Characterization of classical static noise via qubit as ...

Characterization of classical static noise via qubit as probe Muhammad Javed2 , Salman Khan1 ,∗ and Sayed Arif Ullah2 1

Department of Physics, COMSATS Institute of Information Technology, Park road, Islamabad 45550, Pakistan. and

2

Department of Physics, University of Malakand, Chakdara Dir, Pakistan (Dated: July 17, 2017)

Abstract The dynamics of quantum Fisher information (QFI) of a single qubit coupled to classical static noise is investigated. The analytical relation for QFI fixes the optimal initial state of the qubit that maximizes it. An approximate limit for the time of coupling that leads to physically useful results is identified. Moreover, using the approach of quantum estimation theory and the analytical relation for QFI, the qubit is used as a probe to precisely estimate the disordered parameter of the environment. Relation for optimal interaction time with the environment is obtained and condition for the optimal measurement of the noise parameter of the environment is given. It is shown that all values, in the mentioned range, of the noise parameter are estimable with equal precision. A comparison of our results with the previous studies in different classical environments is made. keywords: quantum information, decoherence, estimation theory PACS numbers: 03.67.a, 03.65.Yz, 03.65.Ta Keywords: quantum information, decoherence, estimation theory

∗

Electronic address: [email protected]

1

I.

INTRODUCTION

The processing of quantum systems is a necessary action for the successful implementation of every quantum protocol. More precisely, the realization of the foreseen quantum technology purely depends on the processing of quantum systems under a number of different circumstances. However, during the processing procedure the quantum system of interest may interact with the neighboring environment, and this is a well established fact that the interaction of environment results in the loss of encoded information from the principle quantum system [1]. The influence of different environments in their quantum pictures have been extensively studied and have provided deep insight into the dynamics of physically important quantities [2–5]. Many other studies have shown the possibility of describing a quantum environment in the classical picture [6–10]. In fact, in the limit of large dimensions, the classical picture of environment becomes more reliable than its quantum description. Open quantum systems from different perspective have been studied using classical picture of different environments and very important results have been obtained [11–14]. This means, the delineation of environment by itself is very pertinent to the successful implementation of quantum protocols. Due to impossibility of complete isolation of a quantum system from the surrounding environment, an alternative good strategy for constructing robust quantum protocols against the decohering effects of the environment might be the characterization of the environment itself. Several studies have focused on this point and have tried different ways of modeling for probing the environment via the decohering quantum system of interest [15–19]. A good approach to achieve this goal is to benefit from the techniques of quantum estimation theory (QET). The aim of QET is to first determine an unknown parameter labeling the quantum system from a measured data and then use it to increase the precision of resolution [20–26]. The techniques of QET have already been used to estimate quantum correlations [27, 28], interferometric phase-shift [29–31], spectral properties of Gaussian and non-Gaussian classical environments [18, 32] and the estimation of static parameters of quantum environments [33–38]. In these studies for the estimation of static parameters of quantum environments, the approach of symmetric logarithmic derivative, which, in turn, defines quantum fisher information (QFI) is used. It is important to mention that in the quantum mechanical picture of an environment, the noise parameter η(t) is embedded inside an operator KE (η), which 2

