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Mathematical Preliminaries
In this Section we will present some basic notions of linear algebra that we will be using during the rest of the thesis. Two readings that provide a more detailed introduction to the mathematical foundations of quantum mechanics are the book of Nielsen and Chuang [34] and the lecture notes of John Watrous [35]. The first definition we need to give is that of a complex vector space V. A complex vector space is formed by a set of vectors that can be added together or multiplied by complex numbers (called scalars). For V to be a complex vector space, 8 u, v,w 2 V and 8 a, b 2 C, the following properties are necessary for the two operations, + : V V !V and : C V !V.
Bit Commitment and No-go Theorems
A bit commitment scheme is a two-party protocol that is used in order for one party (the Sender) to commit himself towards another party (the Receiver) to a specific bit value. It comprises of two phases, the Commit and the Reveal phase, and has to be:
1. Concealing – At the end of the Commit phase, the Receiver does not obtain any knowledge of the Sender’s bit value.
2. Binding – At the end of the Reveal phase, and given all communication transactions between the two parties, there exists at most one value that the Receiver accepts as a valid “opening » of the commitment. Bit Commitment schemes play an important role in multiparty computation, since they allow to construct Zero-Knowledge and Coin Flipping protocols and are therefore essential for any type of Secure Function Evaluation (see [9] for an extensive study of multiparty computation). Bit commitment schemes are constructed from one-way functions, and consequently their security also relies on the existence of such functions. It is well known that a classical Bit Commitment scheme cannot be perfectly concealing and perfectly binding at the same time. One of the two parties needs to be computationally bounded, in order for a perfect classical Bit Commitment scheme to be possible. At the dawn of the Quantum Cryptography era, researchers were confident that quantum mechanical effects could be used to strengthen Bit Commitment, and provide a protocol with information-theoretic security. Unfortunately, this was proven to be impossible, due to the no-go theorems (as they are now known) by Lo and Chau [11] and Mayers [10]. Loosely speaking, the reason why Quantum Bit Commitment is impossible, is because any protocol that is perfectly concealing, needs to associate each of two indistinguishable quantum states, i.e. that have the same density matrix, with one of the two values of the bit. Using the Gram-Schmidt decomposition, the Sender can then perform some local operation on his share of an entangled state, in order to transform the state associated to the bit value 0, to the state associated to the bit value 1 (and vice versa), and therefore break the binding condition.
Security assumptions and analysis with standard imperfections.
We first provide a security analysis when the following three assumptions are satisfied:
1. Honest Alice creates each of the four protocol states ji ,ci i (see Fig. 3.1) with the same probability, independently for each pulse and independently of Bob.
2. For the first pulse j that honest Bob successfully measures, the distribution of his measurement basis j and his bit b is uniform and independent of Alice.
3. For the first pulse j that honest Bob successfully measures, if the state of the pulse is , then for each basis j , the probabilities of the two outcomes are hj,0jjj,0i and hj,1jjj,1i. We describe the optimal cheating strategies of malicious Alice and Bob and derive the corresponding cheating probabilities when the above assumptions are satisfied. As we will see in the following sections, the cheating probabilities depend on the number of rounds K, the protocol parameter y and the mean photon number . We can make the protocol fair, by changing the parameter y.
Table of contents :
1 Introduction
1.1 Thesis Contribution
1.1.1 Quantum Coin Flipping
1.1.2 Entanglement Verification
1.1.3 Quantum Game Theory
1.2 Thesis Outline and Scientific Production
1.3 Scientific Collaborations
2 Preliminaries
2.1 Mathematical Preliminaries
2.2 The Postulates of Quantum Mechanics
2.3 Entanglement
2.3.1 The CHSH Game
2.3.2 The Mermin-GHZ Game
2.4 Cryptography
2.4.1 One-way Functions
2.4.2 Bit Commitment and No-go Theorems
2.5 Experimental Tools
2.5.1 Polarisation Encoding
2.5.2 Phase Encoding
3 Coin Flipping
3.1 Introduction
3.2 Previous Work
3.3 Our work
3.3.1 The Protocol
3.3.2 Honest Player Abort
3.4 Experimental Setup
3.5 Security assumptions and analysis with standard imperfections.
3.5.1 Malicious Alice
3.5.2 Malicious Bob.
3.6 Satisfying the security assumptions with the plug&play system.
3.6.1 Malicious Alice
3.6.2 Malicious Bob
3.7 Results
3.8 Enhancing the security of protocols against bounded adversaries
3.8.1 Computationally bounded quantum coin flipping
3.8.2 Noisy storage quantum coin flipping
3.9 Conclusion
4 Entanglement Verification
4.1 Previous Work
4.2 Entanglement Verification under perfect conditions
4.2.1 The Basic Verification Protocol
4.2.2 Correctness of the Basic Verification Protocol
4.2.3 Security in the Honest Model
4.2.4 Security in the Dishonest Model
4.3 Entanglement Verification with imperfections
4.3.1 Enhanced Verification Protocol
4.3.2 Correctness of the Enhanced Protocol
4.3.3 Security in the Honest Model
4.3.4 Security in the Dishonest Model
4.3.5 Losses
4.3.6 Noise
4.4 Experimental Procedure
4.4.1 Tests on an n-party GHZ state
4.4.2 Comparing experimental results
4.5 Conclusion
5 Quantum Games
5.1 Introduction
5.2 Bayesian Games
5.3 Nash Equilibria
5.4 The Game
5.5 Classical Strategies
5.6 Quantum Strategies
5.6.1 Maximally Entangled Strategy
5.6.2 Non-maximally entangled strategy
5.7 Experiment
5.8 Conclusion
6 Summary and Perspectives
6.1 Summary
6.2 Future Perspectives
A Entanglement Verification
A.1 Security of the Enhanced Verification Protocol
A.1.1 Security in the Honest Model
A.1.2 Security in the Dishonest Model
Bibliography
