Physics Letters A 354 (2006) 71–78

www.elsevier.com/locate/pla

Squeezed coherent states as feasible approximationsto phase-optimized states

Alfredo Luis

Departamento de Óptica, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain

Received 8 November 2005; received in revised form 15 January 2006; accepted 16 January 2006

Available online 24 January 2006

Communicated by P.R. Holland

Abstract

We determine the squeezed coherent states with minimum phase uncertainty for fixed mean energy using diverse measures of phase fluctuations.We show that the phase of suitably prepared squeezed coherent states is as well defined as the phase of abstract phase states, allowing robust andaccurate measurements approaching quantum limits.© 2006 Elsevier B.V. All rights reserved.

PACS: 42.50.Dv; 42.50.St; 03.65.-w

Keywords: Nonclassical states; Squeezed states; Quantum phase; Quantum limits; Precision measurements

1. Introduction

The correct description of phase in quantum optics has along history being still an open problem [1–7]. Most of the pe-culiarities that the phase of a single mode field encounters inquantum mechanics can be ascribed to the lack of an standardoperator representing this variable and to its periodic charac-ter (the case of the phase difference is slightly different sinceit admits an standard operator description in the conventionalHilbert space [6,8]). Because of this, the phase concept mani-fests in quantum optics under different perspectives, mutuallyconnected but not equivalent, dealing with issues such as theproper quantum observables representing phase and phase mea-surements, the efficient detection of phase shifts, the proper as-sessment of phase fluctuations, and the generation of states withminimum phase uncertainty. For example, it has been shownthat the states that determine the phase statistics via projectionmay or may not coincide with the states having minimum phaseuncertainty depending on the measure of fluctuations adopted[9].

E-mail address: alluis@fis.ucm.es (A. Luis).

0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2006.01.040

An inconvenient feature of most approaches to quantumphase is the lack of experimental schemes reproducing in prac-tice ideal phase concepts. In this Letter we focus on the approx-imation of phase states by feasible quantum states. The mostclear candidates for feasible quantum phase states that can begenerated in practice are the squeezed coherent states [7,10–14]. Their phase properties have been studied, for example, in[15–27].

In this regard a pictorial procedure has been proposed inRefs. [28–30] suggesting that the squeezed coherent states withoptimum phase properties are those for which the major axisof the quadrature uncertainty ellipse touches the origin of thecomplex amplitude plane. This uncertainty ellipse refers to agiven contour line of the Wigner function at a definite height(otherwise the uncertainty of rotated quadratures is an ellipticallemniscate [31]). This approach is ambiguous since the choiceof the height is arbitrary, and different heights lead to differentresults. This is the case in Refs. [28–30] that prescribe differentheights leading to very different conclusions. On the other hand,the arguments supporting these prescriptions are merely pictor-ial, lacking a definite quantitative relation with phase properties.

In this Letter we examine a complete quantitative formula-tion of the approximation of phase states by squeezed coherent

72 A. Luis / Physics Letters A 354 (2006) 71–78

states using different strategies of comparison. Feasible phasestates can be obtained via two main procedures: (i) Finding thesqueezed coherent states with minimum phase fluctuations for afixed mean number of photons, or (ii) finding the squeezed co-herent states with maximum overlap with phase states. To thisend, in Section 2 we recall proper measures of phase fluctua-tions and overlap. In Section 3 we show that there is a measureof phase fluctuations and a definition of overlap with phasestates that agree with the pictorial approach in Refs. [28,29].On the other hand, in Section 4 we show that the proposal inRef. [30] provides states allowing maximum accuracy in phase-shift detection in two-mode interferometers being close to an-other definition of overlap with phase states.

2. Uncertainty measures

There is a wide consensus concerning that the correct prob-ability distribution P(φ) for the phase φ of a single mode fieldis given by projection of the state of the system ρ on the idealphase states |φ〉 [1–5]

(1)P(φ) = 〈φ|ρ|φ〉, |φ〉 = 1√2π

∞∑n=0

einφ |n〉,

where |n〉 are number states and ρ is the density matrix.Among other possibilities, this phase formalism can be de-

rived from a phase difference operator by computing the phasedifference between ρ and a coherent state of high enough inten-sity [6,8,32]. Note that this procedure does not define a phaseoperator, but the positive operator measure in (1) (the vectors|φ〉 are not orthogonal), so that there is no inconsistency be-tween the existence of a phase-difference operator and the lackof single-mode phase operators.

