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Mixed state entanglement and generalized Schwinger model

Veena Adiga, Swarnamala Sirsi

Published Jun 26, 2009• 0 citations
OpenAlex

Abstract

: A new scheme for constructing a mixed spin density matrix has already been introduced to study spin squeezing in some states of practical interest. Such a construction takes its inspiration from Schwinger's idea of realising a state as being made of (j+m) “up spinors” and (j-m) “down spinors”, all defined with respect to a single axis in space. The new scheme is employed here to study some of the well known entanglement measures which characterise the quantumness of a state. Some well known states are considered for an in depth analysis

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: A new scheme for constructing a mixed spin density matrix has already been introduced to study spin squeezing in some states of practical interest.
Such a construction takes its inspiration from Schwinger's idea of realising a state as being made of (j+m) “up spinors” and (j-m) “down spinors”, all
defined with respect to a single axis in space. The new scheme is employed here to study some of the well known entanglement measures which characterise
the quantumness of a state. Some well known states are considered for an in depth analysis
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.
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.
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.
We propose that the entanglement of mixed states is characterised properly in terms of a probability density function P(E)\mathcal{P}(\mathcal{E})P(E). There is a need for such a measure since the prevalent measures (such as \textit{concurrence} and \textit{negativity}) are rough benchmarks, and not monotones of each other. Considering the specific case of two qubit mixed states, we provide an explicit construction of P(E)\mathcal{P}(\mathcal{E})P(E) and show that it is characterised by a set of parameters, of which concurrence is but one particular combination. P(E)\mathcal{P}(\mathcal{E})P(E) is manifestly invariant under SU(2)×SU(2)SU(2) \times SU(2)SU(2)×SU(2) transformations. It can, in fact, reconstruct the state up to local operations - with the specification of at most four additional parameters. Finally the new measure resolves the controversy regarding the role of entanglement in quantum computation in NMR systems.
A new “classical” formulation of conventional quantum mechanics is used to construct a description of quantum states with spin j =1/2 which uses only positive classical probabilities of observables instead of a complex density matrix. These probabilities are functions of points on a unit sphere which determine the directions relative to which the spin projections are measured. The proposed construction is a generalization of a system for tomography of the spin states of a single particle and may be used as a new method of measuring two-particle spin states. For pure spin states this method is exactly the same as the case of a single particle. Additional probabilities must be defined to describe mixed states.
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.