shared between participants, it is not yet clear how the teleportation task can be utilized to quantitatively characterize the quantum correlations underlying the network.
In order to tackle this problem, we introduce the concept of a genuinely classical process (GCP) to simulate the ideal quantum teleportation process, χ I , and provide a strategy for mimicking teleportation by classical physics. The proposed formalism not only provides a benchmark of faithful teleportation, but also gives the means to classify the quantum correlations between the quantum nodes. In contrast to existing theories, which utilize the state characteristics to verify the teleportation process [9][10][11] and quantum correlations (e.g., Bell nonlocality [12], nonbilocality [13][14][15][16] and non-N locality [17][18][19]), the proposed formalism is truly task-oriented, and is thus well suited to the characterization of gen- Implementation of teleportation. The sender node (Alice) and receiver node (Bob) of the network first share a qubit pair from an entanglement source (E). Alice performs jointstate measurement on the transmitted qubit with a state ρ and half of the entangled pair held by her in the basis of Bell states (not shown). She then sends her measurement result to Bob. Finally, depending on Alice's measurement outcome, Bob performs local operations on his half of the entangled pair to recover the unknown state. (b) Notably, this inputoutput procedure acts as an identity quantum process χI . (c) The proposed formalism of a genuinely classical process (GCP) χGC is used to simulate χI and derive faithful criteria for experiments. (d) The GCP formalism is sufficiently general to encompass the local hidden variable (LHV) model for mimicking teleportation using a classical source (S). eral many-node networking teleportation and its underlying quantum resources. Moreover, the formalism can be readily implemented in a wide variety of present experiments on teleportation, as will be shown in subsequent sections.
Genuinely classical processes.-We define a GCP as a set of three steps describing the system state and its evolution. In particular, the input system is considered to be a physical object with properties satisfying the assumption of realism [12]. The system then evolves in accordance with classical stochastic theory [20]. Finally, once the process is complete, the system resides in its final output state with properties satisfying the assumption of realism.
Since a GCP treats the initial system as a physical object with properties satisfying the assumption of realism, the system can be modeled as a state described by a fixed set of physical properties v ξ . Assume that the system is described by three properties, say V 1 , V 2 and V 3 , where each property has two possible states. There therefore exist 2 3 = 8 sets underlying the classical object: , −1} represent the possible measurement outcomes for V 1 , V 2 and V 3 , respectively. The subsequent classical evolution of the system changes the system from an initial state v Notably, this description of a system and its evolution extends the idea of the quantum output in a classical process (CP) [21]. Therefore, the relationship between a specific input state of the ith physical property of the system, e.g., v i = v can be characterized by See Fig. 1(c). Let process tomography (PT), a particular application of the quantum operations formalism [22,23], be used to systematically exploit the experimentally measurable quantities given in Eq. (1). Suppose that the GCP can be completely characterized with a positive Hermitian matrix, called the process matrix, of the form [24] i =−1 for i = 1, 2, 3. Using Eq. (1) and state tomography [25], the density operator of the output system conditioned on a specific initial state v (a) i is given by whereÎ denotes the identity operator and the observableŝ V and are complementary to each other. Bell nonlocality.-Suppose that a process of interest is created and its normalized process matrix, χ expt , is derived from experimentally available data using the PT procedure, as described above. Suppose further that the process fidelity of χ expt and χ I is used to evaluate the performance of the experimental process. For a given set of observables {V then the experimental process χ expt is qualified as truly nonclassical and is close to teleportation. The overriding goal of Eq. (4) is to rule out the best classical mimicry of ideal teleportation χ I . Such a capability of genuinely classical mimicry can be evaluated by performing the following mathematical maximization task via semi-definite programming with MATLAB [26,27]: max χ GC tr(χ GC χ I ), such that χ GC ≥ 0, tr(χ GC ) = 1, Ω ξµ ≥ 0 ∀ ξ, µ. The above constraints ensure that the GCP matrix χ GC satisfies both the definitions of process fidelity and a density operator. Since the observables for the PT procedure are chosen asV = Z for the input states andV = U ZU † for the output states, where U = |0 0| + exp(iπ/4) |1 1| and X, Y , and Z are the Pauli matrices, the clearest distinction possible is obtained between the classical result and the quantum mechanical prediction [24]. The closest similarity to teleportation is under the above measurement setting. (Note that the same measurement setting is used for all the networking cases presented in the remainder of the text.) The performance inspection described here relies only on the preparation of four different input states and the relevant output state tomography for PT. Therefore, existing reported experiments on teleporting qubits are sufficient for checking the teleportation performance [3][4][5][6][7][8].
