We introduced new schemes for solving the multiparty communication tasks of com-munication complexity and secret sharing. Unlike other quantum-assisted methods employing multi-particle entangled states, our protocols involve only the sequential communication and transformation of a single qubit. As single-qubit operations using linear optical elements and the analysis of photon polarization states are efficiently realizable with today’s technology, we were therefore able to present the first success-ful experimental demonstrations of the protocols for as many as five and six parties, respectively. This is to our knowledge the highest number of actively performing parties in a quantum protocol ever implemented. In principle, we see no experimen-tal barrier to extend the performed protocols to even significantly higher number of participants.
From theoretical point of view, our work unambiguously demonstrates the power of qubit communication in multiparty tasks. Even though a qubit can be used to communicate only one bit of information between two unentangled parties [114], the full continuous nature of the quantum state still helps when performing some quan-tum information tasks. The trick is not to require that the information encoded in a qubit is actually read out, but rather substituted for a large amount of classical
12The initial polarization of the heralded photons is fixed in the experiment by putting the polar-ization filter in the path to the trigger detector, see Fig. 4.4(b). Since the photons form polarpolar-ization entangled EPR pairs, detection of a trigger photon behind a polarization filter collapses the initially undefined polarization state of the heralded one to the required|+xi. All other photons, since no trigger event accompanies them, remain unpolarized.
information, which is sequentially processed by every party.
(i) In CCPs with restricted amount of communication this allows to achieve the exponential separation between the quantum/classical performance with increasing number of parties. The performance of the quantum protocol and of its implemen-tation is so high that even without any correction for detector efficiency, loss in the set-up etc., our experimental results significantly outperform the best classical pro-tocols for both studied tasks.
(ii) In QSS this allows to induce among the players the GHZ-like correlations, which are essential for achieving secret sharing task. Still, due to the different architectures of the GHZ-assisted and single-qubit schemes one has to be aware of possibly new security dangers. Since eavesdropper might have an access to input and output ports of the partners, particulary the attacks of Trojan Horse type have to be carefully analyzed in our scheme. Yet, they can be precluded by the players with a reasonable technological effort like, e.g., discussed in [147]. Also, very recently it was shown that by using a quantum memory and a teleportation protocol the cheater/eavesdropper can learn a minor random part of the key (the size depends on her/his position in the communication chain) in our protocol [148]. To close this subtle security loophole various classical methods, like e.g. hash functions, have to be applied or the quantum protocol has to be further refined [149].
Interestingly, our single-qubit communication protocols for CCPs and QSS can be considered as a direct adaptation of the corresponding entanglement-based protocols using GHZ states. While the entanglement-based protocols use a non-local quantum phase, which can be operated on locally by each party, the single-qubit protocols use a quantum phase of a qubit to sequentially encode local information as it flies by the parties towards the last party, who performs a suitable measurement. The expectation value of such a measurement has the same structure as a correlation function of GHZ state (defined as the expectation value of the product of the local measurement results by parties). This explains why the tasks of secret sharing and communication complexity are efficiently solvable with protocols involving only the communication of a single qubit. It appears rather plausible that our single-qubit approach may be a practical solution for many other problems, which were believed to be possible only with the use of multiparty entangled states until now.
In summary, by successfully solving and implementing a cryptographic task as well as a task originating in computer science, we clearly illustrate the potential to introduce multiparty communication problems into real life.
Chapter 5
Conclusions and Outlook
The work presented in this thesis addresses two topics related primarily to the de-velopment of technologies and systems for practical quantum communication. The first topic is the efficient generation of photonic entanglement and the second is the simplification and implementation of quantum-assisted solutions to multiparty com-munication tasks.
Generation of photonic entanglement. In the last decade we witnessed a dramatic progress of photon pair sources in terms of the output flux and the practicality of systems. It appears safe to say that some of the concepts have come to reach an upper bound, forcing one to develop novel methods and techniques for the generation of photonic entanglement. This gradual progress is also clearly perceptible in this thesis, which describes the work on two distinct sources of polarization-entangled photon pairs, spanning the period of over four years.
The first type of source was designed and implemented using the well established concept of degenerate non-collinear emission from a single type-II nonlinear crys-tal. With a blue laser diode as a pump source of SPDC, this configuration yields maximally hundreds of detected pairs per second and mW of pump power. More importantly, no substantial increase of the brightness is to be expected in the future.
This is due to the limitation of the crystal length to a few mm range and due to the fact that only a small fraction of the total SPDC output flux is polarization-entangled. However, the main goal of this design, to demonstrate the feasibility of sources using blue laser diodes as replacement of costly large frame ion lasers, was achieved. Moreover, based on this design, a setup for a lab course on entanglement and the EPR-Bell-problem could be built.
Deficiencies of the first setup are overcome in the second type of source, which features a novel configuration with only a single spatial mode for collecting non-degenerate photon pairs generated via SPDC in a double-crystal geometry. This fully collinear configuration greatly enhances the practicality of the source. Particularly, it allows one to use very long nonlinear crystals, resulting in a highly increased
photon-111
pair flux. With a pair of 16-mm-long crystals we observed almost two order of magnitudes higher photon-pair flux compared to the non-collinear type II source.
The usage of the only spatial mode for coupling of the photons also precludes the occurrence of any intrinsic spatial effect limiting the quantum-interference visibility.
