15.2.1 Setting Up the Problem and the Role of Entanglement

Consider an observer, Alice, who has a quantum system in a particular state , unknown to her. We first assume the simplest case, a two‐level system in a quantum superposition state known as a qubit, say a spin‐1/2 particle. A general and unknown pure state of such a system can be described as a linear combination of its two possible orthogonal basis states:

15.1

where and are unknown constants. Alice wants to transfer to a receiver, Bob. We assume that it is not possible (or not desirable) to directly send the quantum system to Bob, for example, due to losses in the available quantum channel between Alice and Bob.

Now we need to observe certain requirements. We want Bob to have an exact – and unknown to him – copy of the original quantum state. We are not allowed to “peek” into the state at any stage of the protocol, otherwise we will affect it. In other words, we are not allowed to learn anything at any time about the state we are sending. The original copy of the state should be naturally destroyed when the receiver obtains his copy of the teleported state; otherwise, the process will violate the no‐cloning theorem. The use of nonlocal correlations intrinsic to entangled states gives us a perfect tool to accomplish these tasks.

Entanglement is a puzzling property of quantum mechanics, which in the last two decades enfolded as an efficient unique tool in QIP. Let us consider a composite system . A state of a composite system described by the density matrix is said to be entangled if it is nonseparable, that is, if and only if it cannot be fully written as a convex decomposition of the product states of the subsystems (17) in all possible splittings :

15.2

In this chapter, for simplicity, we consider pure states. The equation above then simplifies to

15.3

where is the state of a composite system and , are states of the corresponding subsystems. Conceptually, such an entangled state has different facets. In the following discussion, for the sake of clarity of presentation, we consider a pure entangled state and assume that the state is maximally entangled.

First of all, the two subsystems of are linked together closer than any classical systems can be linked. The state of the system as a whole is pure. However, if we calculate a reduced state (or ) of the total state , we will find that both subsystems are in a mixed state with maximal possible disorder (maximal entropy for the dimension of the problem). That is, the state of a subsystem is completely undefined. The subsystems do not have identity of their own and only a global state (state of the system taken as a whole) is well defined.

Next, this inner link between subsystems manifests itself in perfect quantum correlations between the subsystems. Let us illustrate it with an example of a two‐level system. Assume that, when measuring some relevant observable, the two possible measurement outcomes on subsystems and are . If we will independently perform measurements on and , the measurement results on will be a random sequence , with on average probability for each of the possible outcomes. The same for . However, the two sequences and will be perfectly correlated (or anti‐correlated). If Alice would measure , Bob with probability would measure (or ).

Finally, the nonlocal aspect of these quantum correlations plays a crucial role. The measurement outcome on instantly determines the measurement outcome on with certainty irrespective of the distance between the two observers. That is, the change in the quantum state of subsystem instantly affects the arbitrary remote quantum state of . This is, of course, an idealization in that we assume here that the perfect entanglement is maintained perfect at the arbitrary distance. We ignore decoherence in this discussion for the sake of a clearer argument.

Mathematically, any entangled state can be represented in terms of four mutually orthogonal maximally entangled quantum states known as Bell states (18):

15.4

15.5

These four states form a complete basis set for the quantum system , the Bell basis.

The three main aspects of an entangled state described above render a quantum protocol possible, which meets all the requirements we have set for transferring an unknown quantum state. The corresponding template is depicted in Figures 15.1 and 15.2.

15.2.2 A Template for Quantum Teleportation

We discuss the teleportation protocol first qualitatively and without specifying the exact form of the quantum states involved – to highlight the main principle.

Step 1 – Figure 15.1a: Sender Alice and receiver Bob share an entangled pair of two quantum systems and (two circles with gray gradient in Figure 15.1a). This is the quantum channel. Alice also holds a quantum system in an unknown quantum state to be teleported (dark circle in Figure 15.1a).