describes the dynamics of the constituent quantum particles of the environment. On the other hand, in the classical picture the operator KE (η) is replaced with the noise parameter η(t) and the system Hamiltonian becomes a stochastic one. For a static noise η(t) = η. One of the many important and deeply investigated tools of quantum information theory and QET is QFI. In quantum information theory, QFI can be used to investigate the nonMarkovianity of open quantum system and the statistical distinguishability on the space of the density operator in quantum information geometry [39–41]. On the other hand, QFI plays an important role in QET where its inverse provides a lower bound to the error of the estimation [20, 42]. Thus, for precise measurement of unknown parameters pertinent to the environment, the goal is to find ways for increasing QFI of an open quantum system. As mentioned above, this technique has been employed to investigate the unknown parameter of classical Gaussian and non-Gaussian noise while using qubit as a quantum probe [18, 32, 43]. We use two different definitions to derive analytical relations for QFI. In one case, the result is approximate whereas in the other case it is exact. We show that both the relations maximize for the same value of the initial state parameter and the maximum of QFI in both cases agree perfectly. The agreement between them fixes a unique optimal initial state of the qubit exposed to a classical static environment characterized by a flat probability distribution. The optimal initial state of the qubit is then used as a probe for estimating the unknown parameter of the environment. Some studies investigating the effects of such an environment on quantum system in different context are given in [14, 44]. The analysis of the dynamics of QFI provides information about the ranges of values of the parameters space that leads to the precision of physically viable results. We use the tools of QET and the qubit as a probe to characterize the disorder parameter of the environment through the optimal interaction time. Our results show that every value of disorder parameter of the environment in the mentioned range can be estimated with uniform precision.

II.

THE PHYSICAL MODEL

The Hamiltonian describing the interaction of a single qubit with a classical environment can be expressed as H(t) = εI + gχ(t)σx ,

3

(1)

where I and σx , respectively, are the identity and Pauli spin flip operators over the Hilbert space of a qubit, ε is the energy of an isolated qubit, χ(t) is a stochastic variable depending on the nature of classical environment and g is the coupling strength between the qubit and the environment, measured in inverse time units, and we will consider the unit of time arbitrary. In the case of static noise, the stochastic variable χ(t) is time independent and is characterized by flat probability distribution P (χ) = 1/∆χ for |χ − χ0 | ≤ ∆χ /2 and vanishes elsewhere [45–47]. The stochastic variable χ(t) represents different aspects of different physical systems. For example, in the case of a transmitting antenna, the stochastic variable χ(t) can be the noisy measurement of the angular position of a transmitting antenna with respect to a receiving antenna [48]. Similarly, in the case of a coupled array of waveguides, χ(t) represents the effective detuning of a particular waveguide [45]. The parameters χ0 and ∆χ quantify the mean value of the distribution and the disorder of the environment, respectively. The disorder parameter ∆χ can have different values depending on the nature of physical model. For example, in sigma-delta modulator, it can have value as large as 2 [49] and in the case of a hard disk drive rotating at 7200 rpm, the disorder parameter is the wait time defined as the time between the read/write head moving into position and the beginning of the required information appearing under the head, which can have value up to 8. The autocorrelation function of χ depends on the degree of disorder and is explicitly given by ⟨δχ(t)δχ(0)⟩ = ∆2χ /2. This dictates that its power spectrum is described by δ-function that peaks up at zero frequency and the memory effects do not wash out, as a result the static noise becomes non-Markovian in nature. The overall effect of static noise on the time evolved states of a system can be obtained by averaging the final density matrix over all the possible noise configurations. To this end, integration of the final density matrix over the stochastic variable χ between χ− and χ+ with χ∓ = χ0 ∓ ∆χ /2 is employed. The final density matrix reads ∫

χ+

ρ(χ, t) = χ−

( ) dχP (χ) U (χ, t) ρi U † (χ, t) ,

(2)

where U (χ, t) describes the time evolution of the initial density matrix ρi of the system in the presence of environment and is given by U (χ, t) = e−i

4

∫

H(t)dt

, with ~ set to one.