The ideal phase states |φ〉 cannot be normalized so they can-not be generated in practice (proposals for the generation oftruncated versions of these states can be found in Refs. [33–35]). A normalized counterpart is provided by the so-calledphase coherent states |n, φ〉 [1–5]

(2)|n, φ〉 = 1√n + 1

∞∑n=0

(n

n + 1

)n/2

einφ |n〉,

where n is the mean number of photons 〈n, φ|a†a|n, φ〉 = n,being a the corresponding complex amplitude operator. A pro-posal for the generation of these states can be found in Ref. [36].

There is no so wide consensus concerning the proper assess-ment of phase fluctuations. The main drawback is caused by theperiodic nature of the phase, that may lead to serious difficultieswhen using standard measures of uncertainty such as variancesince the powers of φ are not periodic functions so that theirmean values depend on the origin of phase chosen. There arediverse proposals that avoid the periodicity problem, such asthe Süssmann measure, dispersion, and reciprocal peak height[37–56]. For the sake of completeness, these measures will becompared with the result of the phase variance when choosingthe phase origin that minimizes it [2,57].

2.1. Süssmann measure

The Süssmann measure of phase fluctuations is defined as[37–42]

(3)�sφ = 1∫dφ P 2(φ)

,

which can be regarded also as the inverse of the certainty intro-duced in Ref. [43]. This can be interpreted also as a measureof the distance D between P(φ) and the uniform distribution1/(2π)

(4)D =∫

dφ

[P(φ) − 1

2π

]2

= 1

�sφ− 1

2π.

This interpretation agrees with recent approaches to opticalproperties such as visibility, degree of polarization, and polar-ization correlations as suitable distances to uniform or factor-ized distributions [58–65].

From a computational perspective, the value of �sφ in thepure state

(5)|ψ〉 =∞∑

n=0

cn|n〉

can be computed as the inverse of

(6)∫

dφ P 2(φ) = 1

2π

∞∑N=0

∣∣∣∣∣N∑

n=0

cncN−n

∣∣∣∣∣2

.

For example, for the phase-coherent states (2) we get

(7)�sφ = 2π

1 + 2n,

that for n � 1 scales as the inverse of the mean number of pho-tons �sφ � π/n. This can be referred to as the Heisenberglimit. As a matter of fact, in Ref. [44] it has been shown thatthe phase coherent states (2) provide a minimum for �sφ.

2.2. Dispersion

The dispersion is defined as [45–52]

(8)�dφ =√

1 −∣∣∣∣∫

dφ eiφP (φ)

∣∣∣∣2

,

that for the state (5) becomes

(9)�dφ =√√√√1 −

∣∣∣∣∣∞∑

n=0

cnc∗n+1

∣∣∣∣∣2

.

2.3. Reciprocal peak height

Looking for states approaching phase states we may assesssuch a similarity in terms of the fidelity 〈φ|ρ|φ〉 between thesystem state ρ and the ideal phase states |φ〉. The measure ofuncertainty related to this idea is the reciprocal peak height [53–56]

(10)�pφ = 1,

P(φmax)

A. Luis / Physics Letters A 354 (2006) 71–78 73

where φmax is the value of φ that maximizes P(φ). This mea-sure should be used with some care since it depends exclusivelyon the value of the probability distribution at a single point[53–56].

2.4. Fidelity

Similarly, we may consider the fidelity F = 〈n, φ|ρ|n, φ〉between the system state ρ and the phase coherent states |n, φ〉in (2). By analogy with (10) we can define a measure of phaseuncertainty as

(11)�f φ = 1

〈n, φmax|ρ|n, φmax〉 ,

where φmax is the value of φ that maximizes the overlap be-tween |n, φ〉 and ρ, being n the mean number of photons in thesystem state ρ.

2.5. Phase variance

For the sake of completeness and comparison we can com-pute the usual phase variance

(12)�vφ =√⟨

φ2⟩ − 〈φ〉2,

with

(13)⟨φk

⟩ =φ0+2π∫φ0

dφ φkP (φ),

where φ0 is an arbitrary origin of phases [2,37,57]. From a com-putational perspective, for the pure state (5) these mean valuescan be computed as

〈φ〉 = φ0 + π +∑n=n′

−ieiφ0(n−n′)

n − n′ c∗ncn′ ,

(14)

⟨φ2⟩ = φ2

0 + 2πφ0 + 4π2

3

+∑n=n′

2eiφ0(n−n′)[

1

(n − n′)2− i

φ0 + π

n − n′

]c∗ncn′ .

