Abstract
In spite of a long history, the quantification of entanglement still calls for exploration. What matters about entanglement depends on the situation, and so presumably do the numbers suitable for its quantification. Regardless of situational complications, a necessary first step is to make available for calculation some quantitative measure of entanglement. Here we define a geometric degree of entanglement, distinct from earlier definitions, but in the case of bipartite pure states related to that proposed by Shimony (Ann N Y Acad Sci 755:675---679, 1995). The definition offered here applies also to multipartite mixed states, and a variational method simplifies the calculation. We analyze especially states that are invariant under permutation of particles, states that we call bosonic. Of interest to quantum sensing, for bosonic states, we show that no partial trace can increase a degree of entanglement. For some sample cases we quantify the degree of entanglement surviving a partial trace. As a function of the degree of entanglement of a bosonic 3-qubit pure state, we show the range of degree of entanglement for the 2-qubit reduced density matrix obtained from it by a partial trace. Then we calculate an upper bound on the degree of entanglement of the mixed state obtained as a partial trace over one qubit of a 4-qubit bosonic state. As a reminder of the situational dependence of the advantage of entanglement, we review the way in which entanglement combines with scattering theory in the example of light-based radar.
Highlights
As early as Born’s 1926 analysis of scattering [1], the correlations peculiar to quantum physics were noticed, and since have been exploited in applications including cryptography
For states without special symmetry, nothing general can be said in this connection; for bosonic states we show that no partial trace ever increases a degree of entanglement
To show that all these equations have solutions, we study the special case of symmetry under permutation of the 3 qubits, so that a state is defined by complex parameters A0, . . . , A3 with
Summary
As early as Born’s 1926 analysis of scattering [1], the correlations peculiar to quantum physics were noticed, and since have been exploited in applications including cryptography. Many of the cases of pure and mixed states addressed in this paper are for states that are invariant under permutation of particles, states that we call bosonic. These offer the charm of low dimensional vector spaces as a wedge into a complicated subject, along with other advantages noted below. As a reminder of the situational dependence of the advantage of entanglement, we review in Sec. VII the way in which entanglement combines with scattering theory in connection with ladar in what is called Type-III sensing, in which part of the signal generated by a transmitter is stored for later comparison with the echo from the part of the signal incident on a target
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