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Journal of Physics A Mathematical and General

Correlative amplitude–operational phase entanglement embodied by the EPR-pair eigenstate |η⟩

Hong-Yi Fan

Published Jan 21, 2002• 10 citations
Publisher

Abstract

After a penetrating analysis of the known number–phase commutative relations we find that the common eigenvector |η⟩ of two particles' relative coordinate and total momentum also embodies the entanglement with respect to the correlative amplitude–operational phase. Based on the fact that the number-difference operator is the canonical conjugate to the Noh–Fougères–Mandel operational phase operator, the corresponding entanglement is also briefly discussed.

DOI
10.1088/0305-4470/35/4/313

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ENTANGLEMENT OF AMPLITUDE AND OPTICAL PHASE INVOLVED IN THE EPR EIGENSTATE

Hong-Yi Fan, Yue Fan

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>.

Modern Physics Letters B•Dec 10, 2000•5 citations

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

Alfredo Luis

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.

Journal of Physics A Mathematical and General•Dec 12, 2002•1 citation

Noh-Fougères-Mandel operational phase operator as an entangling operator and the corresponding squeezed state

Hong-Yi Fan, Zhihu Sun

We construct a new number-difference and operational phase entangled state by operating the Noh-Fougères-Mandel (NFM) phase operator on the two-mode twin-photon state. Its Schmidt decomposition is derived by virtue of the previously constructed Einstein-Podolsky-Rosen eigenstate. We reach the conclusion that the NFM phase operator is essentially an entangling operator, based on which a new type of number-difference and operational phase squeezed state can be introduced.

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

Hong-Yi Fan, Hu Hai-Peng

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 .

Journal of Physics A Mathematical and General•Dec 12, 2002•2 citations

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Since the commutation relation [x1-x2,a1+a2]=0,their common eigenvector |β,x〉 is constructed by using the technique of integration within an ordered product(IWOP) of operators.This state exhibits the properties of the coherent and entangled states.Its entangled properties and the relations with the EPR entangled state can be confirmed by Schmidt decomposition and the unilateral integral for|β,x〉.The superposition of the states |β=β1+iβ2,x〉 along β1 axis yields the ideal two-mode entangled state.

Journal of Heze University•Jan 1, 2009•0 citations
After a penetrating analysis of the known number–phase commutative relations we find that the common eigenvector |η⟩ of two particles' relative coordinate and total momentum also embodies the entanglement with respect to the correlative amplitude–operational phase. Based on the fact that the number-difference operator is the canonical conjugate to the Noh–Fougères–Mandel operational phase operator, the corresponding entanglement is also briefly discussed.
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>.
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.
We demonstrate that a common eigenvector |η> (Einstein, Podolsky and Rosen, EPR states) of two particles' relatively coordinate and total momentum possesses a property that the projection (a Bell measurement) of two pairs of EPR states |η>1,2⊗| η'>3,4 onto the subspace spanned by 1,4<η″|, yields a EPR state |η″-η*-η′* >2,3 for particles 2 and 3. Entanglement swapping operator is also derived by virtue of the technique of integration within an ordered product of operators.
Since the commutation relation [x1-x2,a1+a2]=0,their common eigenvector |β,x〉 is constructed by using the technique of integration within an ordered product(IWOP) of operators.This state exhibits the properties of the coherent and entangled states.Its entangled properties and the relations with the EPR entangled state can be confirmed by Schmidt decomposition and the unilateral integral for|β,x〉.The superposition of the states |β=β1+iβ2,x〉 along β1 axis yields the ideal two-mode entangled state.