It can be appreciated that �vφ depends on the origin ofphases φ0. This ambiguity can be solved by choosing in eachparticular case the value of φ0 minimizing �vφ.

3. Squeezed coherent states with minimum phaseuncertainty

The squeezed coherent states for a single mode field are [7,10–14]

(15)|ξ 〉 = D(R,θ)S(r,ϕ)|0〉,where |0〉 is the vacuum state, being D and S the displacementand squeezing operators, respectively,

Fig. 1. Illustration of the uncertainty ellipse in the complex plane showing thatits orientation is governed by δ = θ − ϕ/2. For δ = ±π/2 the major axis of theuncertainty ellipse is aligned along the radial direction.

D(R,θ) = eR(eiθ a†−e−iθ a),

(16)S(r,ϕ) = er(e−iϕa2−eiϕa†2)/2.

The coefficients of the expansion (5) of |ξ 〉 in the photon num-ber basis are

cn = (eiϕ tanh r)n/2

√2nn! cosh r

exp

[−R2

2

(1 + e−2iδ tanh r

)]

(17)× Hn

[R√

2

(e−iδ

√tanh r + eiδ

√tanh r

)],

where δ = θ − ϕ/2 and Hn are the Hermite polynomials. Themean number of photons is

(18)n = 〈ξ |a†a|ξ 〉 = R2 + sinh2 r.

The term R2 can be considered as the coherent contributionwhile the term sinh2 r is the squeezed contribution.

The idea is to vary the state parameters R, r , θ , and ϕ forfixed n in order to minimize the phase uncertainty. We can beginwith by optimizing the phase-angle variables θ , ϕ. It can be ap-preciated in (17) that these variables affect the uncertainty onlythrough the combination δ = θ − ϕ/2 that controls the orienta-tion of the uncertainty ellipse in the complex amplitude plane,as illustrated in Fig. 1.

In Fig. 2 we have represented the Süssmann measure �sφ

as a function of θ − ϕ/2 for n = 10 and R = 3 showing thatthe minimum occurs at δ = ±π/2. The same conclusion is ob-tained when using the other measures of phase uncertainty. Itcan be appreciated in Fig. 1 that this corresponds to the align-ment of the major axis of the uncertainty ellipse along the radialdirection [28,29].

In Refs. [28,29] it is proposed that the squeezed coherentstates closer to phase states are obtained when

(19)rmin = lnRmin,

since in this case the uncertainty ellipse in the complex am-plitude plane touches the origin. This ellipse refers to a givencontour line of the Wigner function. For fixed mean number of

74 A. Luis / Physics Letters A 354 (2006) 71–78

Fig. 2. Plot of �sφ as a function of δ = θ −ϕ/2 for n = 10 and R = 3 showingthat the minimum occurs at δ = ±π/2.

photons n the relation r = lnR leads to

(20)Rmin =√

2n + 1 + √(2n + 1)2 − 5

5�

√4

5n,

where the last expression holds for n � 1. According to (18)and (20) the squeezing provides a 20% of the total energy con-veyed by this state while the coherent part provides the remain-der 80%.

Another definition of touching based on a different contourline of the Wigner function can be found in Ref. [30]. This dif-ferent implementation of the same idea actually leads to a verydifferent result

(21)r ′min = ln

(2R′

min

),

which means

(22)R′min =

√2n + 1 + √

(2n + 1)2 − 2

8�

√1

2n,

where the last approximation holds for n � 1. In this casethere is a 50% splitting of the energy between the coherent andsqueezed parts.

Next, we avoid this arbitrariness by developing a fully quan-titative approach defining feasible phase states as the squeezedcoherent states with lesser phase fluctuations or large fidelitywith phase states.

3.1. Süssmann measure

In Fig. 3 we have plotted �sφ as a function of R for asqueezed coherent state with n = 10 and δ = π/2. The valuescorresponding to (20) and (22), are Rmin = 2.9 and R′

min = 2.3,respectively, being represented in Fig. 3 by vertical dashedlines. We can see that Rmin in (20) provides the minimum of�sφ very accurately. This coincidence occurs for all values ofn examined.