Applying a tailored dispersion compensation, we were thereby able to reach nearly 99 % quantum-interference visibility, even though a free-running laser diode as a pump of SPDC was used and no filtering of generated photons was applied. These merits, together with almost free-of-alignment operation, suggest that this type of source is an especially promising candidate for many future practical applications, including quantum cryptography, detector calibration or use in undergraduate lab courses. Apart from the practical applicability, the source might find its relevance also in fundamental tests of quantum mechanics. Particularly, a measured net coupling efficiency of photons pairs of about 90 % offers a unique chance to accomplish a photonic Bell test excluding a fair sampling hypothesis for the first time.
There is a multitude of future research directions following on from the work on photon pair sources presented in this thesis. As the technology for manufacturing quasi-phase-matched materials have progressed, periodically poled crystals like PP-KTP have become commonly available, offering a greater freedom in the choice of wavelengths and an access to the highest nonlinear coefficients. Their integration into the collinear type I source should bring a higher output brightness, reduction of photon bandwidth, and possibly, complete suppression of the spatial walk-off effect, which has to be compensated in the current realization using additional birefrin-gent crystals. Another issue deserving more attention is the usage of ultrashort (fs to ps range) pulse pumped SPDC. The precise timing information inherent to such type of pumping showed to be very useful for the realization of certain experiments in quantum optics including generation of multi-photon entangled states, quantum teleportation or other advanced communication protocols. Although the theoretical model developed in the thesis suggests that the concept of collinear type I source works equally efficient for ultrashort pulse pumping, the experimental verification has to be accomplished.
Turning to applications, it is the next goal for the source of non-degenerate en-tangled photon pairs to be used for a real world experiment such as a demonstration of long-distance quantum cryptography; see also [150]. The preliminary steps in this direction have been already taken by testing the source outside an ideal lab environment, showing no impairment of the output performance and an overall long-term stability of the system. Concerning the applicability of the source for future applications a great deal of effort has to be also made to miniaturize the system, to further minimize the necessary adjustment, and to use a design with superior (thermo-)mechanical stability such as, for example, using fixed components.
Multiparty communication tasks. The use of quantum resources leads to a speed-up in solving many communication tasks or even achieving goals which are classically
not possible at all. While the recent rapid progress in the development of bright entangled photon-pair sources has been followed with ample experimental reports on two-party quantum communication tasks, the practical implementations of tasks for more than two parties have been quite rare, so far. This is primarily due to the re-quirement of multiparty entangled states, which are very difficult to be produced with current methods and, moreover, suffer from high noise and (depending on the num-ber of partners) exponentially decreasing output yield. We show that entanglement is not the only non-classical resource endowing the quantum multiparty information processing its power. Instead, only the sequential communication and transformation of a single qubit can be sufficient to accomplish certain tasks in a highly efficient way.
This we prove for two distinct communication tasks, communication complexity and secret sharing.
Quantum communication complexity addresses the problem of communication re-duction for distributed computation tasks by the utilization of quantum effects. In the thesis we study two CCPs, in whichN separated partners with local random data must determine in common the correct value of a globally defined Boolean function with the highest possible probability of success. In these tasks the parties can com-municate onlyN−1 bits (classical scenario) orN−1 qubits (quantum scenario). We theoretically show that for such tasks with restricted communication the advantage of single-qubit assisted protocols over the corresponding classical ones may increase even exponentially with the number of partners. Furthermore, we successfully imple-ment the quantum protocols forN = 5 partners using a heralded single photon from parametric down-conversion as the carrier of the qubit communicated from one part-ner to the other. For a fair comparison with the classical scenario, no correction for imperfections of the state-of-the-art set-up was done whatsoever. The performance of the protocol and of its implementation is so high that even without any correction for detector efficiency, loss in the set-up etc., we significantly outperform the best classical protocols for both studied tasks.
Secret sharing is a cryptographic task with the aim of distributing a secret in such a way that its reconstruction requires the collaboration of the participating partners.
Classically the problem is solved by splitting the information using some mathemati-cal algorithms and distributing the resulting pieces to the legitimate parties. However, classical communication is not secure and thus integration of quantum cryptography to secret sharing was proposed. In this protocol a shared GHZ-state allows the in-formation splitting and the eavesdropper protection simultaneously. We propose a protocol for N partners, in which only single-qubit communication is used with no need for GHZ-states. As our protocol requires only single qubits it is realizable with the current state-of-the-art technologies and, above all, scalable with respect to the number of participating partners. These merits enabled us to accomplish the first experimental proof-of-principle demonstration of secret sharing for as many as six partners. This is to our knowledge the highest number of actively performing parties in a quantum protocol ever implemented.
By successfully solving and implementing a cryptographic task as well as a task
originating in computer science, we clearly illustrate the potential to introduce mul-tiparty communication problems into real life. Yet, a great deal of effort need to be spent for this to succeed. It was our aim to provide proof-of-principle realizations of single-qubit protocols. Further work should be focused on the demonstration of more elaborate implementations and on the design of optimum system architectures. Also, many theoretical issues remain open. As concerns quantum communication com-plexity, the following questions of considerable significance should be answered: Do all entanglement-based protocols with higher-than-classical performance have single-qubit counterparts? Can one use the single-single-qubit approach to solve a broader class of CCPs? Does the higher-dimensional encoding (qudits) bring an increased per-formance of quantum protocols? Turning to secret sharing, first and foremost, the security of the single-qubit protocol has to be inspected in more detail. The next issue to be investigated is the usage of other types of qubit encoding (in particular time-bin encoding, which is suitable for fiber-based systems). Finally, on a more general note, it appears rather plausible that our single-qubit approach may be a practical solution for many other problems, which were believed to be possible only with the use of multiparty entangled states until now. The quest for such problems should certainly not be left out from the future research.