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Unambiguous discrimination of nonorthogonal quantum states in cavity QED R.J. de Assis a , J.S. Sales b , N.G. de Almeida a,∗ a b

Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia, GO, Brazil Campus de Ciências Exatas e Tecnológicas, Universidade Estadual de Goiás, 75132-903, Anápolis, Goiás, Brazil

a r t i c l e

i n f o

Article history: Received 20 April 2017 Received in revised form 23 June 2017 Accepted 11 July 2017 Available online xxxx Communicated by V.A. Markel Keywords: POVM Generalized measurements Cavity QED

a b s t r a c t We propose an oversimpliﬁed scheme to unambiguously discriminate nonorthogonal quantum ﬁeld states inside high-Q cavities. Our scheme, which is based on positive operator-valued measures (POVM) technique, uses a single three-level atom interacting resonantly with a single mode of a cavity-ﬁeld and selective atomic state detectors. While the single three-level atom takes the role of the ancilla, the single cavity mode ﬁeld represents the system we want to obtain information. The eﬃciency of our proposal is analyzed considering the nowadays achievements in the context of cavity QED. We also analyze the effect of a thermal environment to discrimination of nonorthogonal states. © 2017 Elsevier B.V. All rights reserved.

1. Introduction

2. General measurements

Positive operator-valued measures (POVM) generalizes all possible kind of measurements [1,2] and cannot be reduced to standard projections of the initial state onto orthogonal states spanning the initial Hilbert space alone, pertaining to the system we want to obtain information [3,4]. In fact, although in general POVM can always be realized as standard projective measurements on an enlarged system [3], they are such that the number of outputs may be larger than the dimensionality of the space of states of the system in which we are interest in. POVM is now standard in several areas of quantum mechanics, including quantum optics and quantum information, among others [4–7]. In this paper we show how to accomplish POVM to unambiguously discriminate nonorthogonal ﬁeld states inside high-Q cavities. The goal of unambiguous quantum state discrimination (UQSD) is to discern in which state the system was prepared [8–11], founding many applications in several protocols [9,10], mainly for quantum cryptography [12–15]. Our scheme, employing one three-level atom interacting with a single mode of a cavity ﬁeld, is very simple from the experimental point of view and can be implemented using nowadays techniques in cavity QED. We begin by reviewing the general quantum measurement theory. Next, we present our model and results, comparing with the simple case of projective measurements. Then we present our conclusions.

Consider a quantum system we are interested to measure, and a second quantum system we call the ancilla, which is used to get information about the system of interest [2]. Let the Hilbert space dimension of the system of interest and the ancilla as K and L, respectively. The ancilla is prepared in some known initial state independently of the system of interest, and then the two systems are allowed to interact, getting correlated. Next, a von Neumann measurement is performed on the ancilla, providing us with information about the system of interest, which we are going to call the system from now on. Let us call the initial state of the ancilla as |a0 in the basis {|ak }, k = 0, 1, . . . , K − 1, and denote the initial density operator of ancilla-system as

*

Corresponding author. E-mail address: [email protected] (N.G. de Almeida).

http://dx.doi.org/10.1016/j.physleta.2017.07.017 0375-9601/© 2017 Elsevier B.V. All rights reserved.

ρ A S = |a0 a0 | ⊗ ρ S ,

(1)

where ρ S is the initial density operator of the system. Let U denote the ancilla-system evolution operator. Since U acts in the tensor-product space, it can be written as

U=

|ak ak | ⊗ M kk

(2)

kk

where

M kk =

uklk l |sl sl | ,

(3)

ll

being {|sl }, l = 0, 1, . . . , L − 1, a set of basis states for the system, and uklk l = ak , sl | U |ak , sl are the matrix elements of U . Note that M kk acts in the Hilbert space of the system, and since the

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2

H = h¯ g 1

Fig. 1. Scheme of the experimental setup to implement POVM in cavity QED. A three-level atom interacts with a single mode of a high Q cavity. The cavity mode ﬁeld is prepared in one of two nonorthogonal states |ψ1 and |ψ2 . Our POVM discriminates between these two nonorthogonal states prepared inside the high Q cavity.

space of system has dimension L, each sub-block matrix M kk has dimension L × L. From now on we use M k to refer to the ﬁrst column of the sub-block M k0 of U . Denoting the K × K sub-blocks of the matrix U † U by B kk , and since U † U = I , it is readily seen that

B 00 =

†

Mk Mk = I .

