Frustration of Decoherence Frustration of Decoherence and Entanglement-sharingand Entanglement-sharing

in the Spin-bathin the Spin-bathAndrew Hines

Christopher DawsonRoss McKenzieGerard 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)

SBBS HHHH

Decoherence

00 BSSB

)()log(Tr tttS SSS )(

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,

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.

Central qubit (systems, S), coupled to a bath of N spin-half particles (spin-bath B), each with their own dynamics.

Classical vs. Quantum Correlations

Classical:

Quantum:

Decoherence implies correlation between system and bath, but his doesn’t have to be entanglement.

• could be quantified by the mutual information:

In both cases, the bath state contains information about the state of the system.

Entanglement

Mathematically:Mathematically:

Practically:Practically: cannot be prepared using only LOCC violation of a Bell inequality

Bipartite entanglement measures Bipartite entanglement measures (A-B)(A-B)

Entanglement of formation:

For pairs of qubits: tangletangle

ArtisticallyArtistically

R. BlochAll bipartite states with the same Ef are equivalent up to LOCC

Many-particle Entanglement

For 3 qubits – 2 classes of three-party entanglement

W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 (2000)

• GHZ-class: pure tripartite entanglement

• W-class: bipartite entanglement

There are inequivalentinequivalent classes of entangled states in the multipartite setting:

There can be multiple “units” of entanglement, making a multipartite measure difficult to conceive. Beyond three qubits, the structure is not as simple

Entanglement is monogamous

Unlike classical correlations, entanglement may not be shared arbitrarilyshared arbitrarily amongst many parties.

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

then C cannot become entangled with A, unless some entanglement between A-B is destroyed:

Consider 3 qubits A,B,C, if A-B

Entanglement-sharing Inequalities

3-qubit

example

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 inequalitiesentanglement-sharing inequalities

N-qubits, bipartite entanglement

Coffman, Kundu, Wootters, Phys. Rev. A 61, 052306 (2000)

Maximum pair-wise entanglement in symmetric state

Monogamy of entanglement and other correlations

Koashi and Winter Phys. Rev. A 69, 022309 (2004)

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.

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.

Frustration

Spins with anti-ferromagnetic

coupling

??

High School Chemistry Exams

Distributing entanglement between nearest neighbours around a ring of qubits

Frustration of Decoherence

If a state of the system is evolving under a Hamiltonian such as

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 frustratefrustrate correlations between the central spin and the bath.

SBBS HHHH

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

2

2

12

11

NNN

N

BBB

B

SB

Dawson, Hines, McKenzie, Milburn, Phys Rev. A 71, 052321 (2005)

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.

Zurek Model

N

k

kzzSBH

1

)((0)

2

1 g

(0)2

1(k)

BetBtBN

k zti /g

tBtBtSSSB

)()()( tBtBtr

222)(14)( trtSB

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].

Frustration of Decoherence

Tessieri-Wilkie ModelFrustration of Decoherence

,

L. Tessieri and J. Wilkie, J. Phys. A 36, 12305 (2003).

Tessieri-Wilkie ModelFrustration of Decoherence

Rabi Oscillations

Tessieri-Wilkie ModelFrustration of Decoherence

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