Estimation Theory

Estimation theory is an approach used to find an unknown variable η from an observed data ξi that is somehow linked to η. The goal is to construct a function f from the observed data ξi such that a random variable ηˆ, which works as an estimator, can be defined in terms of the function as ηˆ = f (ξi ). A number of different estimation techniques can be used to view η as a deterministic variable, however, the most commonly used one is the minimum variance unbiased estimator (MVU). The MVU can be obtained from minimum mean square measure by limiting the bias of the data to zero [20, 42]. Putting a lower bound on the variance of any unbiased estimator allows us to assert that an estimator is the MVU estimator. Although, many such bounds on the variance exist, the Cramer-Rao lower (CRL) bound is the easiest of all of them. It is given by var(ˆ η) ≥

1 , M F (η)

(3)

where M is the number of repeated observations and F (η) represents the Fisher information (FI). It is defined as F (η) =

∑

[ ]2 p(ξi |η) ∂η log p(ξi |η) ,

(4)

i

where p(ξi |η) is conditional probability with output ξi when the unknown parameter attains its true value η. In the quantum case, the conditional probability can be obtained from projective measurement on the quantum mechanical state ρ through p(ξi |η) = tr(Πi ρ), where ∑ {Πi } with i Πi = I are the elements of positive operator-valued measure (POVM). The CRL bound is analogously extended to quantum regime, by maximizing FI over all POVMs, for estimating the unknown parameter η, describing a family of quantum mechanical states ρ(η). In this case, it is known as quantum CRL (QCRL) bound and is given by var(η) ≥

1 , M H(η)

(5)

where H(η) is known as quantum Fisher Information (QFI). With the spectral decomposition ∑ ρη = n λn |ψn ⟩⟨ψn | of the density matrix, QFI can be expressed as follows H(η) =

∑ (∂η λp )2 p

λp

+2

∑ (λn − λm )2 |⟨ψm |∂η ψn ⟩|2 . λ n + λm n̸=m

(6)

In Eq. (6), the first term on the right side is constituted by the differential of eigenvalues and is equivalent to the classical FI, whereas the second term comes from the differential 5

of the eigenvectors of the density matrix with respect to the estimable parameter η and is quantum mechanical in nature. If the eigenvalues or the eigenvectors are independent of η, the corresponding term vanishes. An optimal measurement of the parameter is one at which the FI becomes equal to QFI. Moreover, an efficient estimator is one for which the equality in Eq. (5) holds. Also, it has been proven that for a density matrix ρ(η), representing a two dimensional system, QFI can be expressed as follows [50, 51] ( )2 H(η) = Tr ∂η ρ(η) +

( )2 1 Tr ρ(η) ∂η ρ(η) . detρ(η)

(7)

An important measure of the estimability of a parameter is the signal-to-noise ratio (SNR) and is given by SN R = η 2 /var(η). In the quantum limit, using QCRL bound a measure known as quantum signal-to-noise ratio (QSNR) can be defined which bounds SNR. With the help of Eq. (5), it can be expressed as R(η) = η 2 H(η), which represents the ultimate quantum bound to the estimability of a parameter. The parameter η is said to be easily estimable provided the corresponding R is large.

III.

RESULTS AND DISCUSSION

In this part, using the tools developed so far, we present the results of our study. Considering the initial state of the qubit to be |ψ(0)⟩ = cos θ/2|0⟩ + sin θ/2|1⟩ and using Eq. (2), the density matrix of the system after evolving in the noisy environment can be expressed 



as follows

ρ(t) = 

1 (2 4

+ X)

1 (sin θ 2

− iY )

1 (sin θ 2 1 (2 4

+ iY )

− X)

.

(8)

The new parameters appearing in Eq. (8) are defined as cos θ (sin[tg(2χo + ∆χ )] − sin [tg(2χo − ∆χ )]) , tg∆χ cos θ sin(2tgχo ) sin(tg∆χ ) Y = . tg∆χ

X=

(9)

The two eigenvalues of the final density matrix of Eq. (8) are given by λ1,2 = (2 ± Θ)/4 √ with Θ = 2 + X 2 + 4Y 2 − 2 cos(2θ). The corresponding two non-normalized eigenvectors can be expressed as follows |ψ⟩1,2 = −

X ±Θ |0⟩ + |1⟩. 2(iY − sin θ) 6

(10)