U |a, 1 = α cos ( g 1 t ) |a, 1 −

⎛

⎜ − iβ ⎝

where

pm = Tr

pm

⎜ − iα ⎝

(6)

† Mm Mm

ρS

+α

†

3. Model In our proposal, see Fig. 1, a three-level atom in ladder conﬁguration, described by the set of states {|a , |b , |c }, is initially prepared in |a and crosses a Ramsey zone (carrier interaction). Next, the atom enters a cavity interacting on-resonance with a singe mode of a cavity ﬁeld which in turns is prepared either in state |ψ1 or in state |ψ2 , for which ψ1 |ψ2 = 0 (nonorthogonal states). While inside the cavity, the atom suffers a Stark shift in order to lead ωba = ωcb = ω [16,17]. After the atom crosses the cavity, it is detected in one of its three possible states, thus revealing in which state the cavity mode was prepared. The Hamiltonian model of the atom-ﬁeld interaction into the cavity is given by [5]

2g 12 + g 22 t |a, 2 −

2g 12

cos

⎞

√

2g 1

2g 12 + g 22

2g 1 g 2

2g 12 + g 22

× |c , 0 − i β sin

(8)

of operators { M k } satisfying k Mk Mk ≡ k E k = I describes a possible measurement on a quantum system, with the measuring having K outcomes. This gives us a complete description of a quantum system under a general measurement. Next, we use the above results to discriminate one of two nonorthogonal ﬁeld states prepared into a high Q cavity.

⎟ ⎠ sin

+ β cos 2g 2 t |b, 2 + √

is the probability of ﬁnding the ancilla in state |am after the unitary evolution U . It is now promptly recognized that every set

⎟ ⎠ sin

(10)

2g 12 + g 22 t

(11)

|a, 2

−1

2g 12 + g 22 t |b, 1 +

√

(7)

,

⎞

2g 12 + g 22

⎛

†

Mm ρ S Mm

2g 1

2g 12 + g 22

U |a, 2 = α 1 +

From Eq. (6) we can write the normalized state of the system as

ρS ,m =

√

or, in terms of the sub-blocks of U :

ρA S ,m = |am am | ⊗ Mm ρ

(9)

− i α sin ( g 1 t ) |b, 0 +

+ β cos 2g 12 + g 22 t |b, 1 − ⎛ ⎞

g2 ⎜ ⎟ 2 2 − iβ ⎝

2g 1 + g 2 t |c , 0 , ⎠ sin 2g 12 + g 22

The important point to note here is that M k can be chosen to be any set of operators, provided the restriction Eq. (4) above is obeyed. Now, performing a von Neumann measurement on the ancilla states, represented by |am am |, we can write the (unnormalized) collapsed evolved state of both ancilla and system as

† S Mm .

U |a, 0 = α |a, 0 − i β sin ( g 1 t ) |a, 1 + β cos ( g 1 t ) |b, 0 ,

k

(5)

where σην = |η ν | and a† (a) is the creation (annihilation) photon number operator, and g i is the atom-ﬁeld coupling, which we take as real positive for convenience. In this protocol we are interested in discriminating nonorthogonal states which are combinations of the Fock states |0 and |1. Thus, since the maximum number of photons in this case is n = 2, which happens when the atom decays and increases the photon number into the cavity, we can consider a† |2 = 0. After a little algebra, it is straightforward to obtain