It is noted that the criterion proposed above is stricter than the existing criterion used to identify faithful teleportation as a means of ruling out the measure-prepare strategy (a direct classical mimicry strategy) [5,28]. In the measure-prepare strategy, the sender (say, Alice) directly measures unknown state given to her by the verifier and then tells the receiver (say, Bob) the measurement result such that he can prepare the output state. The best capability for the measure-prepare strategy to mimic teleportation is F expt = 0.5, i.e., the average state fidelity of the input and output states,F expt,s = 2/3 ∼ 0.6667, whereF expt,s = (2F expt + 1)/3 [29]. However, according to the fidelity criterion proposed in Eqs. (4,5), the average state fidelity isF expt,s >F GC,s ∼ 0.9024. In other words, the proposed criterion is stricter than the existing measure-prepare criterion. This result implies that there exists an experimental process for teleportation that is identified as faithful by the criterion by ruling out the measure-prepare strategy, but which can still be described by χ GC [24]. This result also implies that not all entangled states can demonstrate teleportation process that goes beyond χ GC [11].
To objectively check teleportation between two parties (e.g., Alice and Bob), it is necessary to use a verifier to collect the measurement outcomes and build the process matrix. In particular, the verifier first tells Alice to prepare one of the eigenstates of {V (a) j }, and then asks her to send it to Bob. (Note that this state should be unknown to Bob in principle.) Bob measures the observablesV (b) j chosen by the verifier for the received state, and then sends the measurement outcomes back to the verifier. After collecting all the data, the verifier constructs a process matrix using the PT algorithm and checks whether or not the teleportation is not a GCP using Eq. (4).
A GCP can be treated as an input-output transformation implemented by sharing local hidden variables (LHVs) between the parties involved (Alice and Bob). To explicitly see the role played by the LHVs, let Eq. (1) be rephrased as P (v i ) and the assumption of an equal probability of the input states P (v (a) i ) = 1/2 have both been applied above. Furthermore, let each of the joint outcome sets (v j . Moreover, the distribution P (λ) determines the process matrix χ GC , by which one can consider the correlation behind χ GC as Bell local. Therefore, since F expt > F GC , the shared pair possesses Bell nonlocality to enable the networking participants to perform qualified experimental teleportation. This approach to testing the Bell local model is different from that of existing Bell tests, which all use Bell-like inequalities [12][13][14][15][16][17][18][19]. (Notably, the manner in which a concrete information task together with its implementation can be described by a LHV model is not included in these standard Bell tests.) Quantum correlations behind three-node network.-Teleporting unknown qubits through a linear network composed of three quantum nodes can be implemented by repeating the bipartite teleportation procedure twice in parallel [ Fig. 2(a)]. For example, assume that it is desired to teleport a qubit with state ρ from the first node in a network (say, Alice) to the end node in the network (say, Charlie) through an intermediate node (say, Bob). Since the overall teleportation procedure consists of two ideal input-output subprocesses connecting Alice and Bob χ I1 , and Bob and Charlie χ I2 , respectively, the resultant process, χ I12 = χ I2 χ I1 , is still an identity unitary transformation with the mapping χ I12 (ρ) = ρ. In other words, the general criterion given in Eq. (4) for FIG. 2. Networking teleportation and its classical mimicry.