The normalized forms of these vectors can be obtained by dividing each of them with its norm given by √ ||ψ1,2 || =

1+

(X ± Θ)2 . 4(Y 2 + sin2 θ)

(11)

Now, using the values of λ1,2 , the first part on the right of Eq.(6) can be expressed as follows H1 (∆χ ) =

( ∆2χ

−4 cos2 θ sin2 (tg∆χ ) (−tg∆χ cos(tg∆χ ) + sin(tg∆χ ))2 ). [ ]2 4 2 4 2 −2(tg∆χ ) + 2 sin (tg∆χ ) + (tg∆χ ) − sin (tg∆χ ) cos(2θ)

(12)

As can be seen from the forms of Eqs. (10) and (11), along with the expression for Θ, it is

0.25

gt =1

0.20

gt =0.8 gt =0.6

gt =0.2

0.15

H

(

)

gt =0.4

0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

FIG. 1: The numerical simulation of Eq.(6) against θ for ∆χ = 2 and gt = 0.2, 0.4, 0.6, 0.8, 1.

very difficult to obtain an analytical result for the second part on the right side of Eq.(6). In order to observe its contribution to QFI, we first simulate Eq.(6) numerically and plot the results against θ in figure (1) for different values of the dimensionless quantity gt. As can be seen from the figure, for all choices of gt the QFI is maximum at θ = 0. Next, we simulate only the second part on the right of Eq.(6) for θ = 0 and our simulations show that the contribution of the second term on the right of Eq.(6) is of the order of 10−31 , which can readily be set to zero. With this approximation, the QFI is given by Eq.(12), that is,

7

H(∆χ ) = H1 (∆χ ). Now, maximizing Eq.(12) with respect to θ shows that the maximum corresponds to θ = 2 nπ, (n = 0, 1, 2 ...). With this choice of θ, Eq.(12) reduces to [ H(∆χ ) =

]2 − tg∆χ cos(tg∆χ ) + sin(tg∆χ ) [ ] . ∆2χ t2 g 2 ∆2χ − sin2 (tg∆χ )

(13)

On the other hand, substituting Eq.(8) directly into Eq.(7), leads to the following expression for QFI H(∆χ ) =

cos2 θ [−tg∆χ cos(tg∆χ ) + sin(tg∆χ )]2 [ ] . ∆2χ t2 g 2 ∆2χ − sin2 (tg∆χ )

(14)

The form of Eq.(14) straightaway dictates that it is maximum for θ = 2 n π, and it reduces to Eq.(13) for this choice of θ. This means that approximating the second term on the right of Eq.(6) to zero does not affect the maximum of QFI and thus is valid in this case. At this point, it is important to mention that Eq.(6) is equivalent to Eq.(8), the analytical form of the second part on the right of Eq.(6) can also be obtained by subtracting Eq.(14) from Eq.(13). The result is given by H2 (∆χ ) = −

4 cos2 θ (−tg∆χ cos(tg∆χ ) + sin(tg∆χ ))2 sin2 θ . ∆2χ (−1 − 2(tg∆χ )2 + 2 cos(2tg∆χ ) cos2 θ + (−1 + 2(tg∆χ )2 ) cos(2θ))

(15)

One can see that the numerator of the above last equation vanishes for θ = 0, which confirms the validity of the approximation made above. Unlike the case for different Gaussian noises where the QFI is maximum for θ = π/2 [32], our result is parallel to the dynamics of QFI in the presence of random telegraphic noise [18], however, the rest of its behavior is completely different in the static noise as we discuss next. Now, for θ = 2 n π, QFI becomes equal to FI and hence the measurement of the unknown parameter becomes optimal and the corresponding optimal initial state of the qubit becomes |ψ(0)⟩ = |0⟩. One can see from Eq.(13) that both the coupling constant and the time of coupling appear on the same footing, they both affect QFI identically, however, this is not true for the parameter ∆χ . Also, like the dynamics of entanglement and quantum discord [14, 44], QFI is independent from χ0 . Figure (2) shows the behavior of QFI against ∆χ and t for two different choices of the coupling strength g = 0.1, 2 with arbitrary units both for g and t. The qualitative behavior in the two cases is similar, overall it decays with the increasing value of ∆χ . However, for every value of ∆χ there exists a time of coupling at which QFI is maximum. The pair of values, for the two parameters, that maximizes QFI are visualized by the thick solid curve 8

c t) (D , max

un it

)