Step 2 – Figure 15.1b: Next, Alice performs a joint measurement on the unknown quantum state and her part of the entangled pair (see also time point in Figure 15.2). The most common form of the joint measurement is the Bell‐state measurement (BSM). In the process of the joint measurement, and interact in a specific way and get intertwined. When the measurement is performed following this interaction, the outcome characterizes the entangled state emerged from the interaction of and . At this stage, the initial entanglement between and is broken. The outcome of the joint measurement on and is recorded, and the record is sent over a classical channel to Bob. This transmission of classical information from Alice to Bob constitutes the classical communication channel. The system is not in an original state any more. The original state is destroyed at the sender location.

Step 2, behind the scene – Figure 15.1b, inset: Before the onset of the entanglement between and and before the initial entanglement link between and is broken, two crucial events occur. In the process of joint measurement, the quantum properties of (dark circle) are mapped on the Alice's part of the entangled pair (upper cartoon in the inset). Further, before entanglement between and is disbanded, the change in translates into the change in due to the intertwined nature of the entangled systems (lower cartoon in the inset). In this way, information about the unknown state of is instantaneously mapped from to to . This mapping from to without disclosing the actual state of is facilitated by the entangled property of the quantum channel.

Step 3 – Figure 15.1c: In the final stage of the protocol (see also time point in Figure 15.2), Alice holds a measured system . Bob holds the system now in a quantum state directly linked to the initial unknown state . The classical information about the measurement outcome of the BSM on reveals Bob which unitary operation to perform on to convert it into , without disclosing any information about .

At the end of the protocol, there is only one copy of the unknown quantum state , now mapped on system . Note that in order to actually reconstruct the state, Bob needs to obtain classical information about measurement outcome of Alice's joint measurement. Importantly, this means that no information is transferred with the speed larger than the speed of light. The quantum‐state transfer does not happen with superluminal velocity. Although mapping of the unknown state from to to happens instantaneously, in order to retrieve the unknown state from the amended state of , classical communication is needed and that is limited by the speed of light.

To describe the protocol above quantitatively, we discuss now the seminal work on quantum teleportation by Bennett et al. (1). The process involves the quantum system, particle , which we want to teleport and an EPR pair, particles and . Initially, Alice holds particle and one of the EPR particles, say particle , while the other EPR particle is given to Bob. Particle is in an unknown state of Eq. 15.1, while the EPR pair is in one of the four Bell states (Eqs. 15.4 and 15.5), for example, in the singlet state . This is the quantum channel. The state of the system as a whole can then be written as

15.6

At this stage, the global system is in a pure state, entangled across – and separable across – bipartition. There are neither classical correlations nor quantum entanglement between the particle to be teleported and the EPR pair. In the next step, Alice couples systems and by performing a complete measurement of the von Neumann type on the joint system in the Bell operator basis (Eqs. 15.4 and 15.5), that is, she performs a Bell‐ state measurement (time point in Figure 15.2). The state given by Eq. 15.6 can be rewritten in the complete orthonormal basis of the four Bell states:

15.7

Now if the BSM is performed on and , the measurement result will be one of these Bell states. This corresponds to the protocol step depicted in panel (b) of Figure 15.1 and by Figure 15.2 for . Via the joint measurement, quantum information about the unknown state of is instantaneously mapped from to to . There are four equally probable outcomes for the joint Bell measurement on . Depending on the particular result of the Bell measurement, the particle is projected into one of four possible quantum states, each equal up to a known unitary to the initial unknown state (Eq. 15.7). The state to be teleported is still unknown for all parties. But the unitary operation, which will convert the current state of the particle into the initial state , is known and is determined by the result of the Bell measurement on (Figure 15.2, ). Here, the classical communication channel comes into play. The state can only be reconstructed, if Alice sends classical information about her measurement outcome to Bob (dashed line in Figure 15.1). This also limits the whole transport process to the speed of light: changes in happen instantly but in order to reveal or use these changes, additional transport of classical information is needed, which can only happen at the finite speed.

