solid state realisation of werner quantum states via kondo spins
DESCRIPTION
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) - PowerPoint PPT PresentationTRANSCRIPT
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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)
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Thanks to
Discussions with• Briggs (RKKY in nanotubes)• Doherty and Y.-C. Liang (Werner states)• Dawson, Hines, and Milburn (decoherence
and entanglement sharing)
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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
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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.
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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.
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Quantum correlations in different regions of Hilbert space
Entangled states
No correlations
Violate Bell
inequalitiesCorrelations but no
entanglement
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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
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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
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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
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Experimental realisation II
• Endohedral fullerenes inside nanotubes
Two impurityKondo system
A. Khlobystov et al. Angewandte Chemie International Edition43, 1386-1389 (2004)
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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
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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
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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
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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
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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 >.
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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
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[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
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Entanglement & Quantum Phase transitionEntanglement & Quantum Phase transition
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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
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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?
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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)
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[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
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Unstable fixed pointUnstable fixed point[B. A. Jones and C. M. Varma, Phys. Rev. B 40, 324 (1989)]
Renormalization group flows
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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
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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
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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)]
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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
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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
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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