Statistics of measurement of noncommuting quantum variables: Monitoring and purification of a qubit

Statistics of measurement of noncommuting quantum variables:

Monitoring and purification of a qubit

Hongduo Wei and Yuli V. Nazarov

Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands

共Received 7 February 2008; revised manuscript received 12 June 2008; published 11 July 2008兲

We address continuous weak linear quantum measurement and argue that it is best understood in terms of statistics of the outcomes of the linear detectors measuring a quantum system, for example, a qubit. We mostly concentrate on a setup consisting of a qubit and three independent detectors that simultaneously monitor three noncommuting operator variables, those corresponding to three pseudospin components of the qubit. We address the joint probability distribution of the detector outcomes and the qubit variables. When analyzing the distribution in the limit of big values of the outcomes, we reveal a high degree of correspondence between the three outcomes and three components of the qubit pseudospin after the measurement. This enables a high-fidelity monitoring of all three components. We discuss the relation between the monitoring described and the algorithms of quantum information theory that use the results of the partial measurement. We develop a proper formalism to evaluate the statistics of continuous weak linear measurement. The formalism is based on Feynman-Vernon approach, roots in the theory of full counting statistics, and boils down to a Bloch-Redfield equation augmented with counting fields.

DOI:10.1103/PhysRevB.78.045308 PACS number共s兲: 03.65.Ta, 03.65.Wj, 03.67.Lx, 73.23.⫺b

I. INTRODUCTION

The theory of quantum measurement, being a foundation of quantum physics, is attracting more and more attention.1

Intrinsic paradoxes2_{are definitely a main reason for studying}

quantum measurements. More motivation comes from the practical needs to understand the real solid-state based devices3,4 _{developed for quantum computing.}5

Measure-ments in solid-state setups may provide access to extra vari-ables that facilitate the read out of the quantum information stored in the elementary two-level quantum systems共qubits兲. The concept of continuous weak linear measurement 共CWLM兲, where the interaction between the detector and the measured system is explicit and sufficiently weak, has been recently elaborated in context of the solid-state quantum computing.6–10 _{CWLM provides a universal description of}

the measurement process and is based on general linear-response theory.11_{It applies to a large class of linear}

tors: From common amplifiers to more exotic on-chip detec-tors such as quantum point contact,12 _{superconducting SET}

transistors,13 _{generic mesoscopic conductors,}14 _{and fluxons}

in a Josephson transmission line to measure a flux qubit.15,16

It is an important feature of CWLM that the 共quantum兲 information is transferred from a quantum system being measured—a qubit—to other degrees of freedom: those of the detector. The outcome of the measurement is thus repre-sented by the detector degrees of freedom rather than those of the qubit. We will address both the statistics of the out-comes and joint statistics of the outout-comes and the qubit de-grees of freedom.

We stress the difference between the detector outcomes and the outcomes of a projective measurement of a qubit. In distinction from the result of a projective measurement, the detector outcome is not discrete since the detector output共for instance, voltage or current兲 is a continuous variable. The outcomes do not even have to correlate with the state of the qubit if the detector is uncoupled. Further, the detector

vari-ables are subject to noise that is not related to the qubit. Owing to the feedback of the detector at the qubit, this noise affects the qubit too.

In comparison with the textbook projective measurement that instantly provides a result and projects the system onto the state corresponding to the result, the CWLM takes time both to accumulate the information and to distort the qubit. The time ␶m required to obtain a sufficiently accurate mea-surement result is called “meamea-surement time” and is a char-acteristic of a CWLM setup. It is not a duration of an indi-vidual measurement in this setup: the latter may vary. The distortion is due to the inevitable back action of the detector and is characterized by the dephasing rate ⌫d. It has been shown6–8_{that for an optimized—quantum limited—detector}

␶m⌫d= 1/2 while the measurement time ␶m greatly exceeds 1/2⌫dfor less optimal detectors.