(4)

ρA S ,m = (|am am | ⊗ I ) U ρ A S U † (|am am | ⊗ I )

σba a + σab a† + h¯ g2 σcb a + σbc a† ,

cos

√

2g 12

+

g 22 t

−1

2g 2 t |c , 1 ,

(12)

where U = U C U R Z , with U C = e −i Ht /¯h , H given by Eq. (9), and U Z R is the evolution operator as given by the carrier or Ramsey zone: U R Z |a = α |a + β |b. Following the standard procedure [4], we must build three POVM elements: the one that discriminates |ψ1 ; the other one that discriminates |ψ2 , and a third one leading to inconclusive results with probability p in . It is to be noted that the only constraint obeyed by the POVM elements is k E k = I . Thus, as soon as the atom state is known, we will know with certainty that the cavity mode ﬁeld was either in state |ψ1 or in state |ψ2 , or that we do not know the initial state as a result of the inconclusive measurement. As explained above, to build the three POVM elements †

E ν = Mν Mν ,

Mν =

ν = a, b, c, we have to calculate n, n = 1, 2, 3 :

u ν nan |n n ,

(13)

nn

and

u ν nan = ν , n| U a, n .

(14)

Using Eqs. (10)–(12), we calculate the following operators in Eq. (13):

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M a = α |0 0| − i β sin ( g 1 t ) |1 0| + α cos ( g 1 t ) |1 1| −

⎛

⎜ − iβ ⎝

√

⎞

2g 12

+α 1+

+

⎟ ⎠ sin

2g 1 g 22

2g 12 + g 22 t |2 1| +

2g 12

cos

2g 12 + g 22

2g 12 + g 22 t

−1

(15)

+ β cos ⎛

⎜ − iα ⎝

2g 12 + g 22 t |1 1| −

⎞

√

2g 1

2g 12 + g 22

+ β cos ⎛

√

g2 2g 12 + g 22

2g 1 g 2

2g 12 + g 22

√

⎟ ⎠ sin

cos

2g 12

+

g 22 t

2g 12 + g 22 t

⎦ sin

2g 12 + g 22 t |1 1| +

+

2g 12 + g 22

2

2g 2 t

−1

g 22 t

2g 12

+

g 22 t

α β ∗ |1 2| − α β ∗ |2 1| +

cos

2g 12 + g 22 t

2 −1

+

|2 2| .

(20)

− 1 |0 2| − (17)

∗

∗

2g 12 2g 12 + g 22

g 22 t

E b = |β|2 cos2 ( g 1 t ) |0 0| −

−1

sented by ρ = q1 |ψ1 ψ1 | + q2 |ψ2 ψ2 |, where q1 (q2 ) is the classical probability related to the frequency of preparing the state |ψ1 (|ψ2 ) and q1 + q2 = 1. Clearly, E c discriminates state |ψ2 , since ψ1 | E c |ψ1 = 0. To discriminate |ψ1 , we impose ψ2 | E b |ψ2 = 0. This imposition leads us withthe conditions: (i)

α = cos ( g1t ), (ii) β = i sin ( g1t ), and (iii) cos

Letting g 2 =

+ i sin ( g 1t ) cos ( g 1t ) α β |0 1| − α β |1 0| + + |α |2 cos2 ( g 1 t ) +

2g 12 2 2 2 2 |1 1| + + |β| sin 2g 1 + g 2 t 2g 12 + g 22 ⎛ ⎞ √ 2g 1 ⎟ ⎜ + i ⎝

⎠ 2 2g 1 + g 22

2g 12 2 2 cos 2g 1 + g 2 t − 1 × 1+ 2g 12 + g 22 × α β ∗ |1 2| − α β ∗ |2 1| +

+

⎤

2g 12 g 22

√

(19)

2

× cos

sin2

32

+

As can be checked, ν Eν = I . To be speciﬁc, let us assume that we want to discriminate the following nonorthogonal ﬁeld states into the cavity: |ψ1 = |0 and |ψ2 = √1 (|0 + |1) [4]. The cavity state is thus repre-

2g 12

2g 12

|0 1| +

E a = |α |2 + |β|2 sin2 ( g 1 t ) |0 0| +

+ |β|2 sin2

From Eqs. (15)–(17) we can calculate the POVM elements E ν = † M ν M ν for ν = a, b, c:

2g 12 + g 22

(16)

2g 2 t |1 2| .

