Solid state realisation of Werner quantum states via Kondo spins

Ross McKenzieSam Young Cho

Reference: S.Y. Cho and R.H.M, Phys. Rev. A 73, 012109 (2006)

Thanks to

Discussions with• Briggs (RKKY in nanotubes)• Doherty and Y.-C. Liang (Werner states)• Dawson, Hines, and Milburn (decoherence

and entanglement sharing)

Big goals for quantum nano-science

• Create and manipulate entangled quantum states in solid state devices

• Understand the quantum-classical boundary, e.g., test quantum mechanics versus macro-realism (Leggett)

• Understand the competition between entanglement and decoherence

Entanglement vs. decoherence

• Interaction of a qubit with its environment leads to decoherence and entanglement of qubit with environment.

• Interactions between qubits entangles them with one another.

• We will also see that the environment can entangle the qubits with one another.

Outline

• Classical correlations vs. entanglement vs. violation of Bell inequalities (Werner states)

• Experimental realisations of two impurity Kondo model

• Competition between Kondo effect and RKKY interaction

• Entanglement between the two Kondo spins• How to create Werner states in the solid state.

Quantum correlations in different regions of Hilbert space

Entangled states

No correlations

Violate Bell

inequalitiesCorrelations but no

entanglement

Reduced density matrix Reduced density matrix

In the Bell basis

Werner states

ps is probability of a singlet

Mixed states of two qubits

No entanglementBell-CSSH inequalities satisfied

ps<0:5ps<0:78

Model system: two Kondo spins interact with metallic environment

via Heisenberg exchange interaction

Two impurityKondo system

Two impurity spins A and B

Conduction electrons C

AS

BS

AB

Experimental realisation ITwo impurityKondo system

N. J. Craig et al., Science 304, 565 (2004)

2DEG between spinsin quantum dots induces an RKKY interactionbetween spins.Gates vary J

Experimental realisation II

• Endohedral fullerenes inside nanotubes

Two impurityKondo system

A. Khlobystov et al. Angewandte Chemie International Edition43, 1386-1389 (2004)

Single impurity Kondo modelSingle impurity Kondo model

ACC SsJHH

)0( Hamiltonian

Conduction electrons

Conduction-electron spin density at impurity site R = 0J is the spin exchange coupling

Low temperature properties determined by single energy scale. Kondo temperature ]/1exp[ FFK JJDT

Band width D and the single particle density of state at the Fermi surfaceF

Single impurityKondo system

Single impurityKondo system

For a review, L. Kouwenhoven and L. Glazman, Physics World 14, 33 (2001)

Conduction electron spin

Impurity spin

Tuneable quantum many-body states: Kondo effect in quantum dots

Kondo temperature can be variedover many orders of magnitude

Two impurity Kondo modelTwo impurity Kondo modelTwo impurityKondo system

Hamiltonian

To second order J, the indirect RKKY (Ruderman Kittel-Kasuya-Yosida) interaction is

R

I RKKY interaction

21)( SSRIHRKKY

Ground state determined by competitionbetween Kondo of single spins and RKKY

221

,,

IrIzyx

A

c.f., Yosida’s variational wavefunction ACACG 2

1

Entanglement in single impurity Kondo modelEntanglement in single impurity Kondo model

[K. Yosida, Phys. Rev. 147, 233 (1966)]

[T. A. Costi and R. H. McKenzie, Phys. Rev. A 68, 034301 (2003)]

Impurity spin A

AS

Conduction electrons C

Subsystem A Subsystem B

Single impurityKondo system

AS

AB Tr BA TrTotal system A+B

1logTr)( 2 AAAE

S=1/2

Ground state Spin singlet

0 r

Spin-rotational invariant!

