Nageswaran Rajendran and Dieter Suter, Institutes for Physics, University of Dortmund, 44221 Dortmund, Germany

Geometric Phases for Mixed StatesGeometric Phases for Mixed States

Introduction

• Birth of Geometric Phases

The idea of geometric phase was firstly demonstrated by Pancharatnam, in his studies on intereference of nonorthogonal polarized lightsThe quantal counterpart of this phase was studied by Berry, for pure quantal states which undergoes adiabatic and cyclic paths in the parametric space.

• Geometric Phase, of a quantum system

When a quantum system undergoes an evolution, the system acquires an extra phase along with the dynamical phases

Reason for this geometric phase arise from the geometry of the path, taken by the system.

ApplicationTo build geometric quantum gates, which lead to error-free quantum computation

1 Uhlmann’s Phase 2

Uhlmann was probably the first to address the issue of mixed state holonomy, but it was studiedas a pure mathematical problem.

Purification, to get a Hilbert space from the density operators.

This Hilbert space is isomorphic to the Hilbert space of the Hilbert-Schmidt operators.

Hilbert – Schmidt operators

Density Operator, in terms of Hilbert-Schmidt operator

Where, U is an arbitrary Unitary operator, can be considered as a phase factor.

The inner product between two Hilbert-Schmidt operator

which gives the geometric phases -Uhlmann‘s Phase - acquired by the density operators

Where, Geometric Phases

Ordered set of Hilbert-Schmidt Operators

Shi and Du Approach 4

In [], Shi and Du have given a general description of geometric phases for mixed states.They give an algorithm, to fiddle the mixed states!

1. Considering a mixed state

2. Dynamics of the mixed state

3. Adding Ancilla to the system and Purifying the s tate initial mixed state

Where bases for the ancilla

4. Force it to undergo a bilocal Unitrary transformation

K = time independent Hamiltonian

Which is the sum of the nonorthognal pure state geometric phases

This result is generalisation of Sjöqvist et al., phase and Uhlmann phase.

:

and

5. Now, make the system to have non-orthogonal pure state mixtures. 6. Transform the basis of system, to remove the time dependency of probability

factors in the mixed states.7. Finally, we end up with the geometric phases of mixed state

(with the combination of non-orthogonal pure state), evolution.

Where,

Sjöqvist, et al., Phase 3

The geometric phase for the mixed states is defined as:

Physical version of Uhlmann phase was studied by Sjöqvist, Pati, Ekert, Anandan, Ericsson, Oi and Vedral in [1]. They used modified Mach-Zehnder experiment.

VisibilityGeometric

Phase

The intensity of the intereference pattern at the dector is given by (for an input of 000 )

The density matrices are mixtures of pure states, with probabilities

Interference oscillations produced by the phase shifter, is shifted by the geometric phase

Interference profile of k th pure state can be expressed asConclusion

The Geometric phases acquired by the mixed quantum system is yet to be realized. Though, Shi and Du‘s approach seems to be solution, still, the bases transformation is not very clear, as it is discussed by them. But further research in this regard would lead us to achieve the error-free quantum processors.

References: 1.Sjöqvist, et al., Phys.Rev.Lett., vol..85, 2845 (2000)

2. Ericsson, et al, Phys. Rev. Lett. vol .91, 090405 (2003)

3.Minguin Shi and Jiangfeng Du, quant-ph/0501006. Acknowledgement

Nageswaran Rajendran acknowledges the GK-726, Materials and Concepts for Quantum information processing.