H (D c)

t (a

Dc

rb.

t (a rb.

un

it)

H (D c)

c t) (D , max

Dc

(a)

(b)

FIG. 2: The behavior of quantum fisher information against ∆χ and t with arbitrary units of g for (a) g = 0.1, (b) g = 2.

over the plots. In the case of weak coupling between the system and environment, the values of the parameters in each pair is large as compared to their values in each pair for the case of strong coupling. As can be observed, for large value of ∆χ , small value of coupling time maximizes QFI, however, the maximum value by itself is extremely small. With reference to Eq. (5), this means that short time coupling between the system and environment is of little importance for estimation purposes as it results in large variance. The same information can jointly be obtained by plotting QFI against the dimensionless quantity gt, such as shown in figure (3). In figure (4) we show the behavior of R(∆χ ) against ∆χ and t for the same two values of g as in Figure (2). A closed observation shows that the behavior in the weak and the strong coupling limits are both qualitatively and quantitatively different. In the weak coupling regime, the frequency of revivals of the maximum is considerably smaller than in the case of strong coupling regime. Similarly, the maximum value of R(∆χ ) in the weak coupling regime is 1.14, which is large than 0.99 for the strong coupling case. The revival of the maximum of QSNR occurs in time. However, a closed and careful observation of figure (4), reveals that the peaks are not equally spaced. For example, in figure (4b), for ∆χ = 3, the first peak happens at π/7, the second at π/3 and the third at π/2, which are not the multiples of each other. Moreover, for every value of ∆χ an optimal time (topt ) exists, which is a 9

H (D c)

c gt) (D , max

gt Dc

FIG. 3: The behavior of quantum fisher information against ∆χ and the dimensionless quantity gt.

function of ∆χ and maximizes R(∆χ ). However, unlike the behavior of R(∆χ ) in Gaussian and non-Gaussian noises [18, 32], where the maximum decreases with the increasing value of the unknown parameter, the maximum in this case remains the same for every pair of (∆χ , topt ). Again, unlike Gaussian process, this means that for estimation purposes both the smaller and larger values of ∆χ are equally estimable. In other words, the short and long time exposure of the qubit to the environment equally imprints the effects of the environment on the probe. This behavior of QSNR in the presence of static noise is parallel to its behavior in non-Gaussian noises [18]. The optimal time topt is inversely related to ∆χ through the coupling constant g and can be expressed as topt = a/(g∆χ ), (a = 2.74). Its behavior against ∆χ for two different values of g is shown in figure (5a), which shows that for every g the maximum uniformly shifts to short time of interaction with increasing values of ∆χ . However, in strong coupling the maximum appears in short time, which means the environment quickly induces its effects into the qubit. Figure (5b) shows the behavior of QFI for optimal time which varies as inverse of the square of ∆χ and can explicitly be expressed as H(∆χ ) = b/∆2χ , (b = 1.14). With this optimized value of QFI the QSNR becomes constant and is given as R(∆χ ) = ∆2χ H(∆χ ) = b. The constant value of R(∆χ ) as a result of optimized H(∆χ ) is an assertion to our statement that all values of the considered range of ∆χ are equally estimable.

10

t (a rb.

un it)

R (D c )

rb.

un it

)

R (D c )

Dc

t (a

Dc (a)

(b)

FIG. 4: The behavior of quantum signal to noise ratio against ∆χ and t with arbitrary unit of g for (a) g = 0.1, (b) g = 2.