The unitary transformations required to recover state from the state of particle are simple basis rotations and can be expressed in terms of the Pauli matrices:

15.8

Then, the state of the system can be expressed as

When Alice classically communicate the result of her measurement, Bob will know which unitary transformation he must perform to convert the state of his part of the EPR pair into a replica of Alice's original state . In the very first experimental implementation for qubits (2), due to an involved form of the BSM, only one of the four Bell states could be identified. In this case, teleportation protocol is probabilistic, conditioned on the measurement outcome: the state is successfully teleported one out of four times. The first unconditional teleportation has been achieved for quantum continuous variables, for light beams (3), as the BSM for continuous variables is less intricate. In all cases, the transport of the particle's state is achieved by dual channel, quantum channel, and classical channel.

15.2.3 Efficiency and Fidelity

The experimental realization of quantum teleportation is measured in terms of certain features, namely efficiency and fidelity. The efficiency of a particular process concerns its success rate, given an input state. Within Bennett's scheme, efficiency, or , is achieved when all four Bell states of particles can be uniquely determined by Alice. However, if only one or two of these states are distinguishable, teleportation will still be possible, but with a or efficiency, respectively. There are other factors that also determine the success of teleportation, such as the degree of entanglement between the EPR pair, propagation, and detection losses, and so on. Each particular experimental setup diverges in one way or the other from ideal conditions, reducing the efficiency of the process even further.

In the ideal scenario and when teleportation is successful, the unknown state that emerges in Bob's location is an exact replica of the state teleported by Alice. However, in realistic conditions, the input and output states will differ. Even if the particular input state is pure, , it is likely that the outcome will be represented by a mixed state, . Fidelity of teleportation is one of the possible ways to assess the quality of the quantum information transfer. The fidelity is simply given by the overlap between input and output states, . This measure satisfies

15.9

There are several thresholds associated with the fidelity of teleportation. First of all, this is the criterion whether the transfer of information overperforms the purely classical procedure. By means of classical communication alone, it is possible to reach a maximal fidelity of . Any value that exceeds this must therefore involve some sort of entanglement and consequently quantum communication (19,20). However, teleportation should not only beat the classical limits on measurement and transmission, but should also reach limit stemming from the no‐cloning theorem ( 20,21). This is also related to the question of security of teleportation. If we assume that along with the legitimate parties Alice and Bob, there is a third party (or even many other parties) who wants to obtain a copy of , and a better copy than Bob, this imposes additional requirements on fidelity. Decisive is the distinction between nonclonable quantum information and classical information. If the process needs to ensure that neither Alice or Bob has in some way acquired information, which would allow them to second guess the possible state of particle , then the fidelity for a single qubit must be over (for a comprehensive explanation, see Ref. (19)). This will ensure that the output state is a result of true quantum teleportation. Furthermore, fidelity of ensures that Bob holds the best copy of the original quantum state and guarantees that any copy produced by a third party will have less fidelity ( 20, 21). We will return to the question of fidelity in the next section when we discuss the continuous‐variable teleportation.

15.3.1 The First Quantum Teleportation Experiment

The first reported quantum teleportation was performed by Bouwmeester et al. (2), Figure 15.3. The experiment accomplishes the quantum transfer of the state of a photon in a unknown polarization state, that is, an unknown superposition of the vertically and horizontally polarized states. The EPR sources involved are based on parametric downconversion process in a nonlinear crystal with the second‐order nonlinearity. Here, an outline of the experiment is given; more details of the setup can be found in the original paper (2).

The experiment follows very closely the protocol by Bennett et al. described above. The state is completely unknown, and, furthermore, it is undefined, as the particle itself is a part of an EPR pair (source 1). Its EPR partner heralds the generation of the photon . If this EPR pair is generated by a type II nonlinear crystal, then is in state, orthogonal to (note that we do not know the exact states of ).

