Quantum Communication with Continuous Variables

Abstract

Many quantum communication schemes rely on the resource of entanglement. For example, quantum teleportation is the transfer of arbitrary quantum states through a classical communication channel using shared entanglement. Entanglement, however, is in general not easy to produce on demand. The bottom line of this work is that a particular kind of entanglement, namely that based on continuous quantum variables, can be created relatively easily. Only squeezers and beam splitters are required to entangle arbitrarily many electromagnetic modes. Similarly, other relevant operations in quantum communication protocols become feasible in the continuous‐variable setting. For instance, measurements in the maximally entangled basis of arbitrarily many modes can be accomplished via linear optics and efficient homodyne detections. In the first two chapters, some basics of quantum optics and quantum information theory are presented. These results are then needed in Chapter III, where we characterize continuous‐variable entanglement and show how to make it. The members of a family of multi‐mode states are found to be truly multi‐party entangled with respect to all their modes. These states also violate multi‐party inequalities imposed by local realism, as we demonstrate for some members of the family. Further, we discuss how to measure and verify multi‐party continuous‐variable entanglement. Various quantum communication protocols based on the continuous‐variable entangled states are discussed and developed in Chapter IV. These include the teleportation of entanglement (entanglement swapping) as a test for genuine quantum teleportation. It is shown how to optimize the performance of continuous‐variable entanglement swapping. We highlight the similarities and differences between continuous‐variable entanglement swapping and entanglement swapping with discrete variables. Chapter IV also contains a few remarks on quantum dense coding, quantum error correction, and entanglement distillation with continuous variables, and in addition a review of quantum cryptographic schemes based on continuous variables. Finally, in Chapter V, we consider a multi‐party generalization of quantum teleportation. This so‐called telecloning means that arbitrary quantum states are transferred not only to a single receiver, but to several. However, due to the quantum mechanical no‐cloning theorem, arbitrary quantum states cannot be perfectly copied. We present a protocol that enables telecloning of arbitrary coherent states with the optimal quality allowed by quantum theory. The entangled states needed in this scheme are again producible with squeezed light and beam splitters. Although the telecloning scheme may also be used for "local'' cloning of coherent states, we show that cloning coherent states locally can be achieved in an optimal fashion without entanglement. It only requires a phase‐insensitive amplifier and beam splitters.

Realistic continuous-variable quantum teleportation with non-Gaussian resources

“…Although several alternative formalisms are available for the description of the B-K CV teleportation protocol [35,36,37,38,39,40], the characteristic function representation proves to be particularly convenient when considering the nonideal case and non-Gaussian resources. The description of nonideal teleportation re- In the first step, the input mode is mixed by Alice with one of the two beams (modes) of the entangled resource; the ensuing state is then subject to a realistic Bell measurement.…”

Enhanced multipartite quantum correlation by non-Gaussian operations

“…As is known, the gain g = (e 4r − 1)/(e 4r + 1/2) yields the optimal fidelity for the CV GHZ states, with the classical fidelity 1/2 achieved even in a zerosqueezing limit r → 0 [57]. When maximized over g, the CV GHZ states with photon subtraction on two modes still achieves better fidelities over the CV GHZ states and it also approaches the classical fidelity 1/2 as r → 0.…”

“…Charlie obtains the output state after displacing his mode by the amount of √ 2(x u + ip v ) + igp 2 , where g is a gain factor that can be adjusted to give an optimal fidelity. Without the information on p 2 , the output fidelity may go below the classical bound as the squeezing parameter r increases [57]. A high fidelity that can be achieved only with other remaining party's help can be regarded as an evidence of multipartite entanglement.…”

“…This is not only because there are few protocols exploiting multipartite nature of a system but also because it is difficult to characterize multipartite correlations in general, especially, for non-Gaussian states in CV quantum information regime [55]. Here, extending two-mode systems into multipartite systems, we investigate the effects of photon operations on quantum correlations of a class of multipartite continuous variable systems, the CV GHZ states introduced by van Loock and Braunstein [56][57][58]. Quantum nonlocality already has been investigated on this class and the CV GHZ states have shown violations of local realism [59,60].…”

Entanglement Swapping between Discrete and Continuous Variables

“…However, due to the finite squeezing of both initial entanglement sources, the final entanglement after swapping in the CV scheme is inevitably degraded by excess noise. In fact, the entanglement drops exponentially [17,18] and in practice, CV entanglement swapping can always be replaced by a direct transmission through a lossy channel [19]. Moreover, purification techniques for this type of degraded entanglement are not so advanced at present [20,21].…”