+ | α |2 1 +

× cos

2g 12 + g 22 t |1 2| +

2g 1 g 22

2g 12 + g 22 t

|2 2| ,

2g 2 t

√

2g 12 + g 22

+ |α | 2 ⎞

√

− i β sin

⎟ ⎠ sin

2g 2 t |2 2| ,

⎜ M c = −i β ⎝

+α

+i⎣

√

g 22

2

⎡

M b = β cos ( g 1 t ) |0 0| − i α sin ( g 1 t ) |0 1|

2

E c = |β|

sin2

2g 12 + g 22

+ |β| cos |2 2| ,

2g 12

2

2

+ |α |

3

θ=

2g 12 + g 22 t

= 0.

κ g1 , the third condition can be rewritten as g1t ≡

(m+ 12 )π # , m 2+κ 2

= 0, 1, 2, .... On the other hand, E a is inconclusive,

since ψ1 | E a |ψ1 = 0 and ψ2 | E a |ψ2 = 0, meaning that we must discard this measurement. Using these three conditions, we can write the effective POVM elements in the following way:

E a = cos2 θm + sin4 θm |0 0| +

+

1 4

sin2 (2θm ) (|0 1| + |1 0|) +

+ cos4 θm + Eb =

1 2

Ec =

2 |2 2| ,

(18)

− i sin ( g 1 t ) cos ( g 1 t ) α β ∗ |0 1| − α β ∗ |1 0| +

|1 1 | − + |α |2 sin2 ( g 1t ) + |β|2 cos2 2g 12 + g 22 t ⎛ ⎞ √

2g 1 ⎜ ⎟ − i ⎝

2g 12 + g 22 t cos 2g 12 + g 22 t ⎠ sin 2g 12 + g 22 ∗ × α β |1 2| − α β ∗ |2 1| +

2 2 + κ2

%$

sin2 θm |1 1| ,

sin2 (2θm ) ψ2⊥ ψ2⊥ ,

(21) (22)

κ2 sin2 θm |1 1| , 2 + κ2

(23)

where have neglected terms containing the state |2 and put ⊥ we ψ = √1 (|0 − |1). 2 2

The probabilities related to the success probability rates of POVM elements E b and E c are, respectively,

p b = Tr ( E b ρ ) =

q1 4

sin2 (2θm )

(24)

and

p c = Tr ( E c ρ ) =

q2 2

κ2 sin 2 (θm ) , 2 + κ2

(25)

while the probability for the inconclusive results p in = pa = Tr ( E a ρ ) = q1 ψ1 | E a |ψ1 + q2 ψ2 | E a |ψ2 is

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Fig. 2. Success probability p s versus the coupling ratio κ = g 2 / g 1 for q1 = 0.3. The best choice to UQED is the one that maximizes p s to each integer m.

p in = q1 cos2 θm + sin4 θm +

+

q2

Fig. 3. Success probability p s versus the coupling ratio κ = g 2 / g 1 for q1 = 0.7. The best choice to UQED is the one that maximizes p s to each integer m.

2

1 + cos θm +

2

2

2

2 + κ2

sin θm .

