268 OPTICS LETTERS / Vol. 34, No. 3 / February 1, 2009

Nonexistence of entanglement sudden death indephasing of high NOON states

Asma Al-Qasimi* and Daniel F. V. JamesDepartment of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada

*Corresponding author: alqasimi@physics.utoronto.ca

Received October 3, 2008; revised November 25, 2008; accepted November 26, 2008;posted December 16, 2008 (Doc. ID 102263); published January 26, 2009

We study the dynamics of entanglement in continuous variable quantum systems. Specifically, we study thephenomena of entanglement sudden death (ESD) in general two-mode-N-photon states undergoing puredephasing. We show that for these circumstances, ESD never occurs. These states are generalizations of theso-called high NOON states (i.e., a superposition of N photons in the first mode, O in the second, with Ophotons in the first, N in the second), shown to decrease the Rayleigh limit of � to � /N, which promises greatimprovement in resolution of interference patterns if states with large N are physically realized [Phys. Rev.Lett. 85, 2733 (2000)]. However, we show that in dephasing NOON states, the time to reach some criticalvisibility Vcrit, scales inversely with N2. On the practical level, this shows that as N increases, the visibilitydegrades much faster, which is likely to be a considerable drawback for any practical application of thesestates. © 2009 Optical Society of America

OCIS codes: 270.5290, 270.5585.

Enanglement is a quantum property that, for a longtime, has fascinated those studying the fundamen-tals of quantum mechanics and, more recently, thoseinterested in its powerful applications such as quan-tum information science. Many argue that not only isit a quantum property but rather the only one. Thequestion of when entanglement disappears is an in-teresting fundamental question to consider.

Entanglement sudden death (ESD) is a termcoined by Yu and Eberly [1] to describe loss of en-tanglement in a finite time. The work done so far,which mostly concerns two-qubit systems, hasshown, in one way, how fragile entanglement is in re-alistic systems. Several papers have shown that ESDalways occurs in some very general two-qubit sys-tems. Examples include X states, i.e., states withnonzero parameters (in general) on the diagonal andantidiagonal of the density matrix of the system. In[2], it is shown that for dephasing in X states, there isalways ESD as long as none of the parameters of thedensity matrix are zero; in [3], it is shown that forthese states at finite temperature, and with depopu-lation going on, ESD also always occurs; and in [4], itwas demonstrated that external driving fields on thesystem will enhance ESD. With results like these,one is tempted to make the guess that ESD is actu-ally a universal phenomenon. So far an attempt hasbeen made to prove this in [5].

With all this work showing how prevalent ESD isin qubit systems, it is interesting to ask how commonit is in other quantum systems. For example, is ESDas common in continuous variable quantum systems(CVQS) as it is in qubits? Recently, ESD has beenshown to occur in a system of two free harmonic os-cillators interacting with a Markovian bath [6]. Inaddition, two initially Gaussian states, states withGaussian Wigner functions, coupled to the same(ohmic) environment have been studied in [7], wherethe existence of three phases were demonstrated:

ESD, ESD with revival, and no ESD. ESD in CVQS

0146-9592/09/030268-3/$15.00 ©

has also been studied taking into consideration rela-tivistic effects [8].

In the work we present here, we prove the generalresult that ESD never occurs in two-mode–N-photonstates undergoing dephasing. NOON states (de-scribed in the abstract), which have been shown [9] tobeat the Rayleigh limit in interferometry, fall underthis general class of states. The resolution of interfer-ence patterns improves when the separation betweenthe wave amplitudes falls down to � /N compared tothe minimum of � forced by the Rayleigh limit. Thepower of these NOON states lies in their entangl-ment. It is, therefore, important to study the decay ofentanglement in these systems. The approach we de-scribe here is studying ESD in such systems, tryingto get a feel for the fragility of entanglement. Finally,we touch on the important question: Does the exis-tence of some entanglement for a very long time haveany practical implication for the usefulness of NOONstates?

