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Solid state realisation of Werner quantum states via Kondo spins

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### Solid state realisation of Werner quantum states via Kondo spins

Ross McKenzie

Sam 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

inequalities

Correlations but no

entanglement

0

5

7

8

<

<

p

p

p

s

s

s

:

:

Werner states

Mixed states of two qubits

In the Bell basis

Reduced density matrix

is probability of a singlet

No entanglement

Bell-CSSH inequalities satisfied

spins A and B

Two impurity

Kondo system

Conduction

electrons C

Model system: two Kondo spins interact with metallic environment via Heisenberg exchange interactionKondo system

Experimental realisation IN. J. Craig et al., Science 304, 565 (2004)

2DEG between spins

in quantum dots

induces an

RKKY interaction

between spins.

Gates vary J

Kondo system

Experimental realisation II- Endohedral fullerenes inside nanotubes

A. Khlobystov et al. Angewandte Chemie International Edition

43, 1386-1389 (2004)

Kondo system

Single impurity Kondo model

Hamiltonian

Conduction electrons

J is the spin exchange coupling

Conduction-electron spin density at impurity site R = 0

Low temperature properties determined by single energy scale

. Kondo temperature

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

Kondo system

Tuneable quantum many-body states:

Kondo effect in quantum dots

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

Conduction

electron spin

Impurity spin

Kondo temperature can be varied

over many orders of magnitude

Two impurity

Kondo system

Two impurity Kondo model

Hamiltonian

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

Ground state determined by competition

between Kondo of single spins and RKKY

Kondo system

Impurity spin A

Spin-rotational invariant!

Conduction

electrons C

Spin singlet

Entanglement in single impurity Kondo model

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

J

S=1/2

Subsystem A

Subsystem B

Total system A+B

Ground state

Reduced density matrix for the impurity

von Neumann entropy

The impurity spin is maximally entangled with the conduction electrons

c.f., Yosida’s variational wavefunction

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

~

S

S

<

>

¢

A

B

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

.

$ \langle \vec{S}_A . \vec{S}_B \rangle $

spins A and B

Two impurity

Kondo system

Conduction

electrons C

Reduced density matrix for the impurities

In the Bell basis

Low temperature behaviour of two impurity Kondo model

[B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988)]

Numerical renormalization group calculation shows that

Left:

the staggered susceptibility and the specific heat coefficients diverge.

The spin-spin correlation is continuously varying and approaches at

the critical value of around the divergence of susceptibility.

Right:

T

2

2

\'

K

:

Unstable fixed point- At the fixed point

[Gan, Ludwig, Affleck, and Jones]

- Thus, for the critical coupling there is no entanglement between two qubits.

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)

Low temperature behaviours of two impurity Kondo model

[B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988)]

Numerical renormalization group calculation shows that

Left:

the staggered susceptibility and the specific heat coefficients diverge.

The spin-spin correlation is continuously varying and approaches at

the critical value of around the divergence of susceptibility.

Right:

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

Renormalization group flows

spins A and B

Two impurity

Kondo system

Impurity spin A

Impurity spin B

Conduction

electrons C

Conduction

electrons C

One impurity

spin A

Three types of entanglements

(i)

and

(ii)

and

(iii)

and

Subsystem A

Subsystem B

singlet state

triplet state

Probabilities for spin singlet/triplet states

for singlet state

for triplet state

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.

at ps=1/2

spin-spin correlation

Kondo system

Impurity spin A

Impurity spin B

Entanglement (ii) between the impurities

(ii)

and

Total system A+B+C

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)]

Critical correlation

Concurrence & Critical Correlation

In terms of the Werner state

Hence, at ps=1/2, there exists a critical value of the spin-spin correlation

separating entangled state from disentangled state.

singlet fidelity

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

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

spins A and B

Two impurity

Kondo system

Conduction

electrons C

Entanglement (iii)

S=1/2

Subsystem A and B

Subsystems C

Total system A+B+C

von Neumann entropy

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