GEOMETRIC PHASES FOR MIXED STATES DURING UNITARY AND NON-UNITARY EVOLUTIONS

2003 ◽  
Vol 01 (01) ◽  
pp. 135-152 ◽  
Author(s):  
ARUN K. PATI
Keyword(s):  
Geometric Phase ◽  
Parallel Transport ◽  
Quantum Systems ◽  
Mixed States ◽  
Completely Positive Maps ◽  
Initial State ◽  
Unitary Evolution ◽  
Completely Positive ◽  
Quantum Operations ◽  
Positive Maps

Mixed states typically arise when quantum systems interact with the outside world. Evolution of open quantum systems in general are described by quantum operations which are represented by completely positive maps. We elucidate the notion of geometric phase for a quantum system described by a mixed state undergoing unitary evolution and non-unitary evolutions. We discuss parallel transport condition for mixed states both in the case of unitary maps and completely positive maps. We find that the relative phase shift of a system not only depends on the state of the system, but also depends on the initial state of the ancilla with which it might have interacted in the past. The geometric phase change during a sequence of quantum operations is shown to be non-additive in nature. This property can attribute a "memory" to a quantum channel. We explore these ideas and illustrate them with simple examples.

scholarly journals New encoding schemes for quantum authentication

2005 ◽  
Vol 5 (1) ◽  
pp. 1-12
Author(s):  
P. Garcia-Fernandez ◽  
E. Fernandez-Martinez ◽  
E. Perez ◽  
D.J. Santos
Keyword(s):  
Authentication Protocol ◽  
Completely Positive Maps ◽  
Authentication Protocols ◽  
Completely Positive ◽  
Quantum Operations ◽  
Positive Maps ◽  
General Quantum ◽  
Quantum Authentication

We study the potential of general quantum operations, Trace-Preserving Completely-Positive Maps (TPCPs), as encoding and decoding mechanisms in quantum authentication protocols. The study shows that these general operations do not offer significant advantage over unitary encodings. We also propose a practical authentication protocol based on the use of two successive unitary encodings.

scholarly journals Quantum weakest preconditions

2006 ◽  
Vol 16 (3) ◽  
pp. 429-451 ◽  
Author(s):  
ELLIE D'HONDT ◽  
PRAKASH PANANGADEN
Keyword(s):  
Quantum Computation ◽  
Representation Theorem ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Weakest Precondition ◽  
Positive Maps ◽  
Weakest Preconditions ◽  
Predicate Transformer ◽  
Usual State ◽  
Stone Type

We develop a notion of predicate transformer and, in particular, the weakest precondition, appropriate for quantum computation. We show that there is a Stone-type duality between the usual state-transformer semantics and the weakest precondition semantics. Rather than trying to reduce quantum computation to probabilistic programming, we develop a notion that is directly taken from concepts used in quantum computation. The proof that weakest preconditions exist for completely positive maps follows immediately from the Kraus representation theorem. As an example, we give the semantics of Selinger's language in terms of our weakest preconditions. We also cover some specific situations and exhibit an interesting link with stabilisers.

MODULES OVER MONOTONE COMPLETE C*-ALGEBRAS

1992 ◽  
Vol 03 (02) ◽  
pp. 185-204 ◽  
Author(s):  
MASAMICHI HAMANA
Keyword(s):  
Completely Positive Maps ◽  
Completely Positive ◽  
C Algebra ◽  
C Algebras ◽  
Positive Maps

The main result asserts that given two monotone complete C*-algebras A and B, B is faithfully represented as a monotone closed C*-subalgebra of the monotone complete C*-algebra End A(X) consisting of all bounded module endomorphisms of some self-dual Hilbert A-module X if and only if there are sufficiently many normal completely positive maps of B into A. The key to the proof is the fact that each pre-Hilbert A-module can be completed uniquely to a self-dual Hilbert A-module.

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