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Transition probabilities between quasifree states. (English) Zbl 0986.81054
Summary: The author obtains general formula for the transition probabilities between any state of the $$C^*$$ algebra of the canonical commutation relations (CCR-algebra) and a squeezed quasifree state. Applications of this formula are made for the case of multimode thermal squeezed states of quantum optics using a general canonical decomposition of the correlation matrix valid for any quasifree state. In the particular case of a one-mode CCR-algebra he shows that the transition probability between two quasifree squeezed states is a decreasing function of the geodesic distance between the points of the upper half-plane representing these states. In the special case of the purification map it is shown that the transition probability between the state of the enlarged system and the product state of real and fictitious subsystems can be a measure for the entanglement.

##### MSC:
 81S05 Commutation relations and statistics as related to quantum mechanics (general) 81R30 Coherent states 81V80 Quantum optics
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##### References:
 [1] DOI: 10.1007/BF01649588 · Zbl 0137.45601 [2] Manuceau J., Ann. Inst. Henri Poincaré, Sect. A 8 pp 139– (1968) [3] DOI: 10.1007/BF01654283 · Zbl 0167.55902 [4] DOI: 10.1007/BF01646623 · Zbl 0208.38301 [5] DOI: 10.1007/BF01609051 · Zbl 0338.46050 [6] Holevo A. S., Probl. Peredachi Inf. 6 pp 44– (1970) [7] Holevo A. S., Teor. Mat. Fiz. 6 pp 3– (1971) [8] Holevo A. S., Teor. Mat. Fiz. 13 pp 184– (1979) [9] Holevo A. S., IEEE Trans. Inf. Theory 21 pp 533– (1972) · Zbl 0317.94004 [10] DOI: 10.1016/0375-9601(89)90473-8 [11] DOI: 10.1016/0375-9601(92)90266-O [12] DOI: 10.1063/1.528207 · Zbl 0674.46039 [13] DOI: 10.1016/0029-5582(65)90550-X [14] Bures D. J. C., Trans. Am. Math. Soc. 135 pp 199– (1969) [15] DOI: 10.1016/0034-4877(76)90060-4 · Zbl 0355.46040 [16] DOI: 10.1007/BF00403278 · Zbl 0515.46063 [17] DOI: 10.1016/0003-4916(91)90360-K [18] DOI: 10.1103/PhysRevA.49.52
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