(a) A three-node teleportation process is implemented with two entanglement sources (E1,2) and two Bell state measurements on Alice's node and Bob's node, respectively (not shown). Two dependent classical sources (S1,2) are used for teleportation simulation under the Bell local model, and hence (b) the resulting process χGC12 is a GCP involving all three participants. By contrast, if the sources are independent, the underlying correlations become bilocal (c), and the resulting process χ GC1|2 is composed of two individual GCPs, i.e., χGC1 and χGC2.
identifying teleportation between two nodes still holds for three-node quantum networks. From a classical viewpoint, the three-node networking task described above can be simulated using the same LHV model as that used for the two-node case. That is, the distribution of LHV P (λ) determines the resultant GCP, where LHV λ correlates Alice's inputs and Charlie's outputs and then results in a specific process [ Fig. 2(b)]. For the measurement setting given above, the closest similarity to the three-node teleportation process that can be achieved by χ GC12 is quantified as The correlation behind an experimental three-node qubit transmission process is then Bell nonlocal if the experimental process χ expt12 satisfies the criterion F expt12 ≡ tr(χ expt12 χ I12 ) > F GC12 . Ideal three-node teleportation requires two entangled pairs. When transmitting qubits between distant nodes in a general network, these pairs are inevitably generated by two spatially separated independent sources [15,16]. As a result, it is reasonable to infer that the classical strategy using a single LHV λ to mimic teleportation can be modified.
Given the assumption of independent sources, one can reasonably assign an individual LHV to each subprocess. Let λ 1 and λ 2 be the LHVs assigned to the state transmissions between Alice and Bob, and Bob and Charlie, respectively. See Fig. 2(c). The relation between the inputs and outputs of each subprocess is totally determined by the underlying LHVs λ k (k = 1, 2). The distribution of the LHV P (λ) in the original LHV model for two-node state transmission thus becomes a product of the joint probability of these LHVs, P (λ 1 )P (λ 2 ), in the threenode case, which implies that P (v . Therefore, P (λ 1 ) and P (λ 2 ) determine their individual GCPs, say χ GC1 and χ GC2 , respectively. The resulting GCP in the three-node network is specified by χ GC1|2 ≡ χ GC2 χ GC1 , where the correlation behind χ GC1|2 is referred to as bilocal.
The maximum fidelity of χ GC1|2 and χ I12 can be regarded as a threshold for the bilocal model. For the present case, the fidelity threshold is given as [24] When each subprocess matrix is experimentally measured by PT as χ exptk for k = 1, 2, the correlation behind the joint process If the receiver, Charlie, manipulates the received system under quantum operations and trusts his measurement equipment, then the above-mentioned bilocal model becomes a LHV-LHS (local hidden state [30]) hybrid model provided that the sources are independent. The transmission between Bob and Charlie can then be described by a classical process matrix [21], χ C2 , while the subprocess for Alice and Bob is specified by χ GC1 . However, if the two subprocesses share the same pair (i.e., the sources are dependent), then the LHS model specifies the resulting process as being classical by χ C12 . (Note that χ C2 and χ C12 are not genuinely classical.) For the present measurement setting, the closest similarities between χ C12 and χ I12 , and between the hybrid process χ GC1|C2 ≡ χ C2 χ GC1 and an ideal teleportation process, are given as follows [24]: tr(χ GC1|C2 χ I12 ) 0.5985. (9) Thus, F expt12 > F C12 implies steering, while F expt1|2 > F GC1|C2 implies that the networking process is powered by sources with a nonlocality-steering hybrid correlation. The fidelity thresholds in Eqs. (6)-(9) suggest the existence of the following hierarchy between the Bell nonlocality, nonbilocality, steering and nonlocality-steering hybrid correlations behind the teleportation process, i.e., Bell nonlocality, Compared with the process χ GC1|2 under the bilocal model, the process χ GC12 under the Bell local model achieves a better simulation of ideal teleportation in terms of the process fidelity. Thus, the correlations behind a networking process χ expt12 that are identified as nonlocal through (6) can always go beyond the bilocal description χ GC1|2 . However, nonbilocality of χ expt1|2 does not necessarily imply the existence of Bell nonlocality. (Note that the nonbilocality, steering and nonlocalitysteering hybrid correlations can be compared and analyzed in an analogous manner.) In addition to the correlation