Moreover, numerical evaluations up to n = 15 reveal a linearrelationship between the inverse of the minimum of �sφ and n,as illustrated by the triangles in Fig. 4. More specifically,

(23)�sφmin � 1

0.31n + 0.18,

Fig. 3. Plot of �sφ as a function of R for a coherent squeezed state withδ = π/2 and n = 10. The values Rmin = 2.9 and R′

min = 2.3 determined by(20) and (22), respectively, are represented by vertical dashed lines. It can beappreciated that Rmin coincides very approximately with the minimum of �sφ.

Fig. 4. Plot of the inverse of the minimum uncertainty as a function of n whenusing the Süssmann measure (triangles), the dispersion (squares), and the recip-rocal peak (stars).

with a correlation coefficient 0.9999993. There is also a linearrelation between the value of R leading to minimum �sφ and√

n

(24)Rmin,s � 0.89√

n + 0.06,

with correlation coefficient 0.999996. This agrees very wellwith (20) so that the linear fit (24) corroborates that Rmin,s �Rmin as suggested by Fig. 3.

It must be appreciated that for n � 1 we get that �sφmin �3.2/n which is actually very close to the minimum �sφ � π/n

in (7) reached by the phase coherent states |n, φ〉 in (2). Thissuggests that the phase optimized squeezed coherent states |ξ 〉are actually very close to the phase coherent states |n, φ〉. Thisis confirmed also by the computation of the fidelity below.

3.2. Fidelity

Next, we examine the maximum of the fidelity F =|〈ξ |n, φ〉|2 in (11) between the squeezed coherent state |ξ 〉 andthe phase coherent state |n, φ〉. Concerning the variation of φ,for δ = π/2 we get that the maximum fidelity F occurs forφmax = θ , as illustrated in Fig. 5 for R = 3, ϕ = 0, θ = π/2,and n = 10. In Fig. 6 we have represented �f φ as a functionof R for δ = π/2, and n = 10. It can be appreciated that theminimum of �f φ occurs very approximately for the optimumstates with Rmin determined by (20).

A. Luis / Physics Letters A 354 (2006) 71–78 75

Fig. 5. Plot of the inverse 1/F of the fidelity F = |〈ξ |n, φ〉|2 as a function ofφ for a coherent squeezed state |ξ 〉 with θ = π/2, ϕ = 0, R = 3, and n = 10. Itcan be appreciated that the optimum phase φ corresponds to φmax = θ .

Fig. 6. Plot of �f φ as a function of R for a coherent squeezed state withδ = π/2 and n = 10. The values Rmin = 2.9 and R′

min = 2.3 determined by(20) and (22), respectively, are represented by vertical dashed lines. It can be ap-preciated that Rmin coincides very approximately with the minimum of �f φ.

Fig. 7. Plot of �f φmin as a function of n for coherent squeezed states withδ = π/2.

In this case we have found no relation between �f φmin andn equivalent to (23). As a matter of fact, Fig. 7 shows that�f φmin increases slightly for increasing n, contrary to the be-havior illustrated in Fig. 4, which is satisfied by all the othermeasures of phase uncertainty considered in this Letter.

Fig. 8. Plot of �dφ as a function of R for a coherent squeezed state withδ = π/2 and n = 10. The values Rmin = 2.9 and R′

min = 2.3 determined by(20) and (22), respectively, are represented by vertical dashed lines. It can beappreciated that they do not coincide with the minimum of �dφ.

3.3. Dispersion

The dispersion �dφ for δ = π/2 and fixed n presents a min-imum that does not corresponds neither to Rmin in (20) nor toR′

min in (22), as illustrated in Fig. 8 for n = 10.Numerical calculations up to n = 15 show that also in this

case there is a linear relationship between the inverse of theminimum �dφ and n, as illustrated by the squares in Fig. 4,

(25)�dφmin � 1

0.72n + 0.80,

with a correlation coefficient 0.9999997. There is also a linearrelation between the values of R giving minimum dispersionand

√n

(26)Rmin,d � 0.97√

n + 0.05,

which a correlation coefficient 0.9999993. This value is farfrom both Rmin and R′

min in (20) and (22). According to (26)minimum dispersion requires a 6% of squeezing which is lesserthan the amount required to reach the minimum of the Süss-mann measure and the maximum fidelity (20%).

