1. 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)

2. Thanks to Discussions with Briggs (RKKY in nanotubes) Doherty and Y.-C. Liang (Werner states) Dawson, Hines, and Milburn (decoherence and entanglement sharing)

3. 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

4. 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.

5. 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.

6. Quantum correlations in different regions of Hilbert space Entangled states No correlations Violate Bell inequalities Correlations but no entanglement

7. Reduced density matrix In the Bell basis Werner states p s is probability of a singlet Mixed states of two qubits No entanglement Bell-CSSH inequalities satisfied p s < 0 : 5 p s < 0 : 78

8. Model system: two Kondo spins interact with metallic environment via Heisenberg exchange interaction Two impurity Kondo system Two impurity spins A and B Conduction electrons C

9. Experimental realisation I Two impurity Kondo system N. J. Craig et al., Science 304, 565 (2004) 2DEG between spins in quantum dots induces an RKKY interaction between spins. Gates vary J

10. Experimental realisation II Endohedral fullerenes inside nanotubes Two impurity Kondo system A. Khlobystov et al. Angewandte Chemie International Edition 43, 1386-1389 (2004)

11. Single impurity Kondo model Hamiltonian Conduction electrons Conduction-electron spin density at impurity site R = 0 J is the spin exchange coupling Low temperature properties determined by single energy scale. Kondo temperature Band width D and the single particle density of state at the Fermi surface Single impurity Kondo system

12. Single impurity Kondo 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 varied over many orders of magnitude

13. Two impurity Kondo model Two impurity Kondo system Hamiltonian To second order J, the indirect RKKY (Ruderman Kittel-Kasuya-Yosida) interaction is RKKY interaction Ground state determined by competition between Kondo of single spins and RKKY

14. c.f., Yosida’s variational wavefunction Entanglement 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 Conduction electrons C Subsystem ASubsystem B Single impurity Kondo system Total system A+B S=1/2 Ground stateSpin singlet Spin-rotational invariant! The impurity spin is maximally entangled with the conduction electrons Reduced density matrix for the impurity von Neumann entropy J

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

16. Reduced density matrix for the impurities Two impurity Kondo system Two impurity spins A and B Conduction electrons C In the Bell basis

17. [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

18. Entanglement & Quantum Phase transition

19. 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 : 2 T K

20. 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?

21. 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)

22. [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

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

24. Three types of entanglements Two impurity spins A and B Conduction electrons C One impurity spin A Conduction electrons C Two impurity Kondo system and (i) (ii) (iii) Subsystem ASubsystem B Impurity spin AImpurity spin B

25. Probabilities for spin singlet/triplet states spin-spin correlation for singlet state for triplet state singlet state 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 S imp =0 and S imp =1. spin-spin correlation at p s =1/2

26. Entanglement (ii) between the impurities Total system A+B+C Two impurity Kondo system and (ii) Impurity spin AImpurity spin B 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)]

27. Concurrence & Critical Correlation In terms of the Werner state Concurrence Hence, at p s =1/2, there exists a critical value of the spin-spin correlation separating entangled state from disentangled state. Critical correlation

28. Comparison 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

29. Entanglement (iii) Subsystem A and BSubsystems C Total system A+B+C S=1/2 von Neumann entropy Two impurity Kondo system Two impurity spins A and B Conduction electrons C