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Journal of Physics A Mathematical and General

Comment on Correlative amplitude-operational phase entanglement embodied by the EPR-pair eigenstate

Alfredo Luis

Published Dec 12, 2002• 1 citation
Publisher

Abstract

In a recent paper in this journal Fan (Fan H 2002 J. Phys. A: Math. Gen. 35 1007 ) discards the possibility of using a genuine phase-difference operator to investigate number-phase entanglement because of the lack of unitarity of the Susskind–Glogower phase operators. However, Fan overlooked the existence of a bona fide unitary operator exponential of the phase difference. Here we find the amplitude–phase maximally entangled states as the simultaneous eigenstates of the total number and the phase-difference operators.

DOI
10.1088/0305-4470/36/1/320

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Reply to the comment on Correlative amplitude operational phase entanglement embodied by the EPR-pair eigenstate

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We compare and contrast our amplitude–phase entanglement with that of Luis in his comment. Luis's entangled state is defined in a finite Fock space. His comment on the operational phase operator seems to be contradicting the original meaning of Mandel et al .

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In a recent paper in this journal Fan (Fan H 2002 J. Phys. A: Math. Gen. 35 1007 ) discards the possibility of using a genuine phase-difference operator to investigate number-phase entanglement because of the lack of unitarity of the Susskind–Glogower phase operators. However, Fan overlooked the existence of a bona fide unitary operator exponential of the phase difference. Here we find the amplitude–phase maximally entangled states as the simultaneous eigenstates of the total number and the phase-difference operators.
By noticing that although single-mode Susskind–Glogower phase operator does not commute with photon number operator, two-mode nonlinear phase operator commutes with number-difference operator. We reveal that the common eigenstate <η| of two-particles' relative coordinate and total momentum involves the entanglement between phase and amplitude. We also reveal entanglement between number difference and amplitude inherent in the state |D,|η|>, where [Formula: see text] is a deductive state of |η>.
Interpretational difficulties of a quantum measurement [J. W. Noh, A. Foug\`eres, and L. Mandel, Phys. Rev. A 45, 424 (1992)] are pointed out. The nonunitarity of the respective exponential phase operator reflects an enhanced noise, which can be reduced by discarding ambiguous output data. Consequently, dispersion of such measurements does not represent a meaningful performance measure relevant to the input state. Moreover, the proposed model of quantum phase measurements provides a discrete spectrum of the quantum phase operator, whereas the phase-shift parameter is continuous.
We define a Susskind–Glogower operator that is unitary and acts as a true ladder operator for photons of a given frequency and any polarization. The operator acts on a Hilbert space consisting of all number states in both polarizations. The phase operator derived from this ladder operator satisfies the canonical commutation relations with the number operator. The eigenstates of this ladder operator are also found. Our method based on all the observable states enables us to write the operators for objects such as beam splitters, phase shifters or polarizers, as constant matrices; those used to study classical polarized beams. The Pegg–Barnett operator and the two-mode operator of Ban are special cases of our operator. We compared the phase state defined here with that obtained from the original Susskind–Glogower case, as well as that obtained by the method of Pegg and Barnett in the limit of a very small number of photons being present, and found that there are differences which may enable a direct experimental comparison of these operators.