The quantum channel is created by the entangled photon pair , a pair of photons generated by the type II parametric downconversion nonlinear optical process (source 2). In such a process, a photon interacting with a nonlinear crystal can decay into two photons, which are in the singlet state , one of the Bell states (Eq. 15.4). Alice's BSM measurement is performed by linear interactions alone. In this method, only one particular Bell state can be discriminated, giving only absolute efficiency. When successful, the photons are projected onto . This is achieved by superposing the two photons at a 50/50 beam splitter (BS) Figure 15.3. Two detectors are located at each of its outputs, and . The photons are projected into the Bell state whenever the two detectors and fire simultaneously (when coincidence counts are registered). Teleportation, which occurs simultaneously as the coincidence is detected, results in photon being projected in the same polarization state as the initial state of photon , in accordance with Eq. 15.7. Verification of the process is carried out by passing photon through a polarizing beam splitter (PBS) with detectors and in each output. The PBS selects and polarizations in these two outputs, respectively. Then, recording a threefold coincidence ( analysis) together with the absence of a threefold coincidence ( analysis) is a proof that the polarization of photon has been teleported to photon . The polarization of photon using detectors and is verified to ensure a positive result of teleportation. However, this destroys the state of .

To ensure a true quantum teleportation, many nuances should be observed. We name some to give reader a flavor. Photons are generated in such a manner that they cannot be distinguished by their arrival time at the BS. The experiment is first carried out for a known linear polarization basis ( ), which is not in the preferred direction to the setup. Then, quantum teleportation is performed on a superposition of such states, such as circular polarization. This guarantees that any unknown quantum state can be teleported. At the verification stage, there is the possibility of threefold coincidences when no teleportation has occurred. This is due to a two‐pair downconversion on source 2 while no photon is present. These spurious threefold coincidences can be excluded by conditioning to the detection of photon , which effectively projects photon onto a single‐particle state.

The experiment by Bouwmeester et al. (2) has successfully achieved for the first time quantum teleportation of an arbitrary and unknown state. It was an important step, which has demonstrated the feasibility of quantum‐state transfer. Naturally, this first implementation has some drawbacks. Most importantly, the experiment (2) is not an unconditional teleportation. Teleportation was achieved probabilistically with 25% probability of success (due to incomplete BSM). Further, Braunstein and Kimble (22) and Bouwmeester et al. (23) raised the question that due to the nature of the experimental process, not always a teleported photon is observed conditioned on a coincidence recording. This affects the fidelity, to a level where identical results could be obtained by classical channels. Since 1997, when Bouwmeester et al. first succeeded in carrying out Bennett's scheme, further experiments have been performed, in pursuit of closing all loopholes and improving quantum teleportation further toward technological level, in terms of distance, reliability, and efficiency.

In 2001 Shih's team reported a quantum teleportation (7), where a complete Bell‐state measurement was achieved, that is, unconditional teleportation for photonic qubits. Following Bennett's scheme, this quantum teleportation experiment resolves the problem of accurately discriminating between all four Bell states using the BSM analyzer based on the nonlinear optical process known as sum frequency generation (SFG). Photons and are directed into four (two type I and two type II) nonlinear crystals, which generate a higher‐frequency photon . The crystals are positioned in a particular arrangement and with the aid of four detectors all four states, , , , and , can be distinguished. The rest of the scheme is unchanged. Alice sends the outcome of the BSM analyzer via a classical channel to Bob, who, therefore, can perform the necessary unitary transformation to the state of particle and the teleportation of the input state is then deterministically accomplished. In this case, to ensure that quantum teleportation has taken place, the joint detection rates between Alice's four detectors and Bob's two detectors are measured, resulting in a fidelity of (7).

