18.3.1 Gaussian Transformations

All devices that map Gaussian states onto other Gaussian states can be concisely described by simple linear input–output relations in the Heisenberg picture. From the Hamiltonian for the device in question, , we deduce the unitary evolution operator from which we derive the input–output relation using the transformation:

18.6

The input–output relation of a beam splitter, a phase shifter, a single‐mode squeezer, a two‐mode squeezer (entangler), and a phase‐ insensitive amplifier are presented in Figure 18.2.

The ubiquitous device in quantum optics is the beam splitter. It is for example used to build interferometers, which are unavoidable devices in almost all experimental setups. The beam splitter has two modes at the input, interferes them and creates two output modes. The quadratures of these modes are combined via the input–output relation shown in Figure 18.2.

A phase shifter is a device that changes either the relative phase between two spatially separated modes or the phase between two orthogonal modes in the same spatial mode in order to change the polarization state of light. In the laboratory, the relative phase between two spatially separated modes is either accomplished using a mirror attached to a piezo ceramic, which moves as a function of an applied voltage, or by an electrooptic modulator through which the beam is transmitted. The polarization state of light can be controlled also by an electrooptic modulator but normally if no fast switching times are required a half‐wave or a quarter‐wave plate is used for convenience.

The displacement operation corresponds to a shift of the uncertainty area in phase space. The most important displacer in the laboratory is a laser: the input to the laser is a vacuum state and the output is, ideally, a coherent state; thus, a displaced vacuum state (see also Problem 18.1b). The laser is however a complex device, which is not, in practice, performing a perfect displacement operation. Alternatively, one can use a very asymmetric beam splitter, which is almost perfectly transmitting the state to be shifted and reflecting a small part of a laser beam enabling the displacement. The size of the displacement is controlled by the power of the auxiliary beam. It is also clearly seen that the transformation (Figure 18.2) for the beam splitter reduces to that of a displacement transformation for a very asymmetric beam splitting ratio. Fast displacement operations are obtained using a phase or an amplitude modulator, which displaces the state with a speed given by the bandwidth of the modulator. It is also possible with these modulators to displace only a certain frequency mode (a sideband), by applying an electronic modulation with a frequency equal to that of the sideband mode.

The next device in the figure is the squeezer, a device that squeezes the state in phase space (see Problem 18.1c). A squeezer requires a nonlinear optical interaction (26). The standard way of squeezing a state is by using a degenerate optical parametric amplifier. Such a device is pumped by a strong field and produces two output modes (the so‐called signal and idler modes), which are degenerate in polarization, thus indistinguishable. Under the parametric approximation where the pump field is assumed to be treated classically, the amplifier accomplishes a Gaussian squeezing operation. In many cases, the parametric amplifier is embedded in a cavity, which supports a comb of resonantly enhanced frequency modes, hereby making the process more efficient for these particular frequencies. The parametric amplifier is mediated by a second‐order nonlinearity, but a third‐order nonlinearity such as the Kerr effect (or four‐wave mixing) can be also used. Also in this case the operation is identical to the Gaussian squeezing transformation if the pump beam is treated classically. Normally, the Kerr effect is generated by propagating short pulses through a long optical fiber. Note that the squeezing operation can also be placed off‐line so that squeezed vacuum only serves as an off‐line resource for accomplishing the squeezing transformation on an arbitrary input state. Details about such a scheme can be found in (27). A summary of various squeezing experiments is given in (28).

When the parametric amplifier or the four‐wave mixing process operate in a nondegenerate configuration (either in polarization or direction), it produces entanglement in two different modes (29). Another, but theoretically identical, way to produce CV entanglement is to interfere two squeezed beams on a symmetric beam splitter, which produces entangled output beams ( 8,30) (see Problem 18.2c).

Phase‐insensitive amplification is also an important device in quantum communication. In contrast to the other devices in the table, the amplifier is not unitary: Excess noise is evitable introduced to the amplified state, rendering the amplified state in a mixed state (31). There are numerous examples of devices that, in principle, accomplish ideal phase‐insensitive amplification, for example, the fiber‐based Er‐doped amplifier, parametric amplifiers, processes involving four‐wave mixing, and solid‐state laser amplifiers. In practice, however, none of these devices operates at the ideal quantum limit. Another approach that comes arbitrarily close to the ultimate quantum limit, in particular in the low gain regime, has recently been proposed and demonstrated. It relies solely on linear optical components, homodyne detection and feedforward (see Problem 18.2b). Details about such a scheme can be found in (32).

