1. Experimental Quantification of Entanglement in low dimensional Spin Systems Chiranjib Mitra IISER-Kolkata Quantum Information Processing and Applications 2015, December 7-13, HRI Allahabad

2. 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

3. 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)

4. 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

5. 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?

6. Copper Nitrate Cu(NO 3 ) 2. 2.5H 2 O Is an Heisenberg antiferromagnet alternate dimer spin chain system with weak inter dimer interaction as compare to intra dimer interaction. J j J >>j Our system 6 J

7. Heisenberg model No Entanglement for Ferromagnetic ground state Energy E Magnetic Field H Singlet (AF )

8. The Hamiltonian for two qubit : (Bipartite systems)

9. In the ground state (at low temperatures) the system is in the pure state and is in the state Maximum Mixing Energy E -3J/4 (singlet) J/4 (triplet) J

10. At finite temperatures the system is in a mixed state

11. At very high temperatures, β 0, the density matrix, Reduces to Panigrahi and Mitra, Jour Indian Institute of Science, 89, 333-350(2009)

12. (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

13. Thermal Entanglement (intermediate temp)

14. 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)

15. B = 0 limit Isotropic system

16. Susceptibility as an Entanglement Witness Wie´sniak M, Vedral V and Brukner C; New J. Phys. 7 258 (2005)

17. D. Das, H. Singh, T. Chakraborty, R. K. Gopal and C. Mitra, NJP 15, 013047 (2013) Entangled Region 17

18. D. Das, H. Singh, T. Chakraborty, R. K. Gopal and C. Mitra, NJP 15, 013047 (2013) 18 Concurrence in Copper Nitrate

19. Theoretical Entanglement Arnesen, Bose, Vedral, PRL 87 017901 (2001)

20. Experimental Entanglement NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra

22. Theoretical Entanglement

23. Entanglement vs Field NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra

24. Heisenberg model No Entanglement for Ferromagnetic ground state Magnetic Field H Energy E Singlet (AF )

25. Quantum Phase Transition H(g) = H 0 + g H 1, where H 0 and H 1 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 = g c 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

26. Level Crossing diverging characteristic length scale ξ Δ ∼ J |g − g c | zν ξ −1 ∼ Λ |g − g c | ν

27. Heisenberg model No Entanglement for Ferromagnetic ground state Magnetic Field H Energy E Singlet (AF )

28. Quantum Information Sharing For Product states

29. (Single Qubit) QP Wiesniak, Vedral and Brukner; New Jour Phys 7, 258(2005)

31. Magnetization NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra

32. Susceptibility

33. Susceptibility as a function of field

34. Q

35. Partial information sharing

37. Theoretical and Experimental P+Q at T=1.8 NJP 15, 013047 (2013); Das, Singh, Chakraborty, Gopal, Mitra

38. Heat Capacity As Entanglement Witness 38 NJP 15, 0 113001 (2013); Singh, Chakraborty, Das, Jeevan, Tokiwa, Gegenwart, Mitra

39. The Hamiltonian is related to heat capacity as Theory The measure of entanglement is represented by Concurrence C 39

40. Experimental (heat capacity) H. Singh, T. Chakraborty, D. Das, H. S. Jeevan, Y. K. Tokiwa, P. Gegenwart and C. Mitra NJP 15, 0 113001 (2013) 40

41. Temperature and Field Dependence 41

42. 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)

43. The Hamiltonians and specific heat are related as Specific Heat as an entanglement witness

44. Experimental (heat capacity)….. 44 U =  dC / dT

45. Theoretical 45

46. Temperature and Field Dependence of Internal energy 46

47. Entanglement vs. Temperature vs. Field 47

48. QPT at 0.8K Specific Heat as a function of field at 0.8 K: QPT

49. Heisenberg model No Entanglement for Ferromagnetic ground state Energy E Magnetic Field H Singlet (AF )

50. QPT at 0.8K Quantum Phase Transition ExperimentTheory NJP 15, 0 113001 (2013); Singh, Chakraborty, Das, Jeevan, Tokiwa, Gegenwart, Mitra

51. Entanglement across QPT T=0.8 K NJP 15, 0 113001 (2013); Singh, Chakraborty, Das, Jeevan, Tokiwa, Gegenwart, Mitra

52. NH 4 CuPO 4 ·H 2 O : Spin Cluster compound Tanmoy Chakraborty, Harkirat Singh, Chiranjib Mitra, JMMM, 396, 247 (2015)

53. Magnetization: 3D plots

54. Quantum Information Sharing Tanmoy Chakraborty, Harkirat Singh, Chiranjib Mitra, JMMM, 396, 247 (2015)

55. Quantum Information Sharing

56. Heat Capacity

57. Heat Capacity: 3D plots

58. Capturing the QPT Tanmoy Chakraborty, Harkirat Singh, Chiranjib Mitra, JMMM, 396, 247 (2015)

59. 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 ENTANGLEMENT Dynamics using Microwaves

60. Nature Physics, 11, 255 (2015)

62. Nature Physics, 11, 212 (2015)

64. Collaborators Tanmoy Chakraborty Harkirat Singh Diptaranjan Das Sourabh Singh Radha Krishna Gopal Philipp Gegenwart