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Frustration of Decoherence
and Entanglement-sharing
in the Spin-bath
Andrew Hines
Christopher Dawson
Ross McKenzie
Gerard Milburn
The short version
•
Motivated by paper by Tessieri and Wilkie (J. Phys. A 36,
12305 (2003)), who considered a simple model of a qubit
interacting with a spin-bath, with intra-bath interactions.
They found a suppression of the decoherence resulting
from the intra-bath interactions.
•
In a quantum system, interactions lead to entanglement.
We were interested in entanglement and quantum phase
transitions, so we were interested in entanglement in
many-body systems.
•
In the multi-partite setting, there are bounds on how
entanglement can be distributed amongst subsystems.
•
A quantum bath can be entangled.
•
Entanglement between system and bath decoheres the
system, but entanglement between bath spins can limit
system-bath entanglement.
•
Is the suppression of decoherence a manifestation of
entanglement-sharing?
(Stupid Hat Day 2004)
Decoherence
Central qubit (systems, S), coupled to a bath of N spinhalf particles (spin-bath B), each with their own dynamics.
H  H S  H B  H SB
Initially at t = 0 we take the central spin S to be in a pure state, uncorrelated
with the bath, for some initial state of the bath, usually thermal state, or for low
temperatures, the ground state.
SB 0  
S
   B 0
As the system evolves under H, the central spin becomes correlated with the bath
so at later times is no longer pure. The central spin is said to have decohered, and
the amount of decoherence is typically quantified by the von Neumann entropy of
its reduced density matrix,
S  S ( t )  Tr S (t )log  S (t ) 
Classical vs. Quantum Correlations
Decoherence implies correlation between system and bath,
but his doesn’t have to be entanglement.
Classical:
• could be quantified by the mutual information:
Quantum:
In both cases, the bath state contains information about the
state of the system.
Entanglement
Mathematically:
Practically:
cannot be prepared using only LOCC
 violation of a Bell inequality

Bipartite entanglement measures (A-B)
Artistically
Entanglement of formation:
For pairs of qubits: tangle
All bipartite states with the same Ef are equivalent up to LOCC
R. Bloch
Many-particle Entanglement
There are inequivalent classes of entangled states in the multipartite
setting:
For 3 qubits – 2 classes of three-party entanglement
• GHZ-class: pure tripartite entanglement
• W-class: bipartite entanglement
There can be multiple “units” of entanglement, making a multipartite
measure difficult to conceive. Beyond three qubits, the structure is not as
simple
W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 (2000)
Entanglement is monogamous
Unlike classical correlations, entanglement may not be
shared arbitrarily amongst many parties.
Consider 3 qubits A,B,C, if A-B
then C cannot become entangled with A, unless some entanglement between
A-B is destroyed:
Classical correlations: Consider a collection of random variables
Mutual information between different variables is not bounded.
The correlations between X and Y do not affect correlations
between X and Z
Entanglement-sharing Inequalities
Monogamy places bounds on the amount of entanglement that
may be shared amongst parties in a multipartite setting. These
are quantified in terms of entanglement-sharing inequalities
3-qubit
example
N-qubits, bipartite entanglement
Maximum pair-wise entanglement
in symmetric state
Coffman, Kundu, Wootters, Phys. Rev. A 61, 052306 (2000)
Monogamy of entanglement and other
correlations
Koashi and Winter Phys. Rev. A 69, 022309 (2004)
A perfect classical correlation between A and B will forbid system A
from being entangled to other systems:
The reduced density operator for A must be a pure state. Though this
does NOT restrict classical correlations between A and another party.
Koashi and Winter quantify this trade-off between correlations, using an
operational definition of the classical correlations in a bipartite quantum
state (one-way distillable common randomness).
This describes how it is possible for entanglement to bound potential
classical correlations.
Frustration
Distributing
entanglement
between nearest
neighbours around a
ring of qubits
??
Spins with antiferromagnetic
coupling
High School
Chemistry
Exams
Frustration of Decoherence
If a state of the system is evolving under a Hamiltonian such as
H  H S  H B  H SB
and the bath, initialised in some entangled state, maintains
appreciable entanglement over the evolution, then it follows there is a
restriction on the entanglement between the `central spin' S and the
bath.
For pure states this equivalent to a restriction on the amount that S
may decohere. For mixed states we must also bound the classical
correlations which may be done using the result of Koashi and Winter.
Entanglement between bath spins can frustrate correlations between
the central spin and the bath.
Entanglement-Sharing Inequality
For spin-baths of N particles,
, the situation is more
complicated due to the many different types of entanglement which exist in
these baths, and the absence of good entanglement measures for them.
To overcome this difficulty we will assume the Hamiltonians HB and HSB are
symmetric about permutations of spins.
Here the pair-wise entanglement between any two bath spins is the same,
for all i,j , allowing us to quantify the intra-bath entanglement by a single
parameter.
Our aim is to show how this
constrains the system-bath tangle
.
Entanglement-Sharing Inequality
Use the symmetry in the system, and the characteristics of W-class states to
obtain an entanglement-sharing inequality for a single qubit, with a
permutation symmetric bath of N-qubits.
 SB

1


 N 2  B  N B



1
B  2
N
1
B  2
N
Dawson, Hines, McKenzie, Milburn, Phys Rev. A 71, 052321 (2005)
Frustration of Decoherence
Zurek Model
One simple model of decoherence where the inequality is immediately applicable is
an exactly solvable model introduced by Zurek [8] recently used to investigate the
structure of the decoherence induced by spin environments [9].
r (t )  B (t ) B (t )
H SB
2
1
(0) ( k )
 g  z  z
2 k 1
t  SB   
S
B t    
S
B t 
 (k) / 2

k 1 z
B t   B  t   e
B(0)
igt
N

 SB (t )  4   1  r (t )
N
2
2

Frustration of Decoherence
Tessieri-Wilkie Model
,
L. Tessieri and J. Wilkie, J. Phys. A 36, 12305 (2003).
Frustration of Decoherence
Tessieri-Wilkie Model
Frustration of Decoherence
Tessieri-Wilkie Model
Rabi Oscillations
Entanglement and Chaotic
Environments
Results to date suggest that
chaotic dynamics enhance the
generation of entanglement, and
possibly correlations in general.
Two possible viewpoints for decoherence
1. If chaos is introduced by the
interaction between system and
environment, then this will increase
the generation of entanglement,
leading to greater decoherence
2. Chaos within the bath degrees of
freedom will generate intra-bath
entanglement, which can act to
frustrate decoherence.
Thank you