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arXiv:quant-ph/0204019 · Journal of Physics A Mathematical and General

Squeezing of a coupled state of two spinors

A. R. Usha Devi, K. S. Mallesh, Mahmoud A A Sbaih, K B Nalini, G. Ramachandran

Published Apr 29, 2003• 12 citations
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Abstract

The notion of spin squeezing involves reduction in the uncertainty of a component of the spin vector below a certain limit. This aspect has been studied earlier for pure and mixed states of definite spin. In this paper, this study has been extended to coupled spin states which do not possess sharp spin value. A general squeezing criterion has been obtained by requiring that a direct product state for two spinors is not squeezed. The squeezing aspect of entangled states is studied in relation to their spin- spin correlations.

DOI
10.1088/0305-4470/36/19/310

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The notion of spin squeezing involves reduction in the uncertainty of a component of the spin vector below a certain limit. This aspect has been studied earlier for pure and mixed states of definite spin. In this paper, this study has been extended to coupled spin states which do not possess sharp spin value. A general squeezing criterion has been obtained by requiring that a direct product state for two spinors is not squeezed. The squeezing aspect of entangled states is studied in relation to their spin- spin correlations.
The nature of spin squeezing has been studied earlier for a coupled state of two spinors by Usha Devi et. al. (J. Phys. A: Math. Gen. 36 5333 (2003)). In this paper, we extend this study to a coupled state of two spin-1 systems. Here, we show that the direct product of two spin-1 states exhibits squeezing unlike the direct product of two spinors. We investigate the nature of squeezing for non-separable coupled state of two spin-1 systems and identify the parameters which are responsible for the existence of squeezing. We also provide ways to generate them by external interactions.
The notion of spin squeezing has been discussed in this paper using the density matrix formalism. Extending the definition of squeezing for pure states given by Kitagawa and Ueda in an appropriate manner and employing the spherical tensor representation, we show that mixed spin states which are non-oriented and possess vector polarization indeed exhibit squeezing. We construct a mixed state of a spin 1 system using two spin ½ states and study its squeezing behaviour as a function of the individual polarizations of the two spinors.
Abstract. The notion of spin squeezing has been discussed in this paper using the density matrix formalism. Extending the definition of squeezing for pure states given by Kitagawa and Ueda in an appropriate manner and employing the spherical tensor representation, we show that mixed spin states which are non-oriented and possess vector polarization indeed exhibit squeezing. We construct a mixed state of a spin 1 system using two spin 1/2 states and study its squeezing behaviour as a function of the individual polarizations of the two spinors.
We classify the pure states of a spin- s system into oriented and non-oriented states. A pure state is said to be non-oriented if it is not an eigenstate of S 2 and S z with respect to any axis of quantization. If it is an eigenstate it is oriented. In this paper, we discuss the notion of spin squeezing in these states. Our analysis shows that the oriented states are not squeezed while non-oriented states exhibit squeezing. We also present a new scheme for the construction of spin- s states using 2 s spinors oriented along different axes. Taking the case of s = 1, we show that `non-oriented' nature and hence squeezing arise from the intrinsic quantum correlations that exist among the spinors in the coupled state.
Spin- s assemblies are classified into two mutually exclusive classes: oriented and non-oriented systems. The density matrix ρ, describing oriented systems, can assume diagonal form in the angular momentum basis ( ) defined with respect to the axis of quantization, whereas the eigenstates of ρ for the non-oriented assembly cannot all be identified with states. A new scheme for constructing a mixed, non-oriented spin- s state using s (2 s +1) spinors all pointing in different directions in space and 2 s weights is discussed. Such a construction takes its inspiration from Schwinger's idea of realizing an state as being made up of ( s + m ) 'up' spinors and ( s − m ) 'down' spinors, all defined with respect to a single axis in space. Since the oriented systems are never squeezed, non-oriented spin-1 assemblies which can be prepared in the laboratory with the available NQR technology are examined for signatures of squeezing using our scheme in a frame of reference where the Heisenberg–Robertson uncertainty relation has the same form as the Schrödinger uncertainty relation. It is shown that unlike in the case of the pure spin-1 state where squeezing is synonymous with non-orientedness, a non-oriented spin-1 system need not be squeezed and the existence of entanglement is a necessary but not sufficient condition for the system to be squeezed.
What is the relation between spin squeezing and entanglement? To clarify this, we derive the full set of generalized-spin-squeezing inequalities for the detection of entanglement. These are inequalities for the mean values and variances of the collective angular-momentum components Jk{J}_{k}Jk​. They can be used for the experimental detection of entanglement in a system of spin-12\frac{1}{2}21​ particles in which the spins cannot be individually addressed. We present various sets of inequalities that can detect all entangled states that can be detected based on the knowledge of (i) the mean values and variances of Jk{J}_{k}Jk​ in three orthogonal directions, or (ii) the variances of Jk{J}_{k}Jk​ in three orthogonal directions, or (iii) the mean values of Jk2{J}_{k}^{2}Jk2​ in three orthogonal directions, or (iv) the mean values and variances of Jk{J}_{k}Jk​ in arbitrary directions. We compare our inequalities to known spin-squeezing entanglement criteria and discuss to which extent spin squeezing is related to entanglement in the reduced two-qubit states. Finally, we apply our criteria for the detection of entanglement in spin models, showing that they can be used to detect bound entanglement in these systems.