18.3.2 Homodyne Detection and Feedforward

A homodyne detector, the most important measurement device in CV quantum communication, is shown in Figure 18.3: The signal under interrogation is combined with a much brighter LO at a 50/50 beam splitter (33). The outputs are directed to two balanced PIN photodetectors and subsequently the difference of the two detector outcomes is produced. By using a linearized model, it can be shown that the resulting photocurrent is linearly proportional to a certain quadrature amplitude of the signal; which specific amplitude is detected is determined by the phase of the LO relative to the signal. Thus, by controlling the phase of the LO, any given quadrature can be measured. Another great advantage of the homodyne detector is that the LO selects effectively a certain spatial, temporal, and polarization mode among a general mixture of modes in the signal (21). A dual‐homodyne detector aims at measuring conjugate quadratures simultaneously; for example, it may correspond to a simultaneous measurement of the amplitude and phase quadrature. Such a measurement is performed using a 50/50 beam splitter followed by two homodyne detectors located at the two outputs of the beam splitter (34). One detector is set to measure the amplitude quadrature, whereas the other one is measuring the phase quadrature. Since one cannot simultaneously perform sharp measurements of conjugate quadratures according to the basic laws of quantum mechanics, the accuracy with which the quadratures are determined is intrinsically limited. One unit of vacuum noise is introduced in the measurement, and it can be traced back to the vacuum noise entering the empty port of the 50/50 beam splitter. Note also that if this vacuum state is substituted with a state which is entangled to another mode, the measurement is identical to a CV Bell measurement (8).

An important tool for executing many quantum informational tasks in the CV regime is that of feedback or feedforward. Such a control system measures a certain property of the system (possibly embedded in noise like in the case of a dual‐homodyne measurement), manipulates it by some signal analysis, and finally, based on the gained information, controls some dynamics of the system (35). In Figure 18.3c,d, we give two simple examples: in the first example (Figure 18.3c) a homodyne detector measures a certain quadrature (say ) of a subsystem, and displaces the same quadrature of another subsystem with a magnitude proportional to the measurement outcome and scaled with an electronic gain . In the second example (Figure 18.3d) a dual‐homodyne detector measures conjugate quadrature of a subsystem and drives another subsystem with the measurement outcomes. Simple examples of such feedforward systems can be found in ( 13,36).

18.3.3 Non‐Gaussian Transformations

As we will see in the next section the toolbox in Figure 18.2 consisting of linear transformations provides essential tools for generating and manipulating quantum states. However, some operations cannot be carried out with only this set of transformations. For example, entanglement distillation of highly entangled states from a Gaussian mixture as well as efficient quantum computation cannot be carried out with these Gaussian operations only. To enable these tasks, one must resort to non‐Gaussian transformations. Such transformations are experimentally challenging since an extremely high third‐order (or higher‐order) nonlinearity is required. For example, for the generation of macroscopic superposition states (so‐called Schrodinger's cat states), which might serve as a resource state for CV quantum computing, by exploiting the Kerr effect in standard fiber one needs a loss‐ and noise‐free optical fiber of about 1500 km to reach the required nonlinearity (37,38)! Sufficiently large Kerr type nonlinearities (39–41) might be within reach by interfacing light with mechanical systems. These optomechanical technologies have been dramatically refined over the last decade and are rapidly approaching a stage where quantum effects are becoming visible (42).

A common strategy to enable non‐Gaussian transformation is to couple CV mode to a discrete variable system. For example, a non‐Gaussian nonlinearity can be produced by strongly coupling a CV Gaussian mode to a discrete two‐level atom, describable by the Jaynes–Cummings Hamiltonian. Using this approach, non‐Gaussian transformations have so far been demonstrated only in the microwave regime (43,44). In the optical regime, non‐Gaussian transformations have been realized by coupling the Gaussian CV mode to a DV photon counter ( 22,45). The standard approach is to tap‐off a small part of a Gaussian squeezed mode using an asymmetric beam splitter and subsequently measure this part with a photon counter. Once a photon is measured, it is known that a single photon was subtracted from the squeezed vacuum mode, thus transforming it into a non‐Gaussian state similar to an optical cat state. This method has been further developed and has led to an avalanche of different experiments (see Ref. (6,46) for review).