(26)

The success probability is given by p s = p b + p c :

ps =

q1 4

2

sin (2θm ) +

q2 2

κ2 sin 2 (θm ) , 2 + κ2

(27)

where p s + p in = 1, as should. 4. Discussion Since the result of POVM element E a is the inconclusive one, all we have to do in order to optimize our proposal is either maximize p s or minimize p in . We numerically maximize Eq. (27) with (i) q1 = 0.3, (ii) q1 = 0.7 and, for comparison to other work, (iii) q1 = q2 = 0.5 [4]. As an example, to q1 = q2 = 0.5 we ﬁnd, see Fig. 4, (a) for m = 0, p s = 0.1878 (black line with squares), implying κ = 1, 47, (b) for m = 1, p s = 0.2644 (red line with circles), implying κ = 4.50 (c) for m = 2, p s = 0.2748 (blue line with triangles), implying κ = 7, 50 (d) for m = 3, p s = 0.2779 (green line pentagons), implying κ = 10.55. Values of m ≥ 3 can be used at the expense of greater ratio κ = g 2 / g 1 , see Tables 1–3. The best choice to UQSD is the one that maximizes (minimizes) p s ( p in ) for each integer m, and it is worthwhile to mention that the success probability rate around 0.26, obtained for the ﬁrst values of the integer m, is very close to the best rate of success for this kind of quantum state discrimination predicted theoretically, which is 0.292 when q1 = q2 [4,7]. As can be seen from Figs. 2–3, there are several maxima in p s , depending on the value of κ . Here we have chosen those whose success probability is the greatest one. Note from Table 3, corresponding to q1 = q2 = 0.5, that the best value for the success probability rate is below 0.292, which is the maximum value according to Ref. [4,7]. This is also conﬁrmed by our numerical calculations using much greater values for m and κ . It is to be noted that the simple strategy of choosing whether to project the cavity ﬁeld state on the computational basis |0 or |1 would allow us to discriminate only one state. Indeed, if the result of projection is |1, the cavity mode state could not have been prepared in |ψ1 and was prepared therefore in state |ψ2 ; however, if the measurement result is |0, we can not be sure if the cavity mode state had been prepared in |ψ1 or |ψ2 , this result being inconclusive. As a result of this strategy, we would ﬁnd a success probability of p s = Tr (|1 1| ρ ) = 0.25, thus lesser than the

Fig. 4. Success probability p s versus the coupling ratio κ = g 2 / g 1 for q1 = 0.5. The best choice to UQED is the one that maximizes p s to each integer m.

Table 1 Values for the probabilities according to our protocol to accomplish UQSD in cavity QED. To each integer m, there is a minimum for the inconclusive events p in and a corresponding maximum for the probability of success p s . The table shows p in , p b , p c , and p s separately for q1 = 0.3. m

κ

p in

pb

pc

ps

0

1.52

0.8369

0.0747

0.0884

0.1631

1

3.72

0.7011

0.0366

0.2623

0.2989

2

5.71

0.6728

0.0155

0.3117

0.3272

3

7.63

0.6627

0.0069

0.3304

0.3373

4

9.54

0.6581

0.0032

0.3387

0.3419

5

11.44

0.6556

0.0015

0.3429

0.3444

10

21.23

0.6505

0.0012

0.3483

0.3495

20

41.11

0.6505

0.0001

0.3494

0.3495

50

101.05

0.6501

0.0000

0.3499

0.3499

POVM strategy we developed. In Tables 1–3 we present the values of κ maximizing p s and the corresponding values for p in , p b , p c and p s , Eqs. (24)–(27), for several integers m and q1 = 0.3, 0.7, 0.5. One could ask why to use POVM strategy instead of simply projecting on the computational basis of the cavity states. Three remarks are in order: (i) First, in addition to our protocol presenting a higher probability of success, there is no known technique to directly project the cavity state on the computational basis: usu-

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Table 2 Values for the probabilities according to our protocol to accomplish UQSD in cavity QED. To each integer m, there is a minimum for the inconclusive events p in and a corresponding maximum for the probability of success p s . The table shows p in , p b , p c , and p b separately for p 1 = 0.7. m