12

120

10

100

80

)

g = 1.5

60

(

6

H

topt

8

4

40

g = 2 20

2

0

0

0

1

2

3

0.0

(a)

0.5

1.0

1.5

2.0

2.5

3.0

(b)

FIG. 5: (a) The behavior of optimal time against ∆χ for two different values of the coupling constant g. (b) QFI at the optimal time against ∆χ . IV.

CONCLUSIONS

The behavioral study of different quantum computationally useful quantities in the presence of environment provides significant insight to the successful realization of quantum protocols. QFI is one among such quantities. In this paper we study the behavior of QFI in

11

the presence of classical static noise. We obtain an analytical relation for QFI, which leads us to identify the optimal initial state of the qubit. The analytical relation shows that the time of coupling and the coupling strength identically affect the dynamics of QFI. For the optimal initial state, QFI and FI becomes equal, which is a condition for optimal measurement of the unknown parameter. The results show that short time coupling between the system and environment is of little physical importance. The analytical relation of QFI is used to construct analytical relation for QSNR. From the graphical analysis, comparison with the previous studies on the behavior of QSNR in different Gaussian and non-Gaussian environments is made. It is shown that unlike those studies, the maximum of QSNR does not decay with the estimable parameter of the environment. A relation for the optimal time that maximizes QSNR is obtained. At the optimal time QFI varies as inverse square of the estimable parameter and thus results in a constant value of QSNR at the optimal time. This indicates that all values of the unknown parameter are estimable with equal precision. The approach of our study for the characterization of environment using a simplest quantum system as a probe not only simplifies the procedure but may also make it experimentally realizable.

[1] G. Beneti, G. Casati and G. Strini, Prinicples of quantum computation and information, (World Scientific Publishing,Toh Tuck Link, Singapore 596224, (2005)). [2] T. Yu and J.H. Eberly, Phys. Rev. Lett. 93, 140404 (2004). [3] S. Khan, Math. Struct. Comput. Sci. 23, 1220 (2013). [4] K.K. Sharma, S.K. Awasthi and S.N. Pandey, Quantum Inf. Process. 12, 3437 (2013). [5] S. Khan and M. K. Khan, J. of Modern Optics 58, 918 (2011). [6] I. Neder, M.S. Rudner, H. Bluhm, S. Foletti, B.I. Halperin and A. Yacoby, Phys. Rev. B 84, 035441 (2011). [7] M.J. Biercuck and H. Bluhm, Phys. Rev. B 83, 235316 (2011). [8] G.Y. Neng, F.M. Fa, L. Xiang, and Y.B. Yuan, Chin. Phys. B 23, 034204 (2014). [9] D. Crow and R. Joynt, Phys. Rev. A 89, 042123 (2014). [10] W. M. Witzel, K. Young and S. D. Sarma, Phys. Rev. B 90, 115431 (2014). [11] T. Yu and J.H. Eberly, Opt. Commun. 283, 676 (2010).

