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ACM/IEEE Symposium on Logic in Computer Science (LICS)

A categorical semantics of quantum protocols

Samson Abramsky, Bob Coecke

Published Jan 1, 2004• 581 citations
Publisher

Abstract

Particular focus in this paper is on quantum information protocols, which exploit quantum-mechanical effects in an essential way. The particular examples we shall use to illustrate our approach will be teleportation (Benett et al., 1993), logic-gate teleportation (Gottesman and Chuang,1999), and entanglement swapping (Zukowski et al., 1993). The ideas illustrated in these protocols form the basis for novel and potentially very important applications to secure and fault-tolerant communication and computation (2001,1999,2000).

DOI
10.1109/lics.2004.1319636

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Particular focus in this paper is on quantum information protocols, which exploit quantum-mechanical effects in an essential way. The particular examples we shall use to illustrate our approach will be teleportation (Benett et al., 1993), logic-gate teleportation (Gottesman and Chuang,1999), and entanglement swapping (Zukowski et al., 1993). The ideas illustrated in these protocols form the basis for novel and potentially very important applications to secure and fault-tolerant communication and computation (2001,1999,2000).
We study quantum information and computation from a novel point of view. Our approach is based on recasting the standard axiomatic presentation of quantum mechanics, due to von Neumann, at a more abstract level, of compact closed categories with biproducts. We show how the essential structures found in key quantum information protocols such as teleportation, logic-gate teleportation, and entanglement-swapping can be captured at this abstract level. Moreover, from the combination of the --apparently purely qualitative-- structures of compact closure and biproducts there emerge `scalars` and a `Born rule'. This abstract and structural point of view opens up new possibilities for describing and reasoning about quantum systems. It also shows the degrees of axiomatic freedom: we can show what requirements are placed on the (semi)ring of scalars C(I,I), where C is the category and I is the tensor unit, in order to perform various protocols such as teleportation. Our formalism captures both the information-flow aspect of the protocols (see quant-ph/0402014), and the branching due to quantum indeterminism. This contrasts with the standard accounts, in which the classical information flows are `outside' the usual quantum-mechanical formalism.
Abstract We have come a long way from Chap. 10.1007/978-3-030-61601-4_1 To recap on what we have learnt, we have understood important quantum mechanical phenomena such as superposition and measurement (through the Stern-Gerlach and Mach-Zehnder experiments). We have also learnt that while quantum computers can in principle break classical encryption protocols, they can also be used to make new secure channels of communication. Furthermore, we have applied quantum logic gates to qubits to perform quantum computations. With entanglement, we teleported the information in an unknown qubit to another qubit. This is quite a substantial achievement.
The main contribution of this paper is the introduction of a dynamic logic formalism for reasoning about information flow in composite quantum systems. This builds on our previous work on a complete quantum dynamic logic for single systems. Here we extend that work to a sound (but not necessarily complete) logic for composite systems, which brings together ideas from the quantum logic tradition with concepts from (dynamic) modal logic and from quantum computation. This Logic of Quantum Programs (LQP) is capable of expressing important features of quantum measurements and unitary evolutions of multi-partite states, as well as giving logical characterisations to various forms of entanglement (for example, the Bell states, the GHZ states etc.). We present a finitary syntax, a relational semantics and a sound proof system for this logic. As applications, we use our system to give formal correctness proofs for the Teleportation protocol and for a standard Quantum Secret Sharing protocol; a whole range of other quantum circuits and programs, including other well-known protocols (for example, superdense coding, entanglement swapping, logic-gate teleportation etc.), can be similarly verified using our logic.
Quantum information and computation is concerned with the use of quantum-mechanical systems to carry out computational and information-processing tasks (Nielsen and Chunag, 2000). In the few short years that this approach has been studied, a number of remarkable concepts and results have emerged, most notably:a couple of spectacular algorithms and a number of information protocols, exemplified by quantum teleportation, which exploit quantum entanglement in an essential fashion. The current tools available for developing quantum algorithms and protocols are deficient on two main levels: firstly, they are too low-level and at a more fundamental level, the standard mathematical framework for quantum mechanics (which is essentially due to von Neumann (1932)) is actually insufficiency comprehensive for informatic purposes. In joint work with Bob Coecke, we have recently made some striking progress in addressing both these points. They have recast the von Neumann formalism at a more abstract and conceptual level, using category theory.
The main contribution of this paper is the introduction of a dynamic logic formalism for reasoning about information flow in composite quantum systems. This builds on our previous work on a complete quantum dynamic logic for single systems. Here we extend that work to a sound (but not necessarily complete) logic for composite systems, which brings together ideas from the quantum logic tradition with concepts from (dynamic) modal logic and from quantum computation. This Logic of Quantum Programs (LQP) is capable of expressing important features of quantum measurements and unitary evolutions of multi-partite states, as well as giving logical characterisations to various forms of entanglement (for example, the Bell states, the GHZ states etc.). We present a finitary syntax, a relational semantics and a sound proof system for this logic. As applications, we use our system to give formal correctness proofs for the Teleportation protocol and for a standard Quantum Secret Sharing protocol; a whole range of other quantum circuits and programs, including other well-known protocols (for example, superdense coding, entanglement swapping, logic-gate teleportation etc.), can be similarly verified using our logic.