18.4.1 Quantum Dense Coding

Dense coding is a protocol that enhances the communication capacity of a channel by the usage of entanglement. The protocol was originally proposed (47) and experimentally realized (48) for polarization encoded qubits (see also Chapter 17), and later on the idea was translated to the CV regime (49,50). It was shown that by using CV entanglement the classical channel capacity could ultimately be doubled.

Information is sent by encoding a message with distinguishable symbols onto physical entities, such as optical quantum states, and subsequently transmitting it to the receiver who employs a certain measurement strategy to extract the information. The symbols sent by Alice (A) is defined by the alphabet each member occurring with probability , whereas the alphabet received by Bob (B) is given by with occurrence probabilities (1). Due to fundamental quantum noise of the information carriers as well as noise in the channel, the two alphabets are in general not identical. The interesting information theoretical parameter is the so‐called mutual information, which quantifies the information A and B have in common. When classical information is encoded into quantum states, this quantity is given by

18.7

where is the conditional probability (the probability that A sent the letter if B received the letter ). is an operator that characterizes the measurement strategy applied by B, examples being the standard homodyne and dual‐homodyne detectors for which and , respectively. Finally is the density operator associated with the quantum states.

The channel capacity, which states the maximum achievable channel throughput per usage, is now found by maximizing the mutual information over the input alphabet and all possible measurement strategies:

18.8

In fact, due to the intrinsic indeterminancy of quantum mechanics, there is an upper bound on the mutual information, and therefore on the channel capacity. This famous bound, which is called the Holevo bound (51), puts a fundamental limit to the maximum information transfer. How can this limit be reached for CV states?

Dealing with CV states, the capacity can in principle be infinitely large because the phase space is infinitely large. Therefore, in order to get a finite value on the capacity, we need to place some constraints on the usage of the channel. Because of the ever‐growing traffic on optical communication lines, it is reasonable (and common) to assume that the mean power traveling down the channels per usage is constrained. Using this constraint on the mean number of photons per usage, the best communication strategy, that is, the strategy that reaches Holevo's bound, is the one that use Fock state encoding (52). If the sender uses a Fock‐state alphabet distributed according to a thermal distribution, the channel capacity is , which is the optimal capacity for single‐ channel communication ( is the mean photon number). Other nonoptimum choices of the input alphabets are the coherent state and the squeezed state alphabets. For these alphabets, the maximal throughputs are and , respectively.

Now, if Alice and Bob share an entangled state, the channel throughput per usage can be higher than for the Fock state encoding, a protocol referred to as dense coding: Alice encodes information into her part of the EPR state, sends it to Bob, who obtains information about conjugate observables by combining the two parts of the EPR states on a beam splitter and performing homodyne measurements at the two outputs of the beam splitter. Because this protocol performs better than the Fock‐state protocol, at first sight the dense coding result seems to contradict the result of Holevo! However, it does not violate Holevo's theorem, because the shared state must also be conveyed from A to B, which means that it is a two‐channel protocol. Information is, however, only sent in one channel, so the unencoded half of the EPR state can in principle be sent off‐peak and stored although this technology has still to be developed.

The channel capacity for the dense coding protocol is given by

18.9

This capacity is always larger than that for coherent state communication. The squeezed state protocol is however better than the dense coding protocols for a certain range of mean photon numbers and squeezing degrees. For squeezing degrees greater than 4.77 dB however assures that the dense coding beats the squeezed state protocol. Most interestingly, the optimal single‐ channel capacity is beaten only for a two‐mode squeezing degree higher than 6.78 dB.

There have been some attempts to experimentally implement the dense coding protocol. Li et al. (53) used bright squeezed beams to generate entanglement via a 50: 50 beam splitter. One half of the entangled state was modulated both in amplitude and phase quadrature and subsequently sent to Bob. The state was then combined with the other half of the entangled state on a 50: 50 beam splitter and finally the output states were measured directly and the difference and sum currents were produced to yield information about the amplitude and phase quadrature below the shot noise limit. Mizuno et al. (54) performed a similar experiment, but with “vacuum” entangled state (i.e., entangled states without a carrier) rather than bright entangled states. Both experiments performed better than the coherent state protocol; however, dense coding was not demonstrated because of lack of quantum correlations.

