Technical reports

# Representations of quantum operations, with applications to quantum cryptography

**Pablo J. Arrighi**

July 2004, 157 pages

This technical report is based on a dissertation submitted 23 September 2003 by the author for the degree of Doctor of Philosophy to the University of Cambridge, Emmanuel College.

## Abstract

Representations of quantum operations – We start by introducing a geometrical representation (real vector space) of quantum states and quantum operations. To do so we exploit an isomorphism from positive matrices to a subcone of the Minkowski future light-cone. Pure states map onto certain light-like vectors, whilst the axis of revolution encodes the overall probability of occurrence for the state. This extension of the Generalized Bloch Sphere enables us to cater for non-trace-preserving quantum operations, and in particular to view the per-outcome effects of generalized measurements. We show that these consist of the product of an orthogonal transform about the axis of the cone of revolution and a positive real symmetric linear transform. In the case of a qubit the representation becomes all the more interesting since it elegantly associates, to each measurement element of a generalized measurement, a Lorentz transformation in Minkowski space. We formalize explicitly this correspondence between ‘observation of a quantum system’ and ‘special relativistic change of inertial frame’. To end this part we review the state-operator correspondence, which was successfully exploited by Choi to derive the operator-sum representation of quantum operations. We go further and show that all of the important theorems concerning quantum operations can in fact be derived as simple corollaries of those concerning quantum states. Using this methodology we derive novel composition laws upon quantum states and quantum operations, Schmidt-type decompositions for bipartite pure states and some powerful formulae relating to the correspondence.

Quantum cryptography – The key principle of quantum cryptography could be summarized as follows. Honest parties communicate using quantum states. To the eavesdropper these states are random and non-orthogonal. In order to gather information she must measure them, but this may cause irreversible damage. Honest parties seek to detect her mischief by checking whether certain quantum states are left intact. Thus tradeoff between the eavesdropper’s information gain, and the disturbance she necessarily induces, can be viewed as the power engine behind quantum cryptographic protocols. We begin by quantifying this tradeoff in the case of a measure distinguishing two non-orthogonal equiprobable pure states. A formula for this tradeoff was first obtained by Fuchs and Peres, but we provide a shorter, geometrical derivation (within the framework of the above mentioned conal representation). Next we proceed to analyze the Information gain versus disturbance tradeoff in a scenario where Alice and Bob interleave, at random, pairwise superpositions of two message words within their otherwise classical communications. This work constitutes one of the few results currently available regarding d-level systems quantum cryptography, and seems to provide a good general primitive for building such protocols. The proof crucially relies on the state-operator correspondence formulae derived in the first part, together with some methods by Banaszek. Finally we make use of this analysis to prove the security of a ‘blind quantum computation’ protocol, whereby Alice gets Bob to perform some quantum algorithm for her, but prevents him from learning her input to this quantum algorithm.

## Full text

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## BibTeX record

@TechReport{UCAM-CL-TR-595, author = {Arrighi, Pablo J.}, title = {{Representations of quantum operations, with applications to quantum cryptography}}, year = 2004, month = jul, url = {http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-595.pdf}, institution = {University of Cambridge, Computer Laboratory}, number = {UCAM-CL-TR-595} }