discrimination criteria given in Eq. (10), the experimental process matrices χ expt12 and χ expt1 can be sufficient to verify the underlying correlations for teleportation. For example, consider the process fidelity F expt112 ≡ tr(χ expt12 χ expt1 χ I12 ), and posit that the nonbilocality and nonlocality-steering hybrid correlations behind χ expt12 can be identified according to the following criteria: When the correlation is bilocal, the fidelity F expt112 = F GC1|2 becomes maximal when χ expt12 χ expt1 = χ GC1|2 . If χ expt12 χ expt1 = χ GC1|C2 , the fidelity F expt112 = F GC1|C2 is maximum under the LHV-LHS hybrid assumption. See Fig. 3 for an illustrative example of the correlation discrimination in (10) and identification criteria in (11) for noisy entanglement sources. It is worth emphasizing here that χ expt1 , χ expt2 and χ expt12 provide a complete description of what operations and errors are involved in the three-node experiment. Thus, the resulting process fidelities can be used to reveal the experimental performance by referring to the correlation hierarchy in (10) and identification criteria in (11). Such an inspection is beneficial in evaluating The robustness of the identification protocol is degraded with increasing N . However, compared with the identification protocol using Fexpt11N , that with F expt1|N is more robust against noise as N increases.
and improving primitive operations in networking teleportation from the viewpoints of trusted-untrusted measurement devices and dependent-independent sources in the experiment. Non-N locality.-The correlation discrimination method introduced above can be readily extended to explore the quantum correlations in general many-node teleportation networks. For example, in the following, we demonstrate the quantitative characterization of non-N -local correlations [17][18][19] behind networking teleportation involving N independent entanglement sources. (Note that the nonlocality-steering hybrid correlation can be characterized in the same way.) For ideal (N + 1)-node teleportation, the resultant process remains an identity operation χ I1N (ρ) = ρ, where χ I1N = N k=1 χ Ik and χ Ik denotes the ideal inputoutput subprocesses connecting the kth node and the (k + 1)th node. In the N -local model, the whole process χ GC1|N ≡ N k=1 χ GCk is composed of the subprocesses χ GCk between the kth node and the (k + 1)th node, each having its own underlying LHV λ k . Non-N locality then 0.6250, and lim N →∞ F GC1|N 0.5000 [24]. Notably, this criterion is robust against noise even for large N , as shown in Fig. 4.
Finally, we extend the idea of F expt112 and introduce the following experimental process fidelity F expt11N ≡ tr(χ I1N N k=2 χ expt1k χ expt1 ), where χ expt1k describes experimental teleportation from the first node in the network to the (k + 1)th node through all k − 1 intermediate nodes between them. Since the maximum value of F expt11N predicted by the N -local model is F GC1|N , it can be inferred that F expt11N > F GC1|N implies the existence of non-N locality behind experimental teleportation.
Summary and outlook.-We have proposed a formalism referred to as a genuinely classical process to characterize and identify both true quantum teleportation and the underlying quantum correlations in many-node networking teleportation. We show, for the first time, that quantum-information processing can be employed to quantitatively discriminate quantum correlations. The proposed formalism is well suited to the analysis of existing experiments, and faithfully evaluates the performance of all the operations required for teleportation through quantum networks.
Such a task-oriented approach raises several interesting questions, including how one can identify generic truly quantum networking tasks such as one-way quantum computation in many-node networks and what multipartite quantum correlations exist behind multipartite distributed quantum-information processing.
We thank J.-W. Pan for helpful comments and discussions.
In this supplementary information, we provide the materials required to support the results shown in the main text. We begin by describing the process tomography algorithm and deriving the process matrix for teleportation. We then explain how to find the clearest distinction possible between genuinely classical processes and ideal teleportation. We additionally show how the fidelity criterion proposed in our study is stricter than the existing fidelity criterion for ruling out the measureprepare strategy for mimicking teleportation. Finally, we derive the fidelity threshold and process matrices of a genuinely classical process and hybrid process that have the best capacity to mimic teleportation.