The breakthrough toward practical quantum communication was accomplished in 2004 by the Vienna group led by Zeilinger (8). In this “real‐world experiment,” the goal was to go beyond laboratory conditions and to demonstrate teleportation over long distances and using standard telecommunication links. As in the previous cases, the underlying scheme was the protocol by Bennett et al. But now 800m optical fiber installed in a public sewer running underneath the river Danube played the role of the quantum channel, the shared entanglement link. For simplicity, only linear interactions were used in the BSM analyzer and therefore only two Bell states could be identified (24). The rest of the setup was fairly conventional, a microwave channel was used for classical communications and Bob used a fast electrooptical modulator to perform the necessary unitary transformations corresponding to the two detectable Bell states. The optical fiber reduced the velocity of the EPR photon by a fraction of , a time delay which ensured a successful operation since it allowed the receiver to apply the required unitary transformation. This first long‐distance quantum teleportation was performed with fidelity values above 0.85 for different polarization states.

The current distance record for quantum teleportation is quantum‐state transfer over a 143 km free‐space link between the two Canary Islands of La Palma and Tenerife, off the northwest coast of Africa, accomplished in 2012 (11). In 2016, another technological achievement was reported, quantum teleportation over fiber‐optic networks in 10‐km‐range in the cities of Hefei, China (25), and Calgary, Canada (26). Each of the two experiments involved quantum channels over up to 12.5 km between three distinct locations to simulate the structure of future quantum networks. The most recent achievement is quantum teleportation of independent single‐photon qubits from a ground observatory to a low‐Earth‐orbit satellite over distances of up to 1400 km with an average fidelity of 0.80 ± 0.01 [27].

15.3.2 Quantum Teleportation using Continuous Variables

In protocol devised by Bennett et al. (1) uses quantum systems within a two‐dimensional Hilbert space, for example, photons with two possible polarization states or two‐level atoms with spin up or down. The entangled state employed is then ideally one of the maximally entangled Bell states, (Eqs. 15.4 and 15.5). Quantum teleportation beyond the dichotomic problem in the higher‐dimensional but finite Hilbert space would then be quite complicated. Remarkably, the problem becomes much easier if we go for the limit of the infinite‐dimensional Hilbert space, that is, the case of quantum continuous variables (CV). Examples of such variables are position and momentum of a particle, amplitude and phase quadrature operators of a light mode, continuous polarization variables, the Stokes operators, or collective spin of an atomic ensemble. Vaidman (28) and later Braunstein et al. (29) were first to show theoretically that it is possible to achieve quantum teleportation in systems characterized by continuous variables corresponding to states of infinite‐dimensional systems such as optical fields or the motion of massive particles. Vaidman showed that teleportation is feasible for the wave function of a one‐dimensional particle where the EPR pair shared by Alice and Bob has a perfect correlation in position and momentum (28). Braunstein et al. (29) later extended these results, demonstrating that teleportation can also be accomplished with finite degree of correlation among the relevant particles. This was a very important step, as perfect CV entanglement would mean infinite energy and thus such states are not physical. CV entanglement is cheap but never perfect, if to put it in a very simple way (see (30)and the corresponding chapters in this book: Chapters 3, 10, 18, and 35).

QIP over continuous variables in general represents an alternative approach to quantum communication (30). CV quantum systems and CV quantum measurements are in most cases easier to handle. As already mentioned, in quantum teleportation using discrete quantum variables, nonlinear interactions are needed to perform the complete BSM. This represents a substantial experimental challenge, and as a result of this, unconditional quantum teleportation was achieved first in the continuous‐variable regime in 1998 based on the proposal by Braunstein et al. (29). It was not until 2000 that unconditional DV teleportation involving nonlinear elements for BSM was realized (7), 7 years after the underlying theoretical proposal of Bennett et al. (1).