In the context of quantum information theory, CWLM may be understood as an interaction of the qubit with infi-nitely many ancillary qubits representing the detector de-grees of freedom. Each ancilla is brought to weakly interact with the qubit for a short time and is subsequently measured. Owing to the interaction, the quantum state of the ancillae is entangled with the state of the qubit. The detector output is proportional to the sum of the measurement results of a large set of ancillae. This allows to transfer quantum information from the qubit to the detector without formal projective mea-surement of the qubit. Therefore the peculiarities of the CWLM can be understood in the framework of a projective measurement, although a more complicated one involving the detector degrees of freedom. The CWLM can be thus seen as a buildup of an entanglement between the qubit and the detector. An outcome of an individual CWLM is the de-tector output accumulated during the time interval of a cer-tain duration␶d. Any CWLM can be described as a general-ized quantum measurement, which involves qubit and detector degrees of freedom.

The outcome randomly varies from measurement to surement. We argue here that studying statistics of the

surement outcomes of a CWLM is the best way to un-derstand and characterize such a measurement. This is espe-cially important for the simultaneous measurement of non-commuting variables共say, A and B兲 we concentrate on in this work. In this case, the textbook projective measurement can-not help to predict the statistics of the results; it would de-pend on the order of measurements of A and B. This property of the measurements in noncommuting bases enables most quantum cryptography17_{algorithms and has been extensively}

elucidated in Ref. 18.

One can straightforwardly realize, in experiment, a CWLM of a quantum system where A and B are measured simultaneously. If A and B commute, the statistics of the outcomes of sufficiently long CWLM corresponds to the pre-dictions of projective measurement scheme 共see Sec. III兲. The projective measurement scheme loses its predictive power if A and B do not commute. The reason is that the order of measurement of A and B is not determined in the course of a continuous measurement. The statistics of CWLM outputs thus cannot be straightforwardly conjectured and has to be evaluated from the quantum-mechanical treat-ment of the whole system consisting of the qubit and the detectors.

In a sharp contrast to the case of commuting variables, the most probable outcome of a sufficiently long CWLM of non-commuting variables does not depend on the qubit state. Therefore, it provides no information about the qubit. The information is, however, hidden in the statistics of random outcomes. Recently, the simultaneous acquisition of two noncommuning observables was investigated in the frame-work of CWLM,9_{and the correlation of the random output of}

two detectors was found to be informative. Not only noise, but the whole full counting statistics 共FCS兲 of the noncom-muting measurements has been recently addressed for an ex-ample of many spins traversing the detectors.19

The structure of the article is as follows. We develop the necessary formalism in Sec. II. Our approach stems from the FCS theory of electron transfers20 in the extended Keldysh formalism,21_{which has been recently discussed}22_{in the}

con-text of the quantum measurement. At first step, we obtain a Feynman-Vernon action to describe the fluctuations of the input and output variables of the detector共s兲. In the relevant limit, the action is local in time. So at the second step we reduce the path integral to the solution of a differential equa-tion that appears to be a Bloch-Redfield equaequa-tion augmented with the counting field. In Sec. III we exemplify the formal-ism addressing a relatively simple case of quantum non-demolition共QND兲 measurement.23_{We evaluate the}

distribu-tion of the outcomes for a single detector and understand the statistics of a recently proposed quantum undemolition measurement.10 _{The main results are presented in Sec. IV}

where we discuss statistics of measurement of noncommut-ing variables for the case of three independent detectors mea-suring the three components of the qubit pseudospin. We find the statistical correspondence between the three outcomes and three wave-function components after the measurement. The correspondence is characterized by a fidelity that gener-ally increases with the magnitude of the outcomes reaching the ideal value 1 in the limit of large magnitudes. Since very large outcomes are statistically rare and require long waiting

times, this result could be of a purely theoretical value. To prove the opposite, we have evaluated the fidelity at moder-ate magnitudes of outcomes and measurement durations ␶d and we were able to demonstrate the fidelity of 0.95 for ␶d ⯝7␶m. We term this “quantum monitoring.” Ideally, the re-sult of the quantum monitoring is a pure state of the qubit and three numbers共detector outputs兲 giving the polarization of the state. The same result can be also achieved by prepar-ing the qubit state of the known polarization, for instance, by a projective measurement along a certain axis. The difference is that in the case of preparation the polarization axis is known to the observer in advance, while in the case of moni-toring it is not so; both the three numbers and the state emerge from dynamics of the quantum system that encom-passes the qubit and the detectors.

We discuss the relation between the quantum monitoring proposed and the quantum algorithms that use the results of partial measurements that we summarize in Sec. V. We evaluate the detector action in the Appendix A. We prove in the Appendix B that our approach correspond to a Lindblad scheme for a system consisting of the detectors and the qubit.

