Solid state realisation of Werner quantum states via

Solid state realisation of Werner quantum states via Kondo spins Ross Mc. Kenzie 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 macrorealism (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 No correlations Violate Bell inequalities Correlations but no entanglement Entangled states

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

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

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

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

Single impurity Kondo model Single impurity Kondo system 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

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 Single impurity Kondo system Kondo temperature can be varied over many orders of magnitude

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

Entanglement in single impurity Kondo model [T. A. Costi and R. H. Mc. Kenzie, Phys. Rev. A 68, 034301 (2003)] J S=1/2 Single impurity Kondo system Impurity spin A Subsystem A Conduction electrons C Subsystem B Total system A+B Ground state Spin singlet Spin-rotational invariant! 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)]

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 impurities Two impurity Kondo system In the Bell basis Two impurity spins A and B Conduction electrons C

Low temperature behaviour of two impurity Kondo model [B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988)] Left: Numerical renormalization group calculation shows that the staggered susceptibility and the specific heat coefficients diverge. Non Fermi-liquid behaviour Right: The spin-spin correlation is continuously varying and approaches at the critical value of around the divergence of susceptibility.

Entanglement & Quantum Phase transition

Unstable fixed point • At the fixed point I ' 2: 2 TK [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)] Left: Numerical renormalization group calculation shows that the staggered susceptibility and the specific heat coefficients diverge. Non Fermi-liquid behaviour Right: The spin-spin correlation is continuously varying and approaches at the critical value of around the divergence of susceptibility.

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

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

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 Simp=0 and Simp=1. spin-spin correlation at ps=1/2

Entanglement (ii) between the impurities (ii) and Two impurity Kondo system Impurity spin A Impurity spin B 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)]

Concurrence & 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 criteria 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)

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