Classical Structures Based on Unitaries

Starting from the observation that distinct notions of copying have arisen in different categorical fields (logic and computation, contrasted with quantum mechanics) this paper addresses the question of when, or whether, they may coincide. Provided all definitions are strict in the categorical sense, we show that this can never be the case. However, allowing for the defining axioms to be taken up to canonical isomorphism, a close connection between the classical structures of categorical quantum mechanics, and the categorical property of self-similarity familiar from logical and computational models becomes apparent. The required canonical isomorphisms are non-trivial, and mix both typed (multi-object) and untyped (single-object) tensors and structural isomorphisms; we give coherence results that justify this approach. We then give a class of examples where distinct self-similar structures at an object determine distinct matrix representations of arrows, in the same way as classical structures determine matrix representations in Hilbert space. We also give analogues of familiar notions from linear algebra in this setting such as changes of basis, and diagonalisation.Comment: 24 pages,7 diagram

Quantum channels as a categorical completion

We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all quantum channels is a canonical completion of the category of pure quantum operations (with ancilla preparations). More precisely, we prove that the category of completely positive trace-preserving maps between finite-dimensional C*-algebras is a canonical completion of the category of finite-dimensional vector spaces and isometries. Second, we extend our result to give a foundation to the topological relationships between quantum channels. We do this by generalizing our categorical foundation to the topologically-enriched setting. In particular, we show that the operator norm topology on quantum channels is the canonical topology induced by the norm topology on isometries.Comment: 12 pages + ref, accepted at LICS 201

Classification of Multipartite Entanglement via Negativity Fonts

Partial transposition of state operator is a well known tool to detect quantum correlations between two parts of a composite system. In this letter, the global partial transpose (GPT) is linked to conceptually multipartite underlying structures in a state - the negativity fonts. If K-way negativity fonts with non zero determinants exist, then selective partial transposition of a pure state, involving K of the N qubits (K leq N) yields an operator with negative eigevalues, identifying K-body correlations in the state. Expansion of GPT interms of K-way partially transposed (KPT) operators reveals the nature of intricate intrinsic correlations in the state. Classification criteria for multipartite entangled states, based on underlying structure of global partial transpose of canonical state, are proposed. Number of N-partite entanglement types for an N qubit system is found to be 2^{N-1}-N+2, while the number of major entanglement classes is 2^{N-1}-1. Major classes for three and four qubit states are listed. Subclasses are determined by the number and type of negativity fonts in canonical state.Comment: 5 pages, No figures, Corrected typo

A Bestiary of Sets and Relations

Building on established literature and recent developments in the graph-theoretic characterisation of its CPM category, we provide a treatment of pure state and mixed state quantum mechanics in the category fRel of finite sets and relations. On the way, we highlight the wealth of exotic beasts that hide amongst the extensive operational and structural similarities that the theory shares with more traditional arenas of categorical quantum mechanics, such as the category fdHilb. We conclude our journey by proving that fRel is local, but not without some unexpected twists.Comment: In Proceedings QPL 2015, arXiv:1511.0118

On the Optimality of Quantum Encryption Schemes

It is well known that n bits of entropy are necessary and sufficient to perfectly encrypt n bits (one-time pad). Even if we allow the encryption to be approximate, the amount of entropy needed doesn't asymptotically change. However, this is not the case when we are encrypting quantum bits. For the perfect encryption of n quantum bits, 2n bits of entropy are necessary and sufficient (quantum one-time pad), but for approximate encryption one asymptotically needs only n bits of entropy. In this paper, we provide the optimal trade-off between the approximation measure epsilon and the amount of classical entropy used in the encryption of single quantum bits. Then, we consider n-qubit encryption schemes which are a composition of independent single-qubit ones and provide the optimal schemes both in the 2- and the operator-norm. Moreover, we provide a counterexample to show that the encryption scheme of Ambainis-Smith based on small-bias sets does not work in the operator-norm.Comment: 15 page