II. METHOD

We start the outline of the formalism with a simplest setup where a single detector measures a single component of the qubit pseudospin. In this case, the Hamiltonian reads as fol-lows: H = Hq+ Hint+ Hd, 共1a兲 Hq=

兺

i=1 3 Hi␴ˆi;Hint=␴ˆ3Qˆ . 共1b兲 Here, Hqis the Hamiltonian of the qubit generally given by a linear combination of three Pauli matrices␴ˆi共i=1,2,3兲 cor-responding to three components of the qubit pseudospin. Hint gives the coupling between the detector and the third com-ponent of the pseudospin of the qubit, Qˆ being the detector

input variable. Hdstands for the Hamiltonian of the detector. Since we assume linear dynamics of the detector variables, a general form of this Hamiltonian is that of a boson bath,

Hd=

兺

k

ប␻kbˆk †

bˆk.

This encompasses infinitely many boson degrees of freedom labeled by k, bˆkbeing the corresponding annihilation opera-tors. The output of the detector is given by the output vari-able Vˆ . An arbitrary linear dynamics is reproduced if both variables Qˆ and Vˆ are linear combinations of the boson creation/annihilation operators, Qˆ =

兺

k 共Qkbˆk†+ Qkⴱbˆk兲, 共2兲 Vˆ =

兺

k 共Vkbˆk † + Vkⴱbˆk兲. 共3兲 This is in the spirit of Caldeira-Leggett approach.24 In con-trast to the work,24_{we do not assume thermal equilibrium in}

the boson bath. In fact, this assumption would be wrong for most practical detectors since a signal amplification cannot take place in the state of thermal equilibrium. The only re-quirement we impose is that Wick’s theorem holds for the boson operators involved. This guarantees the linear dynam-ics of the detector variables. Besides, this conveniently al-lows us not to specify the coefficients Qk, Vk. All information about the coefficients and the nonequilibrium boson distribu-tion is incorporated into the two-point correlators of the vari-ables explicitly given below关Eqs. 共7兲 and 共8兲兴. By virtue of

Wick’s theorem, the averages of all possible products of the detector variables can be expressed in terms of these two-point correlators.

We are interested in the statistics of the detector output variable Vˆ . We note that this variable is distinct from those of the qubit, and in principle even does not have to correlate with the qubit state共e.g., if Qˆ=0兲. However, since the detec-tor is supposed to measure the qubit, there must be a共high兲 degree of correspondence between the detector output and the qubit state. This sets the goal of our calculation: to access the joint statistics of Vˆ and the qubit variables.

To achieve the goal, we introduce a counting field ␹共t兲 coupled to the output variable Vˆ and use a modified Feynman-Vernon scheme25_{where the evolution of the “bra”}

and “ket” wave function is governed by different Hamilto-nians H−and H+: H⫾= H⫾ប␹共t兲Vˆ/2. ⫾ corresponds to two branches of closed time contour, respectively.26,27 _This

scheme was first employed in the work.28_{The counting field}

␹共t兲 plays a role of the parameter in the generating function of the probability distribution of the detector outcomes V共t兲. This generating function is given by:

Z共兵␹共t兲其兲 = Tr共Tជe−i/ប兰dtH−Rˆ 共0兲Tឈei/ប兰dtH+兲. 共4兲

Tr共¯兲 implying the trace over both detector and qubit vari-ables. Here, Tជ共Tឈ兲 denotes time 共reversed兲 ordering in evolu-tion exponents and Rˆ 共0兲=␳ˆd共0兲丢␳ˆ共0兲 is the initial density matrix of whole system. It separates into␳ˆd共0兲 and␳ˆ共0兲, the initial density matrix of the detector and qubit, respectively. This implies that the detector and the qubit do not interact before the initial time moment t = 0.

