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Second resource: quantum entanglement
The second crucial resource of quantum information is quantum entanglement, which arises when considering composite Hilbert spaces and multi-partite quantum states (shared between different parties). This property is a fundamental resource of quantum communication in that it allows to engineer and exploit correlations between distant states that cannot occur in the classical world. This enables to teleport quantum states across networks for instance, in which case classical communication and local quantum operations only are required to retrieve the state at the given destination [21]. This avoids the problem of sending quantum information through quantum channels to distant parties, where losses and errors inevitably occur as the quantum state interacts with its environment. It also provides the foundation of security for some quantumcryptographic protocols such as quantum key distribution (Section 3.3) and quantum coin flipping (Section 3.4). More formally, let us consider a general pure quantum state jÃ(N)i living in Hilbert space H Æ NN iÆ1Hi, composed of N subsystems. The state is said to be a product state if it can be expressed in the following factorizable form.
EIT and slow light
In order to fully demonstrate the quantum money scheme in Chapter 6, we use Electromagnetically
Induced Transparency (EIT). This scheme creates and exploits the destructive interference process that occurs between two transitions in a ¤-type atomic system, such as the one presented in Fig. 2.1. The quantum information is effectively stored in the collective excitation of a cloud of cold atoms: the group velocity of the incident light in the cloud is considerably reduced, whilst a transparency window is created which results in no photon absorption. While there exist other mechanisms for light storage in atomic ensembles, such as photon echo schemes [35], or using other platforms like single emitters [36, 37], here we simply provide theoretical intuition for how to create EIT in a cloud of atoms. Note that a more complete derivation of the dark state phenomenon may be found in [38], while a thorough interpretation of slow light can be found in [39]. We provide a semi-classical treatment of the dynamics of the ¤-system, displayed in Fig.2.1: the three-level atomic system is treated as a fully quantum system, while the incident electromagnetic fields are treated as classical waves. We assume that the j1i!j2i transition is forbidden. For simplicity, we neglect the decay rates from the upper level to the lower levels caused by spontaneous decays and dephasing, as well as the decays in the two ground levels.
The total Hamiltonian describing the time evolution of the driven system may be written as a sum of the constant unperturbed atomic Hamiltonian Ha, and the interaction Hamiltonian Hi(t), which describes the dynamics due to the interaction with the electromagnetic fields.
No-cloning theorem
The no-cloning theorem embodies a fundamental difference between classical and quantum information, and represents the second founding principle of quantum cryptography. In fact, it is a more general and abstract way of stating the uncertainty principle, and also allows for conjugate coding. The theorem states that an unknown quantum system cannot be perfectly copied with unit probability. This is in contrast with the classical world, in which a bit of information may be read or measured without destroying its properties, and hence reproduced an infinite amount of times with no error. This property arises once again from the linearity of quantum mechanics described in Section 2.1.1, and the proof may be derived by contradiction.
Table of contents :
1 Introduction
1.1 Context
1.1.1 Quantum networks
1.1.2 Security in practice
1.2 Thesis results
1.2.1 Outline
1.2.2 Publications
2 Preliminaries
2.1 Mathematical framework
2.1.1 First resource: quantum coherence
2.1.2 Second resource: quantum entanglement
2.1.3 Quantum measurements
2.1.4 Quantum operations
2.2 Semidefinite programming
2.2.1 Convex cone
2.2.2 Primal problem
2.2.3 Dual problem
2.2.4 Weak and strong duality
2.3 Quantum optics
2.3.1 Quantum states of light
2.3.2 Linear optics
2.3.3 EIT and slow light
3 Quantum cryptography
3.1 Foundations of information-theoretic security
3.1.1 Conjugate coding
3.1.2 No-cloning theorem
3.2 Quantum money
3.2.1 Classification
3.2.2 Private-key with quantum verification
3.2.3 Private-key with classical verification
3.2.4 Public-key
3.3 Quantum key distribution
3.3.1 BB84 protocol
3.3.2 Other variants
3.4 Quantum coin flipping
3.4.1 Strong coin flipping
3.4.2 Weak coin flipping
3.4.3 Unified framework with abort cases
4 Proof-of-principle implementation of a quantum credit card
4.1 Motivation
4.2 Protocol and correctness
4.3 Security
4.3.1 Single qubit pair
4.3.2 Extension to n pairs
4.3.3 Weak coherent states with fixed phase
4.3.4 Weak coherent states with randomized phase
4.3.5 Loss tolerance for USD
4.4 Experimental implementation
4.4.1 Proof-of-principle setup
4.4.2 Experimental steps
4.4.3 Results
4.5 Independent work
4.5.1 Results outline
4.5.2 Comparison
4.6 Conclusion
5 Practical security for trusted and untrusted payment terminals
5.1 Motivation
5.2 Protocol and correctness
5.3 Security
5.3.1 Principle and proof outline
5.3.2 Trusted terminal
5.3.3 Untrusted terminal
5.4 Optimization results
5.4.1 Single state
5.4.2 Alternative SDP formulation
5.4.3 Extension to n parallel repetitions
5.5 Independent work
5.6 Conclusion
6 Experimental demonstration of genuine credit card storage
6.1 Motivation
6.2 Protocol
6.3 Experimental principle
6.3.1 Outline
6.3.2 Setup
6.4 Practical security
6.4.1 Secure parameter range (reminder)
6.4.2 Post-selection assumptions
6.4.3 Phase locking and randomization
6.5 Time-dependent security
6.6 Conclusion
7 Quantum weak coin flipping with a single photon
7.1 Motivation
7.2 Protocol and correctness
7.3 Ideal security
7.3.1 Dishonest Bob
7.3.2 Dishonest Alice with number-resolving detectors
7.3.3 Dishonest Alice with threshold detectors
7.4 Noise tolerance
7.5 Loss tolerance
7.5.1 Correctness
7.5.2 Dishonest Bob
7.5.3 Dishonest Alice: outline
7.5.4 Dishonest Alice with lossy delay
7.5.5 Dishonest Alice with perfect delay
7.6 Practical protocol performance
7.6.1 Solving the system
7.6.2 Results
7.7 Extension to lower bias (preliminary results)
7.7.1 Framework
7.7.2 Protocol with n rounds
7.7.3 Correctness for 2 rounds
7.7.4 Dishonest Bob
7.7.5 Dishonest Alice
7.7.6 Numerical results
7.8 Conclusion
8 Conclusion
A Proof-of-principle credit card scheme
A.1 Explicit derivation of the ± parameter
A.2 Phase randomization
A.3 Simulation of the evolution of c as a function of ¹
B Trusted and untrusted payment terminals
B.1 Outline of the squashing model
B.2 Proof of Lemma 5.1
B.3 Explicit expression for phase-randomized states
C Quantum weak coin flipping
C.1 Proof of Lemma 7.1
C.2 Creation operator evolution in the lossy protocol
C.3 Proof of Lemma 7.2
C.4 State evolution in the 2-rounded protocol
C.5 Proof of Lemma 7.3
D Résumé en français (French summary)
Bibliography