These scaling laws can be distinguished from previous eval-uations of the dispersion for squeezed coherent states [30]. Thedifferences arise because the scaling laws in [30] are obtainedafter assuming some specific relations between squeezed andcoherent contributions in the splitting (18) for large enough n

in order to develop suitable approximations leading to analyti-cal results.

3.4. Reciprocal peak height

Concerning the reciprocal peak height �pφ in (10) we getthat the minimum phase uncertainty is obtained for a value ofR different from Rmin in (20), but somewhat close to R′

min in(22), as illustrated in Fig. 9.

Also in this case a numerical evaluation up to n = 15 revealsa linear relationship between the inverse of the minimum of

76 A. Luis / Physics Letters A 354 (2006) 71–78

Fig. 9. Plot of �pφ as a function of R for a coherent squeezed state withδ = π/2 and n = 10. The values Rmin = 2.9 and R′

min = 2.3 determined by(20) and (22), respectively, are represented by vertical dashed lines. It can beappreciated that Rmin does not coincide with the minimum of �pφ while R′

minis somewhat closer.

�pφ and n, as illustrated by the stars in Fig. 4, with

(27)�pφmin � 1

0.64n + 0.42,

with a correlation coefficient 0.9999994. There is also a linearrelationship between the value of R required to reach the mini-mum of �pφ and

√n

(28)Rmin,p � 0.67√

n + 0.06,

with a correlation coefficient 0.999993. It can be appreciatedthat the scaling as 0.67

√n is very similar to the scaling 0.71

√n

in (22), in agreement with Fig. 9.With this criterion we get that minimum phase uncertainty

would require a 55% of squeezing, which is larger than thesqueezing required to reach the minima of the Süssmann (20%)and dispersion (6%) measures.

3.5. Phase variance

For the sake of completeness and comparison the precedingresults can be compared with the predictions of phase variance.For definiteness, we focus on squeezed coherent states with δ =π/2, ϕ = −π , θ = 0. In such a case it can be seen that thephase origin φ0 that minimizes �vφ for all the optimum statesobtained so far (say, R � 0.7

√n for n � 15) is φ0 � −π being

P(φ) almost entirely concentrated in a peak around φ = 0.With this choice the phase variance �vφ becomes almost

identical to the dispersion �dφ. More specifically, for meanphoton numbers up to n = 15 we get

(29)�vφmin � 1

0.72n + 0.67,

with a correlation coefficient 0.999998 and

(30)Rmin,v � 0.97√

n + 0.06,

with a correlation coefficient 0.9999992, which are very similarto (25) and (26).

This coincidence is natural since for narrow enough distrib-utions P(φ) concentrated around φ = 0 we can expand eiφ in

(8) in powers of φ. If we retain terms just up to φ2 we get that�dφ becomes �vφ [45,51].

Moreover, it is worth noting that the scaling law (29) isequivalent to the scaling laws derived in [49,57] for the idealstates of light that minimize the phase variance (or approxi-mately the phase dispersion) for fixed mean number of photons.

4. Optimum squeezed coherent states for phase-shiftdetection

In this section we reexamine the same issue under a slightlydifferent perspective. We look for squeezed coherent statesleading to maximum accuracy in the detection of small phaseshifts in two-beam interferometers [6,7,10–14,66,67].

The detection of output intensities in a two-beam interfer-ometer corresponds to the measurement of a Stokes operatorfor the input field, say Sy = i(a†

1a2 − a†2a1), where a1, a2, are

the complex amplitude operators for the two input modes. Thisobservable is sensitive to phase changes in the complex ampli-tudes a1, a2, so the measurement of Sy allows to infer the valueφ of a phase shift produced in one of the modes, say a1. In turn,the fluctuations of Sy lead to some uncertainty �iφ in the in-ferred value for the phase shift φ. A very simple assessment ofthis uncertainty is provided by the error propagation [10–14,66,67]

(31)�iφ = �Sy∣∣ ∂〈Sy 〉∂φ

∣∣ ,where (�Sy)

2 is the variance of the Stokes operator Sy . We willconsider small phase shifts φ � 1 so that the right hand side ofthe above equation is evaluated at φ = 0.