Next, we employ the path-integral representation for the probability-generating function.28_{According to Feynman and}

Vernon25 _{共see also Ref.29兲:}

Z共兵␹共t兲其兲 =

冕

DX¯+DX¯−eAd_Z

Iq共Q−,Q+兲, 共5兲 here X¯⫾共t兲 are two-dimensional vectors of the detector vari-ables X¯⫾共t兲=关Q⫾共t兲,V⫾共t兲兴T_,DX¯⫾_⬅兿

t

dQ⫾共t兲dV⫾共t兲. Q⫾共t兲, V⫾共t兲 are the path-integral variables. ZIqis called the “influ-ence functional” and is given by

ZIq共Q−,Q+兲 = Trq共Tជe− i

ប兰dt共Hq+␴ˆ3Q+共t兲兲␳_ˆ共0兲

⫻ Tឈeiប兰dt共Hq+␴ˆ3Q−共t兲兲兲, 共6兲

where Trqmeans the trace over qubit space. The action Adin

Eq.共5兲 is bilinear in X¯⫾and␹to conform to linear dynamics of the detector and will be specified below. The advantage of this representation is that the dynamics of infinitely many detector degrees of freedom have been reduced to the dy-namics of only two relevant fields: Qˆ and Vˆ. The influence functional describes a nonlinear response of the qubit on the fields.

Let us turn to a specific model of linear dynamics of the detector. Following common assumptions about CWLM,7,8

we assume instant detector responses and white 共frequency-independent兲 noises. Under these assumptions, a detector is characterized by seven independent parameters: four re-sponse functions and three noises. It is convenient to use an index i taking values 1 and 2 for input and output variables, respectively. With this index, we present four response func-tions aijas a single 2⫻2 matrix. The noises Sijform a simi-lar matrix. By virtue of Kubo formula, the response functions are expressed in terms of expectation values of the operator commutators − i ប具关Qˆ共t兲,Qˆ共t

⬘

兲兴典 = a11␦共t − t

⬘

− 0+兲, 共7a兲 − i ប具关Qˆ共t兲,Vˆ共t

⬘

兲兴典 = a12␦共t − t

⬘

− 0+兲, 共7b兲 − i ប具关Vˆ共t兲,Qˆ共t

⬘

兲兴典 = a21␦共t − t

⬘

− 0+兲, 共7c兲 − i ប具关Vˆ共t兲,Vˆ共t

⬘

兲兴典 = a22␦共t − t

⬘

− 0+兲, 共7d兲 where an infinitesimal small positive number 0+ _{in the ␦} function represents small but finite response time.

The noises correspond to the expectation values of sym-metrized operator products,

冓冓

Qˆ 共t兲Qˆ共t

⬘

兲 + Qˆ共t

⬘

兲Qˆ共t兲 2

冔冔

= S11␦共t − t

⬘

兲, 共8a兲

冓冓

Vˆ 共t兲Vˆ共t

⬘

兲 + Vˆ共t

⬘

兲Vˆ共t兲 2

冔冔

= S22␦共t − t

⬘

兲, 共8b兲

冓冓

Qˆ 共t兲Vˆ共t

⬘

兲 + Vˆ共t

⬘

兲Qˆ共t兲 2

冔冔

= S12␦共t − t

⬘

兲, 共8c兲

冓冓

Vˆ 共t兲Qˆ共t

⬘

兲 + Qˆ共t

⬘

兲Vˆ共t兲 2

冔冔

= S21␦共t − t

⬘

兲, 共8d兲 here,具具AˆBˆ典典⬅具共Aˆ−具Aˆ典兲共Bˆ−具Bˆ典兲典 for any operators Aˆ and Bˆ. Let us discuss the physical meaning of the parameters involved. S11is the noise of the input variable responsible for the back action of the detector and decoherence of the qubit;

S₂₂ is the output noise that prevents a fast measurement of the detector outcome. The cross term S₁₂= S₂₁ presents the correlation of these two noises. The response function a21

determines the detector gain: The proportionality coefficient between the detector output and the third component of the qubit pseudospin,具Vˆ典=a21具␴ˆ3典. Other response functions a12,

a22, a11 are, respectively, related to reverse gain, output and input impedances of the detector and are not of immediate interest for us. The detector is characterized with the dephas-ing rate ⌫d= 2S11/ប2 and the measurement time ␶m= S22/a212.7,30The Cauchy-Schwartz inequality

S11S22− S122 ⱖ ប2

4共a21− a12兲 2

imposes an important restrictions on the possible values of the parameters.7,8Following the common assumption, we as-sume that the reverse gain a12 is much less than the direct gain a21: a21Ⰷa12. This condition is commonly required from a good amplifier. Under these assumptions, ␶m⌫d ⱖ1/2, one cannot measure a qubit without dephasing it.