Thus, in the continuous‐variable regime, the possibility of performing Bell‐state‐like measurements using just linear transforms, for example, BSs and phase shifters, provides an elegant and simple method to extend the conventional teleportation scheme on a single‐photon basis (1) to the case of continuous variables (29). The scheme for CV quantum teleportation is represented in Figure 15.4. The Bell‐state measurement at Alice's station is accomplished by mixing the incoming unknown state to be teleported with one of the entangled beams on a BS and consequently measuring the two conjugate field amplitudes (amplitude and phase quadrature operators) in two different output beams. The resulting photocurrents of these two homodyne detectors are transmitted to Bob via classical channels. Bob uses this classical information to extract the copy of the original state from his part of the EPR beam. The operation equivalent to the unitary transformation in single‐photon case is the mapping of the result of the Bell measurement on Bob's EPR beam via amplitude and phase modulation. Let us describe this teleportation process in the Heisenberg picture by first introducing the pair of continuous variables of the electric field , called the amplitude and phase quadratures (30), which describe the infinite‐dimensional state of the optical fields. These variables are analogous to the canonically conjugate variables of position and momentum of a massive particle. The EPR beams have nonlocal correlations similar to those first described by Einstein et al. (5). Thus, for the combined mode , perfect entanglement is exhibited in the limit case of:

15.10

More on CV entanglement can be found in the original papers of Drummond and Reid (31) and in the recent review on quantum information with continuous variables (30).

Alice wants to teleport an unknown input mode described by a pair of variables . Alice and Bob share a continuous‐variable EPR pair with the entangled modes and , respectively. As a next step, Alice performs the Bell measurement on the mode to be teleported and on her part of the EPR beam. As already mentioned, this is done by combining them on a BS and performing ‐measurement in one output and ‐measurement in the other one, which delivers the following results (for the quantum description of the BS transformations, see (32)):

15.11

The measured values for represent the classical information corresponding to the Bell measurement result in the DV case and are transmitted to Bob via the classical channel. At this stage, a particular feature of the continuous‐variable teleportation comes into play: the classical photocurrents corresponding to can be electronically scaled introducing an electronic gain for each of the variables (amplification or deamplification of the signal), which can improve the fidelity of teleportation. However, using the nonunity gain means a restriction in the type of quantum state to be teleported; a teleporter of an arbitrary unknown state should always have its gains set to unity (33,34).

Before the Bell measurement is performed by Alice, Bob's initial mode can be represented in terms of the original mode , the EPR pair , and the results of the Bell measurement 15.11 (the mode remains unchanged, it is only formally rewritten):

15.12

(see also ( 3, 33)). On receiving the measurement results of Alice , Bob performs the following displacement of his mode:

15.13

The mapping of the Bell measurement results onto Bob's mode is performed by driving the amplitude and phase modulators, placed in Bob's mode, with photocurrents applying electronic gains . This last step accomplishes the teleportation process and (for unity gain) the output mode becomes

15.14

which in the ideal case of perfect EPR correlations 15.10 provides Bob with a perfect copy of the initial state, . In real situations, where unperfect CV entanglement is present, the teleported state has additional fluctuations, which reduce the fidelity and, in the case of nonunity gains, requires some additional measures to quantify the quality of teleportation.

Details of the experimental scheme depend on the nature of the EPR source used. In the Caltech group experiments ( 3, 12) and in the experiment by Takei et al. (33), the entangled state shared by Alice and Bob is a highly squeezed⁰¹ two‐mode state of the electromagnetic field, with quadrature amplitudes of the field playing the roles of position and momentum. For the Bell measurement, Alice uses two homodyne detectors, , including two local oscillators .

In the case where a two‐mode squeezed vacuum is used as an EPR source, one has to use an auxiliary beam on which the modulation with the Bell‐measurement results is achieved (or two beams, to avoid mixing of amplitude and phase modulations). Bob's mode is then combined with the modulated beam(s) on a highly transmissive (for Bob's mode) mirror, a BS. This results in an appropriate displacement in the phase space so that Bob's mode becomes a copy of the original state ( 3, 12, 33).