IDEAL PROCESS MATRIX OF TELEPORTATION
The essence of process tomography (PT) is that a process of interest can be completely characterized by a process matrix [1]. This process matrix is constructed by the probabilities of specific output states conditioned specific input states P (v of the input and output respectively. The output state can be represented in the following decomposition form: whereÎ denotes the identity matrix andV j . The process matrix for the experiment can then be represented in the form If the conditional probabilities satisfy the classical assumption then the process matrices of genuinely classical processes χ GC can be constructed through Eqs. (S1) and (S2). If the process is an ideal teleportation process [2], we get the measurement outcomes shown in Table SI. The observables used for the input states areV = Z. Hence, the input states for the PT algorithm are the eigenstates of the Pauli matrices, The observables used for the output states areV Using state tomography [Eq. (S1)] yields the density matrix of the output states, in which The process matrix of ideal teleportation χ I can then be expressed as (S4)

CLEAREST DISTINCTION BETWEEN χI AND χGC
Genuinely classical processes χ GC have different capability to mimic ideal teleportation given different choices of the observablesV (b) j for PT, i.e., (S5) To find the clearest distinction possible between the quantum mechanical prediction χ I and the classical result χ GC , we process all the complementary observables (V = U ZU † ) using an arbitrary unitary transform [1], i.e., . (S6) The corresponding results for F GC are shown in Fig. S1. It is noted that there exist more than one set of observables that maximize the distinction between the genuinely classical process and the teleportation process. In the main text, we arbitrarily choose θ = 0, φ = π/4 and F GC = 0.8536.

COMPARISON BETWEEN χGC AND MEASURE-PREPARE STRATEGY
One of the most widely used criteria for identifying faithful teleportation and ruling out the measureprepare classical mimic strategy for teleportation is that of F expt > 0.5, i.e.,F s,expt > 2/3 ∼ 0.6667, expressed in terms of the average state fidelity [3]. In the measureprepare strategy, Alice directly measures the unknown state given to her by the referee and tells Bob the measurement result so that he can prepare the output state. The fidelity criterion proposed in our study is stricter than the traditional criterion for ruling out all the measure-prepare strategies. In other words, there exist some experimental processes for teleportation that are faithful since they are considered not to be a measureprepare strategy, but can still be described by χ GC .
Consider the example in Fig. S2, which shows F expt for the case where the entangled source of a teleportation experiment mixes with white noise and becomes ρ E = (1 − p noise ) |φ + φ + | + p noiseÎ /4, where p noise is the noise intensity. In the range 0.5000 < F expt ≤ 0.8536, the experimental process χ expt is not mimicked by a measureprepare strategy, but is still not faithful since it could be mimicked by χ GC . For example, given a white noise intensity p noise = 0.3, the experimental process under the state ρ E : can be described by To support the above result and check if an experimental process χ expt can be fully described by χ GC , we propose the following composition concept: where α denotes the maximum amount of χ GC that can be found in χ expt , and can be obtained by maximizing the following quantity via semi-definite programming (SDP) with MATLAB [6,7]: such that χ expt −χ GC =χ ≥ 0,χ GC ≥ 0, Ω ξµ ≥ 0 ∀ξ, µ, (S12) whereχ andχ GC are both unnormalized process matrices. For an experimental process χ expt with p noise = 0.3, , meaning that the maximum amount of χGC that can be found in χexpt is α = 1. See Eqs. (S10,S11). It is worth noting that the above criteria for achieving faithful teleportation can be rephrased as the visibility used for evaluating the source of teleportation [4]. a value of α = 1 implies that while χ expt can be fully described by χ GC , it is not mimicked by the measureprepare strategy since F expt = 0.7750 > 0.5 andF s,expt = 0.8500 > 2/3. Note that for an experimental process with F expt > 0.8536, α must be less than one since there is no χ GC that can fully describe it.