The action Ad corresponding to the model reads

Ad=

冕

dt

冋

− 1 2¯x T_{共t兲共aˇ}−1_兲T S ˇaˇ−1_¯x共t兲 + iX¯T共t兲aˇ−1¯x共t兲 + i¯␹T_{共t兲X¯共t兲}

册

, 共9兲

the derivation is outlined in the Appendix A. Here we switch to the “quantum” x¯ and “classical” X¯ variables defined as

follows: x¯ =共X¯+_{− X}¯−_{兲/ប, X¯=共X¯}+_{+ X}¯−_{兲/2. ¯ =␹ 共0,␹兲}T

. The 2 ⫻2 matrices aˇ and Sˇ are, respectively:

a ˇ =

冉

a11 a12 a21 a22

冊

, 共10a兲 S ˇ =

冉

S11 S12 S21 S22

冊

. 共10b兲

It is important for further advance that the action 共9兲 is

local in time. This allows for reducing the path integral to a differential equation. The procedure is completely similar to the standard reduction of the corresponding path integrals to either Schrödinger or Fokker-Planck equations.31 _{One slices}

time axis into intervals共tk, tk+1兲 共tk= k⌬t. k is an integer兲, and takes the path integral in Eq. 共5兲 without tracing over the

qubit indices slice by slice. The result of the integration at tk is a matrix in qubit indices,␳ˆ共tk兲. Integrating over x,X in the next slice, one finds a linear relation between␳ˆt_k+1and␳ˆt_k:

␳ˆ共␹;tk+1兲 =

冕

兿

tk⬍t⬍tk+1 Dx¯共t兲DX¯共t兲eA_d Sˆ+␳ˆ共␹;tk兲Sˆ−; 共11兲 Sˆ_⫿= exp

冢

⫾i ប

冢

Hq⌬t +␴ˆ3

冕

t_k t_k+1 dtQ⫿共t兲

冣冣

. 共12兲 Since the slice is thin, the exponents may be expanded,

Sˆ_⫿⬇ 1ˆ ⫾ i ប

冢

Hq⌬t +␴ˆ3

冕

t_k t_k+1 dtQ⫿共t兲

冣

− 1 2ប2

冕

t_k t_k+1 dtdt

⬘

Q⫿共t兲Q⫿共t

⬘

兲 + ... ,

and the integration is reduced to evaluation of the averages and the correlators of the fields Q⫾with the action Ad. Col-lecting terms ⬀⌬t and taking the limit ⌬t→0, we obtain a differential equation for␳共t兲 that resembles a familiar Bloch-Redfield equation but essentially depends on the counting field␹.32 _{The resulting equation for␳共␹; t兲 reads:}

⳵␳ˆ ⳵t = − i ប关Hq,␳ˆ兴 − ␹2_共t兲 2 S22␳ˆ + ia21␹共t兲 2 共␳ˆ␴ˆ3+␴ˆ3␳ˆ兲 −S12 ប ␹共t兲共␳ˆ␴ˆ3−␴ˆ3␳ˆ兲 − S11 ប2共␳ˆ −␴ˆ3␳ˆ␴ˆ3兲. 共13兲 How to apply the equation? Let us consider a single mea-surement first. Let␶dbe the duration of the measurement. We collect the detector output during the time interval共0,␶d兲 and normalize it by␶d Vo= 1 ␶d

冕

0 ␶d V共t兲dt. 共14兲

To get the statistics of Vo, we should assume that ␹ is a constant in the interval共0,␶d兲. Indeed, expanding of the gen-erating function in terms of␹gives the averages of products of Vo. Let us suppose that the initial density matrix of the qubit is ␳ˆ共0兲. We solve the Eq. 共13兲 with the initial

condi-tions ␳ˆ共␹, 0兲=␳ˆ共0兲. The output is a ␹-dependent matrix ␳ˆ共␹兲⬅␳ˆ共␹, t =␶d兲 after the measurement.

