experimental quantification of entanglement and quantum discord in spin chains chiranjib mitra...
TRANSCRIPT
![Page 1: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/1.jpg)
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
![Page 2: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/2.jpg)
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
![Page 3: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/3.jpg)
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)
![Page 4: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/4.jpg)
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
![Page 5: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/5.jpg)
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?
![Page 6: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/6.jpg)
6
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
![Page 7: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/7.jpg)
7
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
![Page 8: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/8.jpg)
Exchange coupled pair model (Dimer)
. . . .
E S MS
H 1 -1
0 1 0
-Ji 0 0
H 1 +1
Ener
gy E
Magnetic Field H
Spin S=1/2
Jisinglet
triplet
H = 2 ΣJi Si•Si+1 + g μBH Σ Si
![Page 9: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/9.jpg)
The Hamiltonian for two qubit : (Bipartite systems)
![Page 10: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/10.jpg)
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
![Page 11: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/11.jpg)
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
![Page 12: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/12.jpg)
At finite temperatures the system is in a mixed state
![Page 13: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/13.jpg)
At very high temperatures, β 0, the density matrix, Reduces to
Panigrahi and Mitra, Jour Indian Institute of Science, 89, 333-350(2009)
![Page 14: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/14.jpg)
(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
![Page 15: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/15.jpg)
Thermal Entanglement (intermediate temp)
![Page 16: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/16.jpg)
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)
![Page 17: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/17.jpg)
B = 0 limit
Isotropic system
![Page 18: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/18.jpg)
Susceptibility as an Entanglement Witness
Wie´sniak M, Vedral V and Brukner C; New J. Phys. 7 258 (2005)
![Page 19: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/19.jpg)
19D. Das, H. Singh, T. Chakraborty, R. K. Gopal and C. Mitra, NJP 15, 013047 (2013)
Entangled Region
![Page 20: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/20.jpg)
20D. Das, H. Singh, T. Chakraborty, R. K. Gopal and C. Mitra, NJP 15, 013047 (2013)
Concurrence in Copper Nitrate
![Page 21: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/21.jpg)
Theoretical Entanglement
Arnesen, Bose, Vedral, PRL 87 017901 (2001)
![Page 22: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/22.jpg)
Experimental Entanglement
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
![Page 23: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/23.jpg)
![Page 24: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/24.jpg)
Theoretical Entanglement
![Page 25: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/25.jpg)
Entanglement vs Field
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
![Page 26: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/26.jpg)
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)
![Page 27: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/27.jpg)
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
![Page 28: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/28.jpg)
Level Crossing
diverging characteristic length scale ξ
Δ J |g − g∼ c|zν
ξ−1 ∼ Λ |g − gc|ν
![Page 29: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/29.jpg)
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)
![Page 30: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/30.jpg)
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
![Page 31: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/31.jpg)
(Single Qubit)
Q P
Wiesniak, Vedral and Brukner; New Jour Phys 7, 258(2005)
![Page 32: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/32.jpg)
![Page 33: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/33.jpg)
Magnetization
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
![Page 34: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/34.jpg)
Susceptibility
![Page 35: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/35.jpg)
Susceptibility as a function of field
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
![Page 36: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/36.jpg)
Q
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
![Page 37: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/37.jpg)
Partial information sharing
Wie´sniak M, Vedral V and Brukner C; New J. Phys. 7 258 (2005)
![Page 38: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/38.jpg)
![Page 39: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/39.jpg)
Theoretical and Experimental P+Q at T=1.8
NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra
![Page 40: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/40.jpg)
40
Heat Capacity As Entanglement Witness
NJP 15, 0 113001 (2013); Singh, Chakraborty, Das, Jeevan, Tokiwa, Gegenwart , Mitra
![Page 41: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/41.jpg)
41
The Hamiltonian is related to heat capacity as
Theory
The measure of entanglement is represented by Concurrence C
![Page 42: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/42.jpg)
42
Experimental (heat capacity)
H. Singh, T. Chakraborty, D. Das, H. S. Jeevan, Y. K. Tokiwa, P. Gegenwart and C. MitraNJP 15, 0 113001 (2013)
![Page 43: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/43.jpg)
43
Temperature and Field Dependence
![Page 44: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/44.jpg)
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
10
0 Oe
C
(Jo
ule/
Mol
e-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)
![Page 45: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/45.jpg)
The Hamiltonians and specific heat are related as
Specific Heat as an entanglement witness
![Page 46: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/46.jpg)
46
Experimental (heat capacity)…..
U = dC / dT
![Page 47: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/47.jpg)
47
Theoretical
![Page 48: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/48.jpg)
48
Temperature and Field Dependence of Internal energy
![Page 49: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/49.jpg)
49
Entanglement vs. Temperature vs. Field
![Page 50: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/50.jpg)
-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 (
Ca
l/Mo
l-K
)
H (Tesla)
T= 0.8K
QPT at 0.8K
Specific Heat as a function of field at 0.8 K: QPT
![Page 51: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/51.jpg)
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)
![Page 52: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/52.jpg)
-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 (
Ca
l/Mo
l-K
)
H (Tesla)
T= 0.8K
QPT at 0.8K
Quantum Phase Transition
Experiment Theory
![Page 53: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/53.jpg)
Entanglement across QPT
T=0.8 K
![Page 54: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/54.jpg)
54
Estimation of Quantum Discord In Copper Nitrate
![Page 55: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/55.jpg)
55
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:
![Page 56: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/56.jpg)
56
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).
![Page 57: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/57.jpg)
57
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 c
c cc c
A K Pal and I Bose J. Phys. B 44 045101 (2011).
![Page 58: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/58.jpg)
58
1 0 0 0
0 1 2 0
0 2 1 0
0 0 0 1
G
G G
G 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,
![Page 59: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/59.jpg)
59
![Page 60: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/60.jpg)
60
![Page 61: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/61.jpg)
61
![Page 62: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/62.jpg)
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.
![Page 63: Experimental Quantification of Entanglement and Quantum Discord in Spin Chains Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications](https://reader038.vdocuments.site/reader038/viewer/2022110101/56649ec45503460f94bcf364/html5/thumbnails/63.jpg)
Collaborators• Tanmoy Chakraborty • Harkirat Singh• Diptaranjan Das• Sourabh Singh• Radha Krishna Gopal• Philipp Gegenwart