The system we consider is that of two harmonic os-cillators with N photons shared between them. Thedensity matrix, most generally, describing such a sys-tem is given as follows:

�̂�t� = �k=0

N

�kk�t��N − k,k��N − k,k�

+ �k,m=0,k�m

N

�km�t��N − k,k��N − m,m�. �1�

We deal only with dephasing, since applications ofsuch states tend to be postselective on photon num-ber: processes in which N changes are filtered away.If the system undergoes pure dephasing, due to ran-dom fluctuation of the mode frequency, one expectsthat the off-diagonal terms, i.e., �km�t�, where k�m,in Eq. (1) will acquire decay terms. On the other

hand, the population, which is represented by the di-

2009 Optical Society of America

February 1, 2009 / Vol. 34, No. 3 / OPTICS LETTERS 269

agonal elements, will remain intact; i.e., the photonnumber will be preserved. It can be easily checkedthat the master equation describing the dynamics ofsuch a system, in which there is no correlation be-tween the two fields interacting with the two har-monic oscillators, is given by

��̂�t�

�t= 2��1��n1

ˆ , �̂�t�n1ˆ + �n1

ˆ �̂�t�,n1ˆ

+ �2��n2ˆ , �̂�t�n2

ˆ + �n2ˆ �̂�t�,n2

ˆ �, �2�

where �i and niˆ are the decay rate and the number

state operator of the ith harmonic oscillator. By as-sumption there is no depopulation going on. By Eq.(2), the evolution of Eq. (1) with respect to time in thecase of pure dephasing is

�̂�t� = �k=0

N

�kk�0��N − k,k��N − k,k�

+ �k,m=0,k�m

N

�km�0�e−�1/2��k − m�2��1+�2�t

��N − k,k��N − m,m�. �3�

As with other decoherence mechanisms, one sideeffect of dephasing is the decay of entanglement. Tofind out whether this decay results in ESD or not, weneed to use a reliable measure for entanglement. Intwo-qubit systems, we have good measures such asWootter’s concurrence [10], which can tell with cer-tainty whether a system is entangled or separable.On the other hand, in the general two CVQS, the bestany measure can do is provide a necessary but notsufficient condition for separability [11]. In the caseof CVQS, the sufficiency has only been proven forGaussian states [12,13], and NOON states, the gen-eralization of which we discuss here, do not belong tothis class. Nevertheless, this weakness in the criteriadoes not have to disadvantage the study of entangle-ment, as we will show in our case.

Here is a brief description for Peres’s criterion forentanglement [11]. The density matrix of a bipartitesystem may be written as

�̂ = �i

ci�̂i� � �̂i�. �4�

Taking the partial transpose over one of its sub-systems one obtains

�̂ = �i

ci��̂i��T

� �̂i�. �5�

If �̂ has at least one negative eigenvalue, then weknow with certainty that the system is entangled.However, if none of the eigenvalues are negative,then the system could be entangled or separable. Theconsequence of this weakness to our study of ESD isthat the existence of ESD cannot be proven with cer-tainty, while its nonexistence can be proven with cer-

tainty.

Using this criterion, we obtain the following for thepartial transpose of our state in Eq. (3):

�̂ = �k=0

N

�kk�0��N − k,k��N − k,k�

+ �k,m=0,k�m

N

�km�0�e−�1/2��k − m�2��1+�2�t

��N − k,m��N − m,k�. �6�

Notice that since k and m in the second terms ofthe right-hand side of Eq. (6) are not equal, the totalnumber of photons in the kets and bras are neverequal to N. In other words, N−k+m�N and N−m+k�N. Mathematically, this implies that each pair ofthe matrix elements �N−k ,m��N−m ,k� and �N−m ,k��N−k ,m� fall into a separate subspace distinctfrom the space of the diagonal elements.

This breaks the problem of finding the eigenvaluesof �̂ into finding the eigenvalues of N�N+1� /2 matri-ces; the remaining eigenvalues are just the diagonalelements of �̂. Each of these matrices has the follow-ing form:

��km�0��e−�1/2��k − m�2��1+�2�t�ei��N − k,m��N − m,k�

+ e−i��N − m,k��N − k,m�, �7�

where � is the phase of the matrix element �km andthe vertical bars � � represent the norm of the quan-tity they enclose. It can be easily shown that the fol-lowing state:

��� =1

�2�ei��N − k,m� − �N − m,k�, �8�

is an eigenvector of Eq. (7) with eigenvalue

− ��km�0��e−�1/2��k − m�2��1+�2�t.

This means that Eq. (6) has at least one negativeeigenvalue, which only goes to zero at infinite time,as long as one of the �km, and consequently �mk, isnonzero; i.e., this is true as long as there is some co-herence in the system. If there is not any other deco-herence mechanism, such as depopulation, going onas well, this is always true; there will always be en-tanglement in the system for any finite time. In otherwords, for a general two-mode-N-photon state under-going pure dephasing, there is no sudden death of en-tanglement. This is the main result of this Letter.