We stress that ␳ˆ共␹兲 appearing in the equation is not the

reduced qubit density matrix. It is a more interesting and complicated quantity that reflects the joint probability distri-bution of the qubit pseudospin components after the mea-surement and the detector outcome collected. To see this, let us define the reduced density matrix of the qubit and the outcome Vo, Rˆ 共Vo, Vo

⬘

兲. It is a matrix in qubit indices and in the outcome values Vo, Vo

⬘

共see Appendix B for the details兲. Its diagonal elements give the statistics in question. The re-duced qubit density matrix 共with no regard for the value of the outcome兲 is given by

␳ˆ =

冕

dVoRˆ 共Vo,Vo兲,

the probability distribution of the outcomes共with no regard for the qubit state兲 reads

P共Vo兲 = TrqRˆ 共Vo,Vo兲,

and the joint statistics is expressed in terms of the qubit density matrix conditioned to a certain value of the output,

The quantity in use, ␳ˆ共␹兲, is related to the diagonal

ele-ments of thus introduced density matrix Rˆ by means of Fou-rier transform,28 Rˆ 共Vo,Vo兲 = t 2␲

冕

d␹␳ˆ共␹兲e −i␹Vot . 共15兲

presenting a generating function for the quasidistribution

Rˆ 共Vo, Vo兲. Comparing this with the above definitions, we find convenient relations ␳ˆ共t兲 =␳ˆ共␹= 0兲, P共V0兲 = t

冕

d␹ 2␲e −i␹Vot Trq␳ˆ共␹兲, ␳ˆ共Vo兲 = 兰d␹␳ˆ共␹兲e−i␹Vot 兰d␹Trq␳ˆ共␹兲e−i␹Vot .

We here used the normalization condition Trq␳ˆ共␹兲=1. It is the main technical advantage of our work that Eq. 共13兲 is

similar in form to an elementary Bloch-Redfield equation for the qubit density matrix and not much complicated than that one. However, in its augmented form, it solves a much more challenging task of finding the joint probability distribution of the detector outcome and the qubit state.

It is important to note that Eq.共13兲, as well as Eq. 共17兲 for

multidetector setup, complies with a Lindblad scheme.33,34

We will show this explicitly in Appendix B. This guarantees the positivity of the “big” density matrix Rˆ 共Vo, Vo兲.

The locality in time is a relevant but strong assumption which in fact corresponds to a classical detector共indeed, the action 共9兲 does not contain any ប兲. This is why we do not

have to worry about possible quantum uncertainties of the detector output that could complicate the interpretation of the statistics.28

The scheme can be easily extended to many repetitive 共that is, being constantly repeated兲 measurements to comply with the concept of CWLM. Let us consider 共infinitely兲 many subsequent measurements. For ith measurement, the detector output is collected during the time interval 共ti, ti+1兲. This gives a series of outcomes Vo共i兲. To describe the joint statistics, one solves Eq.共13兲 with a piece-wise constant␹共t兲,

␹共t兲=␹iin the interval共ti, ti+1兲. The solution of the equation at the time moment tM+1 depends on M counting fields: ␳ˆ =␳ˆ共␹₁, . . . ,␹M兲. The Fourier transform with respect to all␹i defines the qubit density matrix conditioned on the outcomes

Vo共i兲of all M preceding measurements. We illustrate two sub-sequent measurements in Sec. III

Importantly, the scheme described can also be easily ex-tended to more qubits and/or detectors: One just adds extra 共counting兲 fields for detectors and extra Pauli matrices for qubits. The case of interest for us is the simultaneous CWLM of three pseudospin projections of a qubit. The coupling term becomes

H_int=␴ˆ₁Qˆ₁+␴ˆ₂Qˆ₂+␴ˆ₃Qˆ₃. 共16兲

Qˆk 共k=1,2,3兲 being the input fields of the three detectors. Three counting fields ␹k are coupled to the corresponding output variables Vˆkof the three detectors.

While it is straightforward to write down the equation for general situation, we employ a specific model at this point. Namely, we assume for simplicity that the detectors are iden-tical and independent. “Ideniden-tical” implies that the noises and response functions of all three detectors are the same. “Inde-pendent” implies that no response function relates inputs/ outputs of two different detectors, neither the noises corre-late. Each detector is described by the action in corresponding variables. Under these assumptions, the setup is conveniently SU共2兲 covariant.

The resulting␹-augmented Bloch-Redfield equation reads ⳵␳ˆ ⳵t = − i ប关Hq,␳ˆ兴 −

冉