Experimental Quantification of Entanglement and Quantum Discord in Spin Chains
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Transcript of Experimental Quantification of Entanglement and Quantum Discord in Spin Chains
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Experimental Quantification of Entanglement and Quantum Discord in Spin Chains
Chiranjib MitraIISER-Kolkata
Quantum Information Processing and Applications 2013, December 2-8, HRI Allahabad
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
NJP 15, 0 113001 (2013); Singh, Chakraborty, Das, Jeevan, Tokiwa, Gegenwart , Mitra
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Plan of the Talk• Introduction to Quantum Spin systems and spin qubits• Entanglement in spin chains• Detailed analysis to extract Entanglement from the data – Magnetic
susceptibility as an Entanglement witness• Variation of Entanglement with Magnetic Field• Quantum Information Sharing through complementary observables• Quantum Phase Transitions in spin systems• Specific Heat as an entanglement witness.• Measurement of specific heat close to quantum criticality• Quantum Discord in Heisenberg Spin chains• Quantum Discord through Susceptibility and Specific heat
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Quantum Magnetic Systems
• Low Spin systems (discrete)• Low Dimensional Systems
– Spin Chains– Spin Ladders
• Models – Ising Model (Classical)– Transverse Ising Model (Quantum)– Heisenberg Model (Quantum)
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The ‘DiVincenzo Checklist’
• Must be able to• Characterise well-defined set of quantum states to
use as qubits• Prepare suitable states within this set• Carry out desired quantum evolution (i.e. the
computation)• Avoid decoherence for long enough to compute• Read out the results
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Natural entanglement
• Entanglement that is present ‘naturally’ in easily accessible states of certain systems (for example, in ground states or in thermal equilibrium)
• Natural questions to ask:– How much is there? Can we quantify it?– How is it distributed in space?– Can we use it for anything?
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Experimental facilitiesMagnetic Property Measurement System(MPMS)1.8K- 350K temperature range0T – 7T magnetic field range Cryogen free system- Resistivity, Heat capacity
Physical Property Measurement System(PPMS).400mK- 300K temperature range0T – 9T magnetic field range
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Copper Nitrate Cu(NO3)2 . 2.5H2O
Is an Heisenberg antiferromagnet alternating dimer spin chain system with weak inter dimer interaction as compare to intra dimer interaction.
J
j J >>j
Macroscopic Quantum Entanglement
J
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Exchange coupled pair model (Dimer)
. . . .
E S MS
H 1 -1
0 1 0
-Ji 0 0H 1 +1
Ener
gy E
Magnetic Field H
Spin S=1/2
Jisinglet
triplet
H = 2 ΣJi Si•Si+1 + g μBH Σ Si
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The Hamiltonian for two qubit : (Bipartite systems)
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Heisenberg model
1i i ii
H B J
No Entanglement for Ferromagnetic ground state
Ener
gy E
Magnetic Field H
2
2
1 ( )2
Singlet (AF)
Jisinglet
triplet
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In the ground state (at low temperatures) the system is in the pure state and is in the state
Maximum Mixing
Ener
gy E
-3J/4 (singlet)
J/4 (triplet)J
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At finite temperatures the system is in a mixed state
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At very high temperatures, β 0, the density matrix, Reduces to
Panigrahi and Mitra, Jour Indian Institute of Science, 89, 333-350(2009)
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(goes to zero, since the Pauli matrices are traceless)
Hence the system is perfectly separable
ρ is separable if it can be expressed as a convex sum of tensor product states of the two subsystems
There exists
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Thermal Entanglement (intermediate temp)
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Concurrence
In Ferromagnet it is zero
For an Antiferromagnet
[W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)]
O’Connor and Wootters, Phys. Rev. A, 63, 052302 (2001)
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B = 0 limit
Isotropic system
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Susceptibility as an Entanglement Witness
Wie´sniak M, Vedral V and Brukner C; New J. Phys. 7 258 (2005)
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19D. Das, H. Singh, T. Chakraborty, R. K. Gopal and C. Mitra, NJP 15, 013047 (2013)
Entangled Region
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20D. Das, H. Singh, T. Chakraborty, R. K. Gopal and C. Mitra, NJP 15, 013047 (2013)
Concurrence in Copper Nitrate
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Theoretical Entanglement
Arnesen, Bose, Vedral, PRL 87 017901 (2001)
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Experimental Entanglement
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
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Theoretical Entanglement
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Entanglement vs Field
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
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Heisenberg model
1i i ii
H B J
No Entanglement for Ferromagnetic ground state
Magnetic Field H
Ener
gy E
2
2
1 ( )2
Singlet (AF)
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Quantum Phase Transition
• H(g) = H0 + g H1, where H0 and H1 commute • Eigenfunctions are independent of g even though the
Eigenvalues vary with g
• level-crossing where an excited level becomes the ground state at g = gc
• Creating a point of non-analyticity of the ground state energy as a function of g
Subir Sachdev, Quantum Phase Transisions, Cambridge Univ Press, 2000
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Level Crossing
diverging characteristic length scale ξ
Δ J |g − g∼ c|zν
ξ−1 ∼ Λ |g − gc|ν