The impurity spin is maximally entangled with the conduction electrons

Reduced density matrix for the impurity

von Neumann entropy

J

Entanglement between the two Kondo spins

• Given by concurrence of the reduced density matrix for the two localised spins (Wootters)

• Ground state is a total spin singlet (S=0) and thus invariant under global spin rotations

• Entanglement is determined by < ~S A ¢ ~S B >.

Reduced density matrix for the impuritiesReduced density matrix for the impurities

Two impurityKondo system

Two impurity spins A and B

Conduction electrons C

AS

BS

AB

In the Bell basis

[B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988)]Low temperature behaviour of two impurity Kondo model

the staggered susceptibility and the specific heat coefficients diverge. Numerical renormalization group calculation shows that

The spin-spin correlation is continuously varying and approaches at the critical value of around the divergence of susceptibility.

Left:

Right:

Non Fermi-liquid behaviour

Entanglement & Quantum Phase transitionEntanglement & Quantum Phase transition

Unstable fixed point

• At the fixed point

[Gan, Ludwig, Affleck, and Jones]• Thus, for the critical coupling there is no

entanglement between two qubits.

I ' 2:2TK

Questions for future• Can the competition between Kondo and

RKKY be better understood in terms of entanglement sharing?

• Why does the entanglement between Kondo spins vanish at the quantum critical point?

• What effect does temperature have?

Conclusions• Two spin Kondo model provides a model system

to study competition between entanglement of two qubits with each other and entanglement of each qubit with environment

• Entanglement between the two Kondo spins vanishes at the unstable fixed point.

• Varying system parameters will produce all the Werner states

S.Y. Cho and RHM, Phys. Rev. A 73, 012109 (2006)

[B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988)]Low temperature behaviours of two impurity Kondo model

the staggered susceptibility and the specific heat coefficients diverge. Numerical renormalization group calculation shows that

The spin-spin correlation is continuously varying and approaches at the critical value of around the divergence of susceptibility.

Left:

Right:

Non Fermi-liquid behaviour

Unstable fixed pointUnstable fixed point[B. A. Jones and C. M. Varma, Phys. Rev. B 40, 324 (1989)]

Renormalization group flows

Three types of entanglementsThree types of entanglements

Two impurity spins A and B

AS

BS

Conduction electrons C

One impurity spin A

AS

Conduction electrons C

BS

Two impurityKondo system

and

and

and

(i)

(ii)

(iii)

Subsystem A Subsystem B

Impurity spin A

AS

Impurity spin B

BS

Probabilities for spin singlet/triplet statesProbabilities for spin singlet/triplet states

13)()( tS ppTPSP

41

43

BA SS

spin-spin correlation

for singlet stateSpSP )(

tpTP 3)( for triplet state

singlet state

0impS triplet state

1impS

For P(S)=P(T)=1/2, the state for the two spins can be regarded as an equal admixture of the total spin of impurities Simp=0 and Simp=1.

41

BA SS

spin-spin correlation at ps=1/2

Entanglement (ii) between the impuritiesEntanglement (ii) between the impurities

CAB TrTotal system A+B+C

Two impurityKondo system

and(ii)

Impurity spin A

AS

Impurity spin B

BS

Although the total system is in a pure state, the two impurity spins are in a mixed state.

Need to calculate the concurrence as a measure of entanglement

[W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)]

Concurrence & Critical CorrelationConcurrence & Critical Correlation

In terms of the Werner state

Concurrence

Hence, at ps=1/2, there exists a critical value of the spin-spin correlation separating entangled state from disentangled state.

Critical correlation

Comparison of criteriaComparison of criteria

[42] R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 200, 340 (1995)

[48] S. Popescu, Phys. Rev. Lett. 72, 797 (1994)

singlet fidelity

Entanglement (iii)Entanglement (iii)

Subsystem A and B Subsystems C

CAB TrTotal system A+B+C

31log)1(log)( 22

SSSSAB

ppppE

S=1/2

von Neumann entropy

Two impurityKondo system

Two impurity spins A and B

AS

BS

Conduction electrons C