Transcript Geometric Phase, of a quantum system

Geometric Phases for Mixed States
Nageswaran Rajendran and Dieter Suter, Institutes for Physics, University of Dortmund,
44221 Dortmund, Germany
Introduction
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Uhlmann was probably the first to address the issue of mixed state holonomy, but it was studied
as a pure mathematical problem.
Birth of Geometric Phases
The idea of geometric phase was firstly demonstrated by Pancharatnam, in his studies on
Purification, to get a Hilbert space from the density operators.
intereference of nonorthogonal polarized lights
The quantal counterpart of this phase was studied by Berry, for pure quantal states which
undergoes adiabatic and cyclic paths in the parametric space.
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Uhlmann’s Phase
This Hilbert space is isomorphic to the Hilbert space of the Hilbert-Schmidt operators.
Geometric Phase, of a quantum system
Hilbert – Schmidt operators
When a quantum system undergoes an evolution, the system acquires an extra phase along
Where, U is an arbitrary Unitary operator, can be considered as a phase factor.
with the dynamical phases
Reason for this geometric phase arise from the geometry of the path, taken by the system.
Density Operator, in terms of Hilbert-Schmidt operator
The inner product between two Hilbert-Schmidt operator
Application
To build geometric quantum gates, which lead to error-free quantum computation
which gives the geometric phases -Uhlmann‘s Phase - acquired by the density operators
Shi and Du Approach
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Geometric Phases
Where,
Ordered set of Hilbert-Schmidt Operators
In [], Shi and Du have given a general description of geometric phases for mixed states.
They give an algorithm, to fiddle the mixed states!
Sjöqvist, et al., Phase
1. Considering a mixed state
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2. Dynamics of the mixed state
Physical version of Uhlmann phase was studied by Sjöqvist, Pati, Ekert, Anandan, Ericsson,
Oi and Vedral in [1]. They used modified Mach-Zehnder experiment.
3. Adding Ancilla to the system and Purifying the s tate initial mixed state
bases for the ancilla
Where
4. Force it to undergo a bilocal Unitrary transformation
:
and
Where,
K = time independent Hamiltonian
The intensity of the intereference pattern at the dector is given by (for an input of 0 0   0 )
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.
Geometric
Phase
Visibility
Which is the sum of the nonorthognal pure state geometric phases
The geometric phase for the mixed states is defined as:
This result is generalisation of Sjöqvist et al., phase and Uhlmann phase.
Interference oscillations produced by the phase shifter, is shifted by f, the geometric phase
Conclusion
Interference profile of k th pure state can be expressed as
The density matrices are mixtures of pure states, with probabilities
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.