A. R. Usha Devi, Sudha and B. G. Divyamani

1. Local Invariant Spin Squeezing Criteria for multiqubit states A. R. Usha Devi, Sudha and B. G. Divyamani Department of Physics, Bangalore University, Bangalore Department of Physics, Kuvempu University, Shankaraghatta Inspire Institute, Alexandria, VA, USA Tunga Mahavidyalaya, Thirthahalli 4th December 2013, HRI, Allahabad

2. Outline of the talk • Introduction: Spin Squeezing • Exchange Symmetry, Local Invariance and Spin Squeezing: The Need for Local Invariant Spin Squeezing Criteria (LISS): LISS for symmetric and non-symmetric multiqubit states • Relationship between LISS and Quantum Entanglement • Connection between local invariants and LISS in symmetric • multiqubit states • Conclusion and Future directions

3. Introduction Recent years have witnessed revolutionary improvement in the production, manipulation, characterization and quantification of multiqubit states because of their promising applications in high precision atomic clocks, atomic interferrometry, quantum metrology and quantum information protocols. Spin Squeezing and Quantum Entanglement are two important concepts in the characterization and quantification of non-classical atomic correlations.

4. Well before the focus on quantifying entanglement begun, the notion of Spin Squeezing had caught much attention both in the context of low-noise spectroscopy as well as in high precision interferrometry. While entanglement has its root in superposition principle, squeezing originates from uncertainty principle. D.J. Wineland et.al., Phys. Rev. A 50, 67 (1994)

5. Spin Squeezing Criteria Several definitions of spin squeezing have been proposed in the literature. In particular, Kitagawa and Ueda developed a spin squeezing criteria based on the uncertainty relation between collective angular momentum components of a multiqubit state K. Wodkiewicz and J. H. Eberly, J. Opt. Soc. Am. B. 2, 458 (1985) M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993) D. J. Winelandet. al, Phys. Rev. A50, 67 (1994) A. Sorenson et.al., Nature (London), 409, 63 (2001)

6. Spin Squeezing--Basic definition

7. Schematic representation of coherent and spin squeezed states Coherent state Spin Squeezed state

8. Kitagawa Ueda spin squeezing criteria Kitagawa and Ueda also made an important observation that identifying a mean spin direction of the multiqubit state is essential for an unambiguous determination of spin squeezing in multiqubit states. M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993)

9. Kitegawa-Ueda Spin Squeezing Criteria

10. Kitegawa-Ueda Spin Squeezing Criteria

11. Wineland Squeezing Criteria All Wineland Squeezed states are Kitegawa-Ueda squeezed There exist Kitagawa-Ueda squeezed states that are not Wineland squeezed

12. Exchange Symmetry, Local Invariance and Spin Squeezing Spin squeezing, in the original sense, is defined for multiqubit states that are invariant under the exchange of particles. Such states, the so-called symmetric states belong to the maximal multiplicity subspace of the collective angular momentum operator. The possibility of extending the concept of spin squeezing to multi-qubit systems that are not necessarily symmetric under interchange of particles and that are accessible not just to .

13. collective operations but also to local operations was explored by Usha Devi et. al. This requires a criterion for spin squeezing that exhibits invariance under local unitary operations on the qubits. It is important to notice that both the spin squeezing parameters are not invariant under arbitrary local unitary transformations on the qubits. . A.R. Usha Devi, X. Wang and B.C. Sanders, Quant. Infn. Proc. 2, 207 (2003)

14. Exchange Symmetry, Local Invariance and Spin Squeezing .

15. Exchange Symmetry, Local Invariance and Spin Squeezing .

16. From these two examples, we can conclude the following: • Both the criteria for spin squeezing do not exhibit local unitary invariance. This aspect makes them inconvenient to relate with the local unitary invariant entanglement criterion. • Both the criteria are defined for symmetric states alone and using them to assess the spin squeezing in non-symmetric states may lead to unphysical results. • Thus there arises a definite need for defining spin squeezing criteria that remains local unitary invariant and applicable for non-symmetric states also. We accomplish this through the Local Invariant Spin Squeezing criteria (LISS). Exchange Symmetry, Local Invariance and Spin Squeezing

17. Local Invariant Spin Squeezing criteria A.R. Usha Devi, X. Wang and B.C. Sanders, Quant. Infn. Proc. 2, 207 (2003) A.R. Usha Devi and Sudha, Asian Journal of Physics,19, 1 (2010)

18. Local Invariant Spin Squeezing criteria

19. Local Invariant Spin Squeezing criteria

20. Local Invariant nature of spin squeezing parameters Having defined the generalized spin squeezing parameters as establishing their local invariant nature is of utmost importance. To do this we only have to establish the local invariant nature of the quantities and

21. Local Invariant nature of spin squeezing parameters

22. Local Invariant nature of spin squeezing parameters

23. Local Invariant nature of spin squeezing parameters

24. Operational approach towards the evaluation of the local invariant spin-squeezing parameters .

25. Operational approach towards the evaluation of the local invariant spin-squeezing parameters

26. Operational approach towards the evaluation of the local invariant spin-squeezing parameters

27. Local Invariant Spin Squeezing parameters for non-summetric states

28. Local Invariant Spin Squeezing parameters for non-summetric states

29. The generalized collective spin operators of a multi-qubit spin squeezed state can be represented geometrically by an elliptical cone of local invariant height centered about the z-axis (common qubit orientation direction in the case of symmetric multiqubit states). The semi-minor and semi-major axes of the ellipse being and the corresponding value respectively. In contrast, the coherent spin state is depicted by a circular cone.