For intense entangled beams, this scheme may be simplified. In the first proposal along these lines by Ralph and Lam (36), two intense squeezed continuous‐wave beams interfere at a BS to produce nonlocal quantum correlations. In this scheme, one of the interfering beams is required to be substantially more intense than the other. The use of bright beams allows one to simplify the inverse Bell‐state‐like measurement at Bob's side: the amplitude and phase fluctuations from the photocurrents are mapped by Bob directly onto the second EPR beam (Figure 15.4). The continuous teleportation scheme of Leuchs et al. (37) is essentially the same as the one reported by Ralph and Lam (36), the difference being that the entangled beams are bright optical pulses of the same intensity and with a more complicated spectral structure. The use of the EPR beams of the same average power allows for a particularly simple detection scheme for teleportation: the amplitude and phase quadrature detectors with local oscillators ( 3, 12, 33, 36) are now replaced by standard direct amplitude detectors in both outputs of Alice's BS, thus avoiding the cumbersome local oscillator techniques.

However, despite its experimental advantages and its higher efficiency, CV information processing is restricted in maximal achievable degree of entanglement. CV entanglement is deterministic, but maximal entanglement would need infinite energy resources, which is in contrast to readily available – but probabilistic – maximal DV entanglement. Lower degrees of entanglement result in lower teleportation fidelity. The first continuous‐variable quantum teleportation of a coherent state has been implemented at Caltech (3) with a fidelity of and repeated later by the same group (12) and the ANU group (38) with an increased fidelity of . In 2005, Takei et al. (33) reported the first CV unity‐gain teleportation of an entangled state (entanglement swapping) and the first teleportation of a coherent state with the fidelity over the no‐cloning limit, . This value was still lower than the fidelity of the unconditional DV experiment, ( 7, 8). Still there is a strong potential in CV quantum teleportation in terms of efficiency and integrability in current telecommunication infrastructure, and new recent achievements testify this. In 2008 Furusawa group demonstrated quantum teleportation of coherent states with a fidelity up to 0.83 (39), and deterministic teleportation between macroscopic atomic ensembles separated by 0.5 m was performed by the Polzik group (40) in 2013. Furthermore, in 2015 a fully integrated CV teleportation device on a photonic chip was reported (41). To benchmark it, the recent DV quantum teleportation breakthroughs are probabilistic teleportation over a distance up to 143 km with a fidelity up to 0.86 for photonic qubits ( 10, 11), teleportation over a ground‐to‐satellite uplink over up to 1400 km (27) and deterministic teleportation over a distance of 3 m with a fidelity up to 0.77 for solid‐state qubits (42).

15.3.3 More about Fidelity and Efficiency

The characterization of the quality of the CV quantum teleportation in terms of fidelity is essentially the same as that already described in Section 15.2.3. The important point here is the choice of gain, which is the issue specific to CV teleportation. The choice of nonunity gain can partially compensate for the additional fluctuations in the teleported state emerging from the unperfect EPR entanglement by an appropriate rescaling of the output state in the phase space. Hence, in some cases, the best fidelity of teleportation is achieved by using optimal, nonunity gains ( 34, 38,43). However, this optimization is state‐specific, that is, such a teleporter is optimal only for a particular class of the input states and the improved fidelity is calculated only for this particular class of states. The general goal, however, is to teleport an unknown arbitrary quantum state. To characterize teleportation of an arbitrary state, the fidelity is averaged over the whole phase space (44). When using nonunity gains, the displacement of the teleported state does not match the original displacement of the input state and the fidelity calculated over the whole phase space goes to zero. Thus, teleportation of an arbitrary state requires the gains in the classical channel to be set to unity ( 33, 44).