In order to fit with the main objective of this work we con-sider that the input state is the product of two identical squeezedcoherent states |ξ 〉1|ξ 〉2, with θ = 0 for simplicity, representingfeasible states with well defined phase difference. In order tocompute �iφ we use the transformation law

(32)U†ajU = cosh r aj − eiϕ sinh r a†j + Reiθ ,

where U = D(R,θ)S(r,ϕ) is the unitary operator in (15). It canbe easily seen that for φ → 0

〈Sy〉 � 2R2φ,

(33)

(�Sy)2 � 2n2 + 2n

− 2(R4 + cosh2 r sinh2 r

− 2R2 cosh r sinh r cosϕ).

Minimum fluctuations would occur provided that ϕ = π ,which is exactly the same condition δ = ±π/2 obtained abovein Section 3. In such a case, and taking into account (18), weget

(34)

(�iφ)2 = 1

2R4

{n2 + n −

[R2 +

√(1 + n − R2

)(n − R2

) ]2}.

Assuming n − R2 � 1 we have

A. Luis / Physics Letters A 354 (2006) 71–78 77

Fig. 10. Plot of �iφ as a function of R for n = 100. The vertical bars repre-sent Rmin = 8.97 and R′

min = 7.1 as specified by (20) and (22), respectively.We can appreciate that R′

min provides the minimum while Rmin provides veryapproximately the same �iφ.√(

1 + n − R2)(

n − R2)

(35)� (n − R2)[1 + 1

2(n − R2)− 1

8(n − R2)2+ · · ·

],

leading to

(36)(�iφ)2 = 1

8R2(n − R2).

A typical plot of �iφ as a function of R is represented in Fig. 10for n = 100.

The minimum in (36) is given by

(37)�iφmin = 1√2n

, Rmin,i = √n/2,

which means a 50% of energy of the squeezed part. We cansee that Rmin,i coincides with R′

min in (22) being different fromRmin in (20). We can see that �iφmin scales as the Heisenberglimit. Incidentally, it can be noticed that the idea of the Heisen-berg limit as the minimum uncertainty in phase shift detectionallowed by quantum physics holds exclusively for phase shiftsgenerated by linear transformations, while nonlinear transfor-mations can beat it as discussed in Refs. [68–72].

Although Rmin,i = Rmin we notice that for Rmin we get that(36) leads to

(38)�iφ = 0.88

n.

This is not the minimum uncertainty �iφmin in (37), but asa matter of fact is very close to the true minimum �iφmin �0.71/n. In this regard Fig. 10 shows that Rmin is optimum in thesense that it combines minimum squeezing (i.e., large R) withalmost minimum phase uncertainty reaching the Heisenberglimit. This result is interesting since it is known that squeez-ing is very sensitive to practical imperfections. Therefore, theRmin in (20) combines small phase uncertainty with large ro-bustness against practical imperfections.

5. Conclusions

We have addressed a complete quantitative determination ofthe squeezed coherent states with minimum phase uncertainty

for a fixed mean number of photons using diverse measuresof phase fluctuations. All these measures reveal that squeezedcoherent states can be actually very close to phase states, as il-lustrated in particular by the fidelity.

Moreover, we have found that both the Süssmann measureand the fidelity with phase coherent states agree with a previ-ous pictorial approach to the problem based on the size of thecontour of the Wigner function at a given height. The Süss-mann measure and the fidelity with phase coherent states some-what reproduce the intuitive idea that phase optimized statesmust have minimum phase uncertainty being simultaneouslythe closer ones to the phase states that determine the phase sta-tistics (but nevertheless note that the statistics is determined byideal phase states instead of coherent phase states). This intu-itive idea is not reflected by the dispersion.

We have examined the performance of the phase optimizedsqueezed coherent states as input states for the interferomet-ric detection of phase shifts. We have shown that the optimumstates differ from the conclusions derived from the Süssmannmeasure and the fidelity with phase coherent states. However,they almost coincide with the fidelity with ideal phase statesand the approach based on the size of the contour of the Wignerfunction at a different height. Therefore, these measures appearto be mutually consistent and intuitive.

We have also found that the optimal states for Süssmannmeasure and fidelity combine very large accuracy (scaling asthe Heisenberg limit) with robustness against practical imper-fections.

The interest of this approximation relies on the fact thatthere are no feasible practical procedures to generate phasestates, while squeezed coherent states can be easily generated.Therefore, this approximation would allow to devise practicalschemes dealing with abstract phase concepts.

Acknowledgement

This work has been supported by project No. FIS2004-01814 of the Spanish Dirección General de Investigación delMinisterio de Educación y Ciencia.

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