30. Coherent State Spin Squeezed State This geometric picture of the spin squeezed state versus that of a spin coherent state illustrates collective spin-squeezing feature in symmetric as well as non-symmetric states.

31. Relationship between spin squeezing and quantum entanglement A deeper understanding between spin squeezing and quantum entanglement has been explored in the literature and it has been established that the presence of spin squeezing essentially reflects pairwise entanglement in symmetric multiqubit states. Here we obtain a relationship between local invariant spin squeezing parameters and quantum entanglement quantified through concurrence in two-qubit pure states. • Sorenson et.al., Nature (London), 409, 63 (2001) • D. Ulam-Orgikh and M. Kitagawa, Phys. Rev. A64, 052106 (2001) • X. Wang and B.C Sanders, Phys. Rev. A68, 03382 (2003) • X. Wang and K. Molmer, Eur. Phys. J. D68, 03382 (2003)

32. 1. Spin squeezing and quantum entanglement in pure 2-qubit states

33. 1. Spin squeezing and quantum entanglement in pure 2-qubit states

34. 1. Spin squeezing and quantum entanglement in pure 2-qubit states

35. Graphs depicting the variation of the LISS squeezing parameters and concurrence as a function of time for the N-qubit spin-down state evolving under one-axis twisting Hamiltonian N>2 N=2

36. Graphs depicting the variation of the LISS squeezing parameters and concurrence as a function of time for the pure state evolving under Ising interaction

37. Spin squeezing in essentially non-symmetric qubit states • We now wish to examine essentially non-symmetric multiqubit states and the nature of spin squeezing in them. • The loss of symmetry in initially symmetric multiqubit pure states with under Ising chain evolution has been established earlier and we consider the 4-qubit spin down state subjected to Ising interaction. Sudha, B,G. Divyamani and A.R. Usha Devi , Chin. Phys. Lett. 2, 020305 (2011)

38. Variation of pairwise entanglement and Kitegawa-Ueda generalized spin squeezing in an essentially nonsymmetric 4-qubit state

39. Entanglement and spin-squeezing in multiqubit states

40. Entanglement and spin-squeezing in multiqubit states

41. Entanglement and spin-squeezing in multiqubit states • That is, fully separable states are not spin squeezed in the Wineland sense. • But this does not mean that all states with no Wineland squeezing are of fully separable form • In particular, there can be entangled states with

42. Connection between local invariants and Kitagawa-Ueda spin squeezing in symmetric multiqubit states It is well known that entanglement of a composite quantum system remain invariant, when the subsystems are subjected to local unitary operations. Any two quantum states are entanglementwise equivalent iff they are related to each other through local unitary transformations. In fact, the non-local properties associated with a quantum state can be represented in terms of a complete set of local invariants. While a set of 18 local invariants are required for the complete description of an arbitrary two-qubit mixed state, the number of invariants reduce to 6, when the two qubit state obeys exchange symmetry. Symmetric states indeed offer elegant mathematical analysis as the dimension of the Hilbert space reduces drastically from to N+1, when N two-level systems respect exchange symmetry. Y. Makhlin, Quantum.Inf.Proc1, 243 (2003) A. R. Usha Devi, M.S. Uma, R. Prabhu and Sudha, J. Opt. B. 7, S740 (2005)

43. Connection between local invariants and Kitagawa-Ueda spin squeezing in symmetric multiqubit states The parameters of a collective phenomena like spin-squeezing, reflecting pairwise entanglement of symmetric qubits, should be expressible in terms of two qubit local invariants. In fact, the local invariant version of Kitagawa-Ueda spin squeezing parameter can be expressed in terms of one of the local invariant quantity associated with the symmetric two-qubit reduced density matrix of the N qubit symmetric state. We proceed to describe an elegant connection between spin squeezing and pairwise entanglement. • A. R. Usha Devi, M.S. Uma, R. Prabhu and Sudha, J. Opt. B. 7, S740 (2005) • A. R. Usha Devi, M.S. Uma, R. Prabhu and Sudha, Int. J. Mod. Phys. B 20, 1917(2006)

44. Connection between local invariants and Kitagawa-Ueda spin squeezing in symmetric multiqubit states

45. Connection between local invariants and Kitagawa-Ueda spin squeezing in symmetric multiqubit states

46. Connection between local invariants and Kitagawa-Ueda spin squeezing in symmetric multiqubit states

47. Connection between local invariants and Kitagawa-Ueda spin squeezing in symmetric multiqubit states

48. Variation of the local invariant parameter as a function of time for the N-qubit spin down state subjected to one-axis twisting Hamilotonian. The reduction in the negativity of as N increases is clearly seen. This supports our observation that squeezing reduces as N increases for this state

49. Variation of the local invariant parameter as a function of time for the state subjected to Ising chain evolution. ti The state under consideration exhibits spin squeezing at all times as is indicated by the negative value of . Maximum squeezing is seen at time t=0 and shows a periodic decrease and increase.

50. Conclusion • We have demonstrated the need for Local Invariant Spin Squeezing criteria that exhibit local invariance in addition to being applicable to non-symmetric states. • We have explicitly obtained the expressions for local invariant spin squeezing parameters for symmetric as well as non-symmetric states. • The connection between quantum entanglement and local invariant spin squeezing has been established (especially for 2-qubit pure states). • Examples from both symmetric and nonsymmetric class of states are worked out.