κ

p in

pb

pc

ps

0

1.45

0.7874

0.1749

0.0377

0.2126

1

5.19

0.7483

0.1693

0.0824

0.2517

2

8.54

0.7443

0.1679

0.0878

0.2557

3

12.27

0.7432

0.1674

0.0894

0.2568

4

15.80

0.7427

0.1673

0.0900

0.2573

5

19.32

0.7426

0.1671

0.0903

0.2574

10

36.91

0.7422

0.1670

0.0908

0.2578

20

72.08

0.7420

0.1670

0.0910

0.2580

50

177.58

0.7420

0.1670

0.0910

0.2580

Table 3 Values for the probabilities according to our protocol to accomplish UQSD in cavity QED. To each integer m, there is a minimum for the inconclusive events p in and a corresponding maximum for the probability of success p s . The table shows p in , p b , p c , and p s separately for p 1 = 0.5. m

κ

p in

pb

pc

ps

0

1.47

0.8123

0.1248

0.0629

0.1877

1

4.50

0.7356

0.1039

0.1605

0.2644

2

7.55

0.7252

0.0989

0.1759

0.2748

3

10.55

0.7221

0.0957

0.1822

0.2779

4

13.70

0.7209

0.0956

0.1835

0.2791

5

16.50

0.7101

0.0950

0.1849

0.2799

10

31.52

0.7191

0.0941

0.1868

0.2809

20

61.50

0.7189

0.0938

0.1873

0.2811

50

151.50

0.7188

0.0930

0.1882

0.2812

5. Effect of loss due to thermal environment In this Section we discuss the feasibility of our scheme in presence of loss. Although our proposal applies very well to other systems, we specialize our analysis to cavity QED in the microwave domain, where we can safely neglect the damping effect in the ﬁeld states, since cavities are now constructed with one-photon damping time T c = 0.129 ± 0.003 s [24]. This is much greater than the radiative time for Rydberg atoms, which is T a = 0.03 s [25]. Three-level atoms were considered in a number of protocols [25–29]. To our goal, we will consider, as in Ref. [25], three circular levels with principal quantum numbers 51, 50, and 49, where the c⇔b and b⇔a transitions are at 51.1 and 54.3 GHz, respectively (compare our Fig. 1 with Fig. 1 of Ref. [25]). Also, as the mean photon number is less than 0.1 in typical experiments [24, 25], we consider the reservoir at zero temperature and therefore in the vacuum state. If we include the reservoir operators and apply the general theory described in& Eqs. ' (1)–(8), it is possible to deduce the new . Now POVM elements E m pm = Tr E m ρS

is the probability of ﬁnding the ancilla in state |am after the unitary evolution U = U RC U R Z , where now U RC = e −i ( H + H R )t /¯h , and

†

(28)

†

H R = h¯ k ωk bk bk + h¯ k λkab σab bk + λkbc σbc bk + h.c . Here b j b j is the creation (annihilation) bosonic operator of the jth resermode, ωk is the kth reservoir oscillator frequency, voir oscillator λkab λkbc is the atom-reservoir coupling for the transition b⇔a (c⇔b). Note that, in principle, the Eq. (28) can provide conditions on POVM elements to unambiguously discriminate nonorthogonal states even in presence of losses. However, this is a formidable task, even for the simple case analyzed here. Therefore, we shall take another approach to estimate the effect of losses due to thermal environments to the discrimination of nonorthogonal quantum states. To this end, we will take advantage of the phenomenological operator approach (POA) introduced in Ref. [30,31], which has been proved to be an extremely simple and effective way to compute environmental errors [32]. Taking advantage of the formal equivalence between the internal three-level states {|a , |b , |c } and the Fock states {|0 , |1 , |2}, we can write the following relations [30] Ut

|a|E −→ | g T0 |E ,

(29)

Ut

ally, measurement of the cavity state requires additional atoms and/or cavities, thus being necessary to build another POVM elements to measure the cavity mode ﬁeld [18,19], (ii) second, the direct projective strategy does not discriminate both states but just |ψ1 = |0, while the POVM strategy allows us to discriminate both |ψ1 and |ψ2 , (iii) the POVM strategy was build using known matter-radiation interaction parameters: it remains an open question if other types of interaction, such as those developed by effective Hamiltonian techniques [20–22], could attain optimal POVM results [23].