18.4.2 Quantum Key Distribution

By means of QKD followed by one‐time pad, two authenticated parties (Alice and Bob) can, in principle, exchange confidential information with unconditional security independent of the technological power of an eavesdropper (Eve) who might interfere with the conveyed signal. Correlations between the legitimate users are established by sending quantum states from Alice to Bob through an insecure channel (controlled by Eve). These quantum correlations are turned into a set of classically correlated symbols, a set which is partly determined by the specific measurement strategy. Subsequently, by the use of an authenticated public channel and classical algorithms, Alice and Bob can distil from their list of partially correlated data a secret key about which Eve has only negligible information.

There are normally two approaches to QKD, one that is relying on shared entanglement between Alice and Bob (55) and one that involves the sending of nonorthogonal states and measurements in conjugate bases (56) (see Chapter ?? for a discussion on these approaches in the discrete variable regime). Both schemes have been implemented experimentally ( 10,57). However, the latter scheme is most widespread due to its inherent simplicity, and, therefore, only this one will be subject to discussion in the following. (We should however note that the two schemes can be treated under equal footing since the correlations obtained by Alice and Bob can be modeled as if they had shared an entangled state (58).) The scheme is referred to as a prepare and measure scheme since Alice prepares nonorthogonal states chosen randomly from a predefined set of states, she sends it to Bob who measures the states in conjugate bases, for example, the amplitude and the phase quadrature bases. In the original CV QKD, prepare and measure proposal information was encoded into a discrete (59–61) (or continuous (62)) set of squeezed or entangled states and randomly measuring the amplitude and phase quadrature (Figure 18.4). Only later it was realized that coherent state encoding and homodyne detection also serve as an interesting route to secure QKD ( 10,63). Note that the new ingredient in these proposals was the detection system at the receiver, namely, homodyne detection. Coherent state encoding was already proposed in 1992 by Bennett (64).

The idea of using coherent state and homodyne detection as a mean of QKD was first put forward by Ralph (65) and further elaborated on by Grosshans and Grangier (66), but they came to the conclusion that the scheme was only secure if the losses in the channel were less than 50%: If the loss exceeds 50%, the mutual information between an eavesdropper (who measured the part that would have been leaking into the environment and replaces the lossy channel with a perfect one) and Alice was higher than that between Bob and Alice, rendering the protocol insecure. However, this apparent “3 dB” penalty was overcome using classical distillation techniques, namely postselection (63) or reverse reconciliation (10).

The first experimental demonstration of CV QKD was performed by Hirano et al. (9) and by Grosshans et al. (10). In the former experiment, information was encoded into four different coherent states in a BB84‐type encoding strategy (which by that time was not proven to be secure), whereas the latter experiment relies on a Gaussian distribution of coherent states (proven to be secure). In this experiment, Alice continuously varies the amplitude and phase quadrature and Bob randomly measures these quadratures using fast homodyne detection. Using the reverse reconciliation algorithm, Bob then converts the continuous data set into a secret binary key. There has been several other implementations of CV QKD: Lorenz et al. (67) used a BB84‐type strategy followed by postselection as did Hirano et al. (9), but instead of using the quadrature amplitudes the Stokes parameters served as the encoding variables. This experiment was simplified by considering only a two‐state protocol (68), and, similarly, a one‐dimensional version of the Gaussian modulation has been put forward (69,70). Lance et al. (71) implemented a protocol where Bob performs dual homodyning; in terms of security, there is no need of switching between ‐ and ‐basis as realized by Weedbrook et al. (72). Lodewyck et al. (73) and Legre et al. (74) have implemented a fiber‐based QKD scheme operating at the telecommunication wavelength of m. In more recent years, the CVQKD technology at the telecomm wavelength has been greatly advanced resulting in long‐distance QKD demonstrated in the laboratory (75,76) as well as in the field (77,78).