Nevertheless, the optimal quantum teleportation of a particular class of input quantum states is of interest and can find its own applications. Such teleportation schemes providing optimal transfer of quantum information for nonunity gains were termed by Bowen et al. (34) nonunity gain teleportation as opposed to unity gain teleportation, delivering output states identical to input ones (with some noise added). For nonunity gain teleportation, fidelity does not provide satisfactory measure of teleportation quality any more (34). The emphasis here lies not on producing an exact copy of the original state but on optimal transfer of quantum information contained in quantum uncertainties of the input state. This leads to introduce new measures ( 34, 36,45,46), borrowed from traditional quantum optics, in particular using concepts of information transfer in quantum nondemolition measurements (QND) (47). The following three figures of merit are suggested to quantify CV teleportation in a wider context of optimal quantum information transfer.

Fidelity ( 19, 20, 29). It is a decisive measure to characterize quantum teleportation of an arbitrary unknown state in a sense of producing the best distant copy of the original state.

diagram ( 34, 36, 45, 46). The measure takes more exact account of transfer of quantum information contained in quantum uncertainties of the original state. It uses the signal transfer coefficients and conditional variance product between the input and output states. These quantities have been originally introduced in the context of QND measurements (47). Signal transfer coefficient is defined as a relation between signal‐to‐noise ratios (SNR) between output and input states. It characterizes how good the original signal is transferred to the output and is directly linked to the teleportation gain. Conditional variance characterizes the quantum correlations between input and output states and is a measure of the noise introduced during the protocol. Transfer coefficients and conditional variances are then calculated for both conjugate variables and are combined in a certain manner to deliver figures of merit and , which build up the diagram. Interpretation of the results is similar to analyzing the QND measurement. Surpassing the limit means that Bob has got over the half of the signal from Alice and thus has more information on the input state that any third party. Surpassing the limit is required for reconstruction of nonclassical features of the input state (squeezing, entanglement, etc.). For unity gains, means surpassing the teleportation no‐cloning limit . The graph is two dimensional and conveys more information about the teleportation process than fidelity. For more details of the diagram analysis, see ( 34, 36, 45, 46).

Gain normalized conditional variance product (34). This measure was specially designed to provide a single number as a figure of merit for nonunity gain teleportation (such as fidelity is a single number characterizing unity gain teleportation). It is directly related to the characterization above.

Recently, He et al. (48) provided a valuable contribution to better understanding of security and efficiency of CV quantum teleportation of a coherent state and proposed quantum teleamplification, preamplification, or postattenuation of a coherent state as extended protocols based on nonunity gain teleportation. Einstein–Podolsky–Rosen steering (49) has been considered as an important quantum resource to achieve a teleportation fidelity beyond the no‐cloning theorem (48). The notion of steering refers to the EPR‐paradox in its original formulation, where one observer appears to adjust (“steer”) the state of the other by local measurements. Quantum steering thus describes a situation where two parties share a bipartite system onto which one of the parties applies measurements that change the state of the other party in a way that cannot be explained by classical means (see (50) for a brief review). Quantum steering has been also interpreted as the task of entanglement detection when one of the parties performs uncharacterized measurements.

There are several advantages of using continuous‐variable systems for quantum communication. Firstly, simple communication schemes with CV can be implemented where only linear operations are considered for BSM; therefore, there is no need for nonlinear interactions beyond the generation of the EPR pair. Further, these methods can also be applied to other quantum computational protocols, such as quantum error correction for CV and superdense coding in optical information. Lastly, it was suggested that CV systems are more suitable for the integration of quantum teleportations into the communication technology arena.

15.3.4 Outlook

The goal is to put quantum teleportation to the service of quantum‐based technologies. Teleportation routines can become building blocks in different practical quantum communication schemes, as well as in quantum computation. For example, a recent work (51) proposes quantum repeaters based on CV quantum teleportation to bridge long distances in quantum communication networks. The ultimate long‐term vision is the functional quantum internet (52). Deterministic teleportation of massive atomic qubits (ions) paves the way to scalable quantum information processing in ion‐trap systems (53,54).