5

† † |b|E −→ |bTb |E + |aTba |E , Ut

†

(30)

†

†

|c |E −→ |c Tc |E + |bTcb |E + |aTca |E .

(31)

Here U t stands for the unitary operation mixing the atom state |i , i = a, b, c to the environment state |E . The operators acting on this environment state and accounting for the atom-environment κ † † † † † coupling are T0 = I , Tb = e − 2 t I , Tba = j β j (t )b j , Tc =

√

† † † † β j (t )b j , Tca = k j βk (t )β j (t )bk b j , 2 − 2 t with , where denotes the damping rate j |β j (t )| = 1 − e

e −2 2 t I , Tcb = †

2e − 2 t

j

to the excited states, I is the identity operator, and t is the time spent after the atom undergoing an excitation. It is to be noted that the phenomenological-operator evolution method leads to the same atomic density operator ρdamped as the one we obtain using an ab-initio master equation approach [30–32]. In order to perform an estimate, we have to do some assumptions. Here we will assume that the Ramsey zone (see Fig. 1) is placed right before the cavity, such that the atom enters the cavity without decaying. Also, since the unitary operation occurs in typical interaction times ∼ 2 × 10−5 s [25], we will neglect possible decaying inside the cavity. Our estimate thus consider decaying during the time ﬂight of the atom until the selective detection. Applying the POA method above, we obtain

U R |a, n, R 0 = |a, n, R 0 U R |b, n, R 0 = e U R |c , n, R 0 = e

− 2 t

−t

(32)

|b, n, R 0 + |a, n, R 1 (33) √ −t |c , n, R 0 + 2e 2 |b, n, R 1 + |a, n, R 2 (34)

with

R 0 | R 0 = 1; R 1 | R 1 = 1 − e−t ; 2 R 2 | R 2 = 1 − e−t ; R i R j = 0, i = j ,

(35)

where we have deﬁned | R = T |E . It is now straightforward to consider the effect of losses when the cavity ﬁeld mode is prepared in either |ψ1 = |0 or |ψ2 = √1 (|0 + |1), which are the †

2

states we want to discriminate. The results are:

U |a, ψ1 , R 0 = cos (θm ) |a, 0, R 0 + i sin (θm ) cos (θm ) |a, 0, R 1 +

+ sin2 (θm ) |a, 1, R 0 +

+ i sin (θm ) cos (θm ) e− 2 t |b, 0, R 0 ,

(36)

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6

and

U |a, ψ2 , R 0 =

cos (θm )

√

2 1

|a, 0, R 0 +

sin (θm ) |a, 2, R 0 + + √ |a, 1, R 0 + √ 2 2 + κ2 1 κ +√ sin (θm ) |a, 0, R 2 + √ 2 2 + κ2 κ + √ sin (θm ) e− 2 t |b, 0, R 1 + 2 + κ2 1 κ +√ sin (θm ) e−t |c , 0, R 0 . (37) √ 2 2 + κ2 If we now remember the POVM strategy, i.e., that of imposing p c (ψ1 ) =Tr (|ψ1 ψ1 | ρs ) = 0 to the state |ψ1 and p b(ψ2 ) = Tr (|ψ2 ψ2 | ρs ) = 0 to the state |ψ2 , we clearly see that, according to Eqs. (36)–(37), even in presence of losses we can impose p c (ψ1 ) = 0. However, due to atomic decay, it is impossible to eliminate all the terms containing |b, n, R . Particularly, the term |b, 0, R 1 , which stem from decaying |c → |b, cannot be eliminated and is responsible for errors when we try to discriminate state |ψ2 . This means that unambiguously discrimination is now impossible for both states: whenever the result E b is obtained, we cannot anymore aﬃrm that the prepared state into the cavity was |ψ1 , since this can occur either because the prepared state was in fact |ψ1 or as a result of decaying |c → |b when the state was |ψ2 . We can estimate the error in trying to discriminate |ψ2 by calculating the new probabilities for ﬁnding p b(ψ2 ) in presence of losses, which is no longer zero. Below we give all the relevant probabilities calculated using the POA method:

p b = q1 p b(ψ1 ) + q2 p b(ψ2 )

(38)

or

pb =

q1 4

sin2 (2θm ) e−t +

+ q2

κ2 sin2 (θm ) e−t 1 − e−t . 2 + κ2

p c = q1 p c (ψ1 ) + q2 p c (ψ2 )

(39)

(40)

or, since p c (ψ1 ) = 0,

q2

2

κ2 sin2 (θm ) e−2t . 2 + κ2

(41)

Likewise, the probability related to inconclusive measurements is

pa = q1 pa(ψ1 ) + q2 pa(ψ2 ) or

(42)

pa = q1 cos2 (θm ) + sin4 (θm ) +

+ +

sin2 (2θm ) 1 − e−t

4 q2 2

+

1

2

1 + cos (θm ) +

can estimate numerically the losses due to thermal environment. As an example, consider Table 3 where q1 = q2 = 0.5, m = 1 and κ = g1 / g2 = 4.5, with g1 ∼ 104 s−1 , ∼ 102 s−1 [16,17,24,25]. As we said previously, the probability p b(ψ2 ) in Eq. (39) is zero only when there are no losses (Γ = 0) or when t = 0. Therefore, we can take p b(ψ2 ) as our error measure p error . In Fig. 5 we show the behavior of p b(ψ2 ) as time goes on. Taking the atom velocity to be around 100 m/s–500 m/s and the distance traveled by the atom inside the apparatus as 20 cm [25], then we can take t ∼ 10−1 to our estimate. According to Fig. 5, this gives an error around 3%, meaning that, as we have said above, instead of unambiguous discrimination, there will be around 3% of error whenever we try to discriminate between both states. We should stress that if we use other parameters to q1 , m, κ the error remains of the same order of magnitude. 6. Conclusion

Also:

pc =

Fig. 5. Error probability p error versus t, where is the damping rate.

Acknowledgements

+ 2

2 + κ2

We have proposed an oversimpliﬁed scheme to build POVM elements allowing to discriminate nonorthogonal ﬁeld states inside a high Q cavity. Besides to circumvent the impossibility to directly project the cavity states onto the computational Fock states without using ancilla, our protocol achieves a rate of success probability greater than the direct projective technique. Our proposal relies on nowadays techniques in the cavity QED domain, making use of just one single three-level atom undergoing a Ramsey zone (carrier interaction) plus one cavity and selective atomic state detectors. This simplicity makes our protocol very attractive from the experimental point of view. Using the phenomenological operator approach technique, we analyzed and estimated the effect of losses due to a thermal environment to the quantum state discrimination proposed here. Finally, we hope our protocol can inspire other POVM strategies based on effective Hamiltonians technique making possible to attain optimal rates of success probability.

We acknowledge ﬁnancial support from the Brazilian agency CNPq, CAPES and FAPEG. This work was performed as part of the Brazilian National Institute of Science and Technology (INCT) for Quantum Information.

2

sin (θm ) +

2 κ2 . sin2 (θm ) 1 − e−t 2 2+κ

(43)

It is now easily veriﬁed that pa + p b + p c = q1 + q2 = 1. It is to be noted that p b + p c → 0 in the limit t → ∞, while the probability for the inconclusive results pa(ψ2 )+ pa(ψ1 ) → 1. If we now use typical values to the atom damping rate used in experiments, we

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