When a certain QKD scheme is designed, the next question that arises is whether the scheme is secure against eavesdropping attacks. Normally, three different levels of attacks are considered: (i) Individual attack: Eve couples each state to a probe and stores the state in a quantum memory until Bob reveals the measurement basis. She then measures each probe independently of the others. (ii) Collective attack: Eve again interacts individually with all the signal states but now all the probes are stored in a big quantum memory and after the classical communication she measures all the probes jointly in complex generalized measurement that extracts maximum information. (iii) Coherent attack: Eve couples a preentangled multimode probe with all the states sent from Alice to Bob. This highly dimensional state is stored in a large quantum memory and after classical authentication, Eve uses an optimal strategy to extract information. In the literature, there are various proofs for security on different levels. Individual attacks are considered in ( 62, 63, 66, 72,79,80), the collective attack in (81,82), and the coherent attacks in (83,84).

When deducing the security against these attacks, it is often assumed that Alice and Bob exchange an infinitely number of states, also known as the asymptotic limit. This is, of course, not realistic, and thus one needs to take into account the fact that only a finite set of states will be available. There has been some recent attempts to refine the security analysis in this direction (85–87).

Although the security proof might tell us that the system is secure against arbitrary attacks, it is often based on the assumptions that the sender and receiver stations (Alice and Bob) are securely isolated from the outside world. However, since side‐channel attacks exist and in fact can be quite effective (88,89), these assumptions are in some cases quite crude, rendering the system insecure although the security analysis tells us it is secure. It is therefore important to consider the practical security of the devices. One solution against side‐channel attacks is to use device‐independent QKD (90) where the security is guaranteed by the violation of Bell's inequality: If Alice and Bob are able to violate Bell's inequality, then an eavesdropper will have gained no information. The violation of Bell's inequality is however extremely challenging and has not yet been accomplished for CV systems. However, a much more practical alternative to device‐independent CVQKD is measurement‐device‐independent QKD (91,92) where the protection is solely targeting the detector station. This protocol has been recently proposed and experimentally realized for CVs (93).

18.4.3 Long‐Distance Communication

Quantum information must be distributed via quantum channels, that is, channels preserving the quantumness (or quantum information) of the state. Examples of quantum communication channels are free space and fibers. However, these channels are in practice imperfect because they are lossy. One way of diminishing the losses is to use an appropriate wavelength: silica fibers possess low loss at m, whereas free‐space communication is best at around 800 nm. For communication outside the earth's atmosphere, where scattering losses are almost nonexisting, any wavelength can be used.

The bottom line is, however, that long‐distance quantum communication is not possible with the losses in present‐day communication channels. Naively one might think that a way around this is to amplify the state. But, according to basic quantum mechanical considerations, amplification is not possible without the introduction of noise, which, in turn, demolishes the quantum coherence.

So what does this mean? Is quantum communication confined to short distances only? The answer is no. By using a quantum repeater one can, in principle, extend the communication channel to arbitrary long distances (94). Such a device, is however, quite challenging since it requires the combination of quantum teleportation (95), a quantum memory, and entanglement distillation. Teleportation is a protocol that enables the communication of quantum information via a classical channel (e.g., via a mobile phone) between two parties that share an entangled state (96), and a quantum memory is a device that can store quantum coherence. The former protocol was described in Chapter 15 and the latter was the subject of Chapter 30. Entanglement distillation is a way to distil, from a large ensemble of weakly entangled states, a smaller ensemble of highly entangled states. This protocol, in contrast to the teleporter and the memory, requires the use of non‐Gaussian operations.

By combining these three protocols, we can built a quantum repeater (94): Let us assume that quantum information is to be sent from A to C. There is a person B in between. An entangled state is then sent to A and B from the midpoint, using a realistic, that is imperfect, channel. The entanglement is subsequently distilled using non‐Gaussian operations. The distilled states are then stored in quantum memories. The same protocols are performed between B and C. Now, the entanglement between B and C is used to teleport perfectly the half of the entangled state that B has, and consequently, A and B share an entangled pair, which finally can be used to faithfully transmit (teleport) quantum information. The full construction of such a quantum repeater is experimentally very challenging, but also a very active field of research. Alternative routes to long‐distance communication include a protocol based on photon storage in atomic ensembles (97) and a scheme that is based on photon emission from solid‐state devices (98).