12

[12] J.Q. Li and J.Q. Liang, Phys. Lett. A 375, 1496 (2011). [13] C. Benedetti, F. Buscemi, P. Bordone, M.G.A. Paris, Phys. Rev. A 87, 052328 (2013). [14] M. Javed, S. Khan, and S. A. Ullah, J. Russ. Las. Re. 37, 562 (2016). [15] J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D.G. Cory, Y. Nakamura, J.S. Tsai and W.D. Oliver, Nat. Phys. 7, 565 (2011). [16] J. Zhang, X. Peng, N. Rajendran and D. Suter, Phys. Rev. A 75, 042314 (2007). [17] G.A. Alvarez and D. Suter, Phys. Rev. Lett. 107, 230501 (2011). [18] C. Benedetti, F. Buscemi, P. Bordone and M.G.A. Paris, Phys. Rev. A 89, 032114 (2014). [19] I. Almog, Y. Sagi, G. Gordon, G. Bensky, G. Kurizki and N. Davidson, J. Phys. B 44, 154006 (2011). [20] C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, (1976)). [21] V. Giovannetti, S. Lloyd and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006). [22] M. G. A. Pairs, Int. J. Quantum Inform. 7, 125 (2009). [23] B. M. Escher, R. L. de Matos Fillo and L. Davidovich, Nat. Phys. 7, 406 (2011). [24] J. Joo, W. J. Munro and T. P. Spiller, Phys. Rev. Lett. 107, 083601 (2011). [25] R. Demkowicz-Dobrzanski, J. Kolodynski and M. Guta, Nat. Commun. 3, 1063 (2012). [26] A. Shaji and C. M. Caves, Phys. Rev. A 76, 032111 (2007). [27] C. Benedetti, A.P. Shurupov, M.G.A. Paris, G. Brida and M. Genovese, Phys. Rev. A 87, 052136 (2013). [28] R. Blandino, M.G. Genoni, J. Etesse, M. Barbieri, M.G.A. Paris, P. Grangier and R. TualleBrouri, Phys. Rev. Lett. 109, 180402 (2012). [29] G.A. Durkin, New J. Phys. 12, 023010 (2010). [30] N. Spagnolo, C. Vitelli, V.G. Lucivero, V. Giovannetti, L. Maccone and F. Sciarrino, Phys. Rev. Lett. 108, 233602 (2012). [31] A. Monras, Phys. Rev. A 73, 033821 (2006). [32] C. Benedetti and M. G.A. Paris, Physics Letters A 378, 2495 (2014). [33] M. Hotta, T. Karasawa, and M. Ozawa, Phys. Rev. A 72, 052334 (2005). [34] A. Monras and M. G. A. Paris Phys. Rev. Lett. 98, 160401 (2007). [35] A. Fujiwara, Phys. Rev. A 63, 042304 (2001). [36] A. Fujiwara and H. Imai, J. Phys. A 36, 8093 (2003).

13

[37] Z. Ji, G.Wang, R. Duan, Y. Feng, and M. Ying, IEEE Trans. Inf. Theory 54, 5172 (2008). [38] V. D’Auria, C. de Lisio, A. Porzio, S. Solimeno and M. G. A. Paris, J. Phys. B 39, 1187 (2006). [39] W. K. Wootters, Phys. Rev. D 23, 357 (1981). [40] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994). [41] X. M. Lu, X. G. Wang, and C. P. Sun, Phys. Rev. A 82, 042103 (2010). [42] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982). [43] M.G.A. Paris, Physica A 413, 256 (2014). [44] C. Benedetti, F. Buscemi, P. Bordone, and M. G. A. Paris, Int. J. Quantum Inform., 10, 1241005 (2012). [45] C. Thompson, G. Vemuri and G. S. Agarwal, Phys. Rev. A 82, 053805 (2010). [46] M. Tchoffo, L. T Kenfack, G. C. Fouokeng, and L. C. Fai, Eur. Phys. J. Plus 131, 380 (2016). [47] X. N. Hao, J.C Hou, and J.Q Li, Quantum Inf. Process. 15, 2819 (2016). [48] A.V. Oppenheim and G. C. Verghese, Signals, Systems and Inference (Pearson Education, 0133944212, (2015)) [49] V. I. Didenko and A. V. Ivanov, Distribution laws of quantization noise for sigma-delta modulator, in: Proc. 16th IMECO TC4 Symp. and 13th Workshop on ADC Modelling and Testing, Florence, Italy, 9951000 (2008). [50] W. Zhong, Z. Sun, J. Ma, X. Wang, and F. Nori, Phys. Rev. A 87, 022337 (2013. [51] J. Dittmann, J. Phys. A 32, 2663 (1999).

14