Toward a general theory of quantum games

We study properties of quantum strategies, which are complete specifications of a given party's actions in any multiple-round interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantum strategies that generalizes the Choi-Jamio{\l}kowski representation of quantum operations. This new representation associates with each strategy a positive semidefinite operator acting only on the tensor product of its input and output spaces. Various facts about such representations are established, and two applications are discussed: the first is a new and conceptually simple proof of Kitaev's lower bound for strong coin-flipping, and the second is a proof of the exact characterization QRG = EXP of the class of problems having quantum refereed games.Comment: 23 pages, 12pt font, single-column compilation of STOC 2007 final versio

Short Quantum Games

In this thesis we introduce quantum refereed games, which are quantum interactive proof systems with two competing provers. We focus on a restriction of this model that we call "short quantum games" and we prove an upper bound and a lower bound on the expressive power of these games. For the lower bound, we prove that every language having an ordinary quantum interactive proof system also has a short quantum game. An important part of this proof is the establishment of a quantum measurement that reliably distinguishes between quantum states chosen from disjoint convex sets. For the upper bound, we show that certain types of quantum refereed games, including short quantum games, are decidable in deterministic exponential time by supplying a separation oracle for use with the ellipsoid method for convex feasibility.Comment: MSc thesis, 79 pages single-space

Quantum interactive proofs and the complexity of separability testing

We identify a formal connection between physical problems related to the detection of separable (unentangled) quantum states and complexity classes in theoretical computer science. In particular, we show that to nearly every quantum interactive proof complexity class (including BQP, QMA, QMA(2), and QSZK), there corresponds a natural separability testing problem that is complete for that class. Of particular interest is the fact that the problem of determining whether an isometry can be made to produce a separable state is either QMA-complete or QMA(2)-complete, depending upon whether the distance between quantum states is measured by the one-way LOCC norm or the trace norm. We obtain strong hardness results by proving that for each n-qubit maximally entangled state there exists a fixed one-way LOCC measurement that distinguishes it from any separable state with error probability that decays exponentially in n.Comment: v2: 43 pages, 5 figures, completely rewritten and in Theory of Computing (ToC) journal forma

Fidelity of Quantum Strategies with Applications to Cryptography

We introduce a definition of the fidelity function for multi-round quantum strategies, which we call the strategy fidelity, that is a generalization of the fidelity function for quantum states. We provide many interesting properties of the strategy fidelity including a Fuchs-van de Graaf relationship with the strategy norm. We illustrate an operational interpretation of the strategy fidelity in the spirit of Uhlmann\u27s Theorem and discuss its application to the security analysis of quantum protocols for interactive cryptographic tasks such as bit-commitment and oblivious string transfer. Our analysis is very general in the sense that the actions of the protocol need not be fully specified, which is in stark contrast to most other security proofs. Lastly, we provide a semidefinite programming formulation of the strategy fidelity

Arcula: A Secure Hierarchical Deterministic Wallet for Multi-asset Blockchains

This work presents Arcula, a new design for hierarchical deterministic wallets that brings identity-based addresses to the blockchain. Arcula is built on top of provably secure cryptographic primitives. It generates all its cryptographic secrets from a user-provided seed and enables the derivation of new public keys based on the identities of users, without requiring any secret information. Unlike other wallets, it achieves all these properties while being secure against privilege escalation. We formalize the security model of hierarchical deterministic wallets and prove that an attacker compromising an arbitrary number of users within an Arcula wallet cannot escalate his privileges and compromise users higher in the access hierarchy. Our design works out-of-the-box with any blockchain that enables the verification of signatures on arbitrary messages. We evaluate its usage in a real-world scenario on the Bitcoin Cash network