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Heisenberg model
1i i ii
H B J
No Entanglement for Ferromagnetic ground state
Magnetic Field H
Ener
gy E
2
2
1 ( )2
Singlet (AF)
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Quantum Information SharingFor Product states
Wie´sniak M, Vedral V and Brukner C; New J. Phys. 7 258 (2005)
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
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(Single Qubit)
Q P
Wiesniak, Vedral and Brukner; New Jour Phys 7, 258(2005)
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Magnetization
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
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Susceptibility
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Susceptibility as a function of field
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
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Q
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
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Partial information sharing
Wie´sniak M, Vedral V and Brukner C; New J. Phys. 7 258 (2005)
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Theoretical and Experimental P+Q at T=1.8
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
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Heat Capacity As Entanglement Witness
NJP 15, 0 113001 (2013); Singh, Chakraborty, Das, Jeevan, Tokiwa, Gegenwart , Mitra
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The Hamiltonian is related to heat capacity as
Theory
The measure of entanglement is represented by Concurrence C
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Experimental (heat capacity)
H. Singh, T. Chakraborty, D. Das, H. S. Jeevan, Y. K. Tokiwa, P. Gegenwart and C. MitraNJP 15, 0 113001 (2013)
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Temperature and Field Dependence
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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
0
2
4
6
8
100 Oe
C
(Jou
le/M
ole-
K)
T (K)
separable region
Entangled Regime
Specific Heat as an entanglement witness
Wie´s niak M, Vedral V and Brukner C; Phys.Rev.B 78,064108 (2008)
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The Hamiltonians and specific heat are related as
Specific Heat as an entanglement witness
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Experimental (heat capacity)…..
U = dC / dT
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Theoretical
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Temperature and Field Dependence of Internal energy
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Entanglement vs. Temperature vs. Field
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-1 0 1 2 3 4 5 6 7 8
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35C
V (C
al/M
ol-K
)
H (Tesla)
T= 0.8K
QPT at 0.8K
Specific Heat as a function of field at 0.8 K: QPT
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Heisenberg model
1i i ii
H B J
No Entanglement for Ferromagnetic ground state
Ener
gy E
Magnetic Field H
2
2
1 ( )2
Singlet (AF)
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-1 0 1 2 3 4 5 6 7 8
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
CV (C
al/M
ol-K
)
H (Tesla)
T= 0.8K
QPT at 0.8K
Quantum Phase Transition
Experiment Theory
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Entanglement across QPT
T=0.8 K
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Estimation of Quantum Discord In Copper Nitrate
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Conditional Entropy : Information about A for a given B (when Y is known or H(Y) is known)
Mutual information : correlation between two random variables(common information of X and Y)
Classical version of mutual information
Quantum Discord
Classically:
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Shanon entropy von Neumann entropy
Conditional Entropy
Mutual information
Classical information
Quantum Discord
Ref: H. Olliver and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001).
Quantum Versions:
Ref: A. Datta and A. Shaji, C. Caves Phys. Rev. Lett. 100 , 050502 (2008).
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The two particle density matrix is of the form
The eigenvalues of are xy
classical correlation given by
Mutual information takes the form
Eventually, the mathematical expression for QD can be written as
1 2 3 2 1 2 3 1 2 3 2 1 2 3
1 2 3 2 1 2 3 1 2 3 2 1 2 3
2 2
1) ( ) ( ) [(1 ) log (1 ) (1 ) log (1 )4
(1 ) log (1 ) (1 ) log (1 )(1 ) (1 )log (1 ) log (1 )]
2 2
QD( YX XY XYI Q c c c c c c c c c c c c
c c c c c c c c c c c cc cc c
A K Pal and I Bose J. Phys. B 44 045101 (2011).
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1 0 0 00 1 2 0
0 2 1 00 0 0 1
GG GG G
G
1 2 1 2 1 2 z z y y x xG
Where G is defined as the two-site spin-spin correlation function given by,
The analytical relations of quantum mutual information (I) and total classical correlation (C) with G can be written as
Hence, with the given value of G, one can easily obtain the amount of QD present in a certain system at finite temperature using the relation
S Luo 2008 Phys. Rev. A 77 042303.M A Yurischev Phys. Rev. B 84 024418 (2011).
considering the standard basis of a two qubit system {|00>, |01>, |10>, |11>}, the density matrix can be defined as,
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Conclusion and future directions• AF Ground state of a quantum mechanical spin system is
entangled• Magnetic susceptibility can be used as a macroscopic
entangled witness• Using quantum mechanical uncertainty principle for
macroscopic observables, it is possible to throw light on quantum correlations close to QPT.
• Specific heat measurements at low temperatures explicitly capture the QPT.
• Specific Heat is an Entanglement Witness• Quantum Discord can be quantified using magnetic
susceptibility and heat capacity data.• Quantum Discord to be used to capture QPT in spin systems.
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Collaborators• Tanmoy Chakraborty • Harkirat Singh• Diptaranjan Das• Sourabh Singh• Radha Krishna Gopal• Philipp Gegenwart