We demonstrate our results using the first realizedNOON state [14], i.e., for a two-mode–three-photonstate, specifically given by

��� =1

�2��N0� + �0N�, �9�

where N=3, but more generally by

��� = a�30� + b�21� + c�12� + d�03�, �10�

where �a�2+ �b�2+ �c�2+ �d�2. Applying the argumentsabove, we find that for the partial transpose of

the density matrix of this system, the negative eigen-

270 OPTICS LETTERS / Vol. 34, No. 3 / February 1, 2009

values are: −�a �b �e−�1/2���1+�2�t, −�a �c �e−2��1+�2�t,−�a �d �e−�9/2���1+�2�t, −�b �c �e−�1/2���1+�2�t, −�b �d ��e−2��1+�2�t, and −�c �d �e−�1/2���1+�2�t. Each of them in-volve a decay term owing to dephasing. However,they only become zero after an infinite amount oftime that renders the negative exponential zero.Therefore, for any finite time, there is always en-tanglement in the system, so ESD does not occur.

Finally, we consider how practical this long-livedentanglement is in NOON states undergoing dephas-ing. In Fig. 1 standard setup for interfering twobeams to produce interference fringes is illustrated.The presence of a phase shifter in the upper path in-duces photons travelling there to acquire a phaseshift ei�. When the “N-photon-NOON-state” is cre-ated inside this interferometer, the phase is acuumi-lated N times, and the state becomes

��� =1

�2��N0� + eiN��0N�. �11�

With dephasing occuring, the density matrix of thestate is as follows:

��t�ˆ = 12 ��N0��N0� + e−N2�teiN��0N��N0�

+ e−N2�te−iN��N0��0N� + �0N��0N��, �12�

where we assume �1=�2=�.The expectaion value of the exposure dosage, �̂�,

displays fringes of visibility V. The exposure dosageoperator ̂ is described in terms of creation and an-nhilation operators acting on the two output (C andD paths) operators in Fig. 1. With simple algebra, itcan be shown, in the case of dephasing of NOONstates, that the expectation value of ̂ is �̂�=1+e−N2�t cos�N��. From which the visibility is found tobe

V =�̂�max − �̂�min

�̂�max + �̂�min

= e−N2�t. �13�

When the visibility becomes vanishingly small, thefringes (and hence the measured phase) become im-possible to measure: we will call this visibility, atwhich measurement becomes impracticable, the criti-

Mirror

Mirror

Screen

BeamSplitter

PhaseShifter (Φ)

CC

DD

Fig. 1. Interface pattern formation adapted from Fig. 1 of[1]. Two photon beams pass through a beam splitter andget reflected off the upper and lower mirrors to form an in-terference pattern on the screen. The upper beam passesthrough a phase shifter before reaching the screen. Thephase aquired depends on the number of photons N thatpass through the upper path, and it equals eiN�.

cal visibility Vcrit; its value will depend on the sensi-tivity of the fringe measurement techniques em-ployed. For the given decay rate �, the time it takesto reach this critical visibility Vcrit is given by

tcrit =1

�N2 ln 1

Vcrit� . �14�

Notice that in Eq. (14), the expression for time de-pends inversely on N2. This implies that the larger Nis, the faster it takes for visibilty to fall down to Vcritand become worse. This is completely the opposite ofwhat was earlier hoped to be acheived in improvingresolution of fringes by creating high NOON states;i.e., states with large N.

We showed that although the criterion for seper-ability has a weakness that can render some studiesof entanglement uncertain, in our case and by usingthis criteria, we proved with certainty that for puredephasing (no photon loss), ESD does not occur in oursystem. We also demonstrate our result using the so-called NOON states to show that there is no suddendeath at NOON. Although this criteria allows us toprove that, even after a long time, there is someenanglement left in the system, it does not give us away to determine how much is left. Therefore, to an-swer the question about the usefulness of thedephased NOON states in interferomentry, we studythe time it takes to reach critical visibility. In doingso, we reveal that the presence of some entanglementdoes not have many practical implications. In fact,we show that for this realistic decohering system, in-creasing N does not improve resolution, but ratherallows it to worsen at a faster rate, which is propor-tionl to N2.

This work was supported by the Natural Sciencesand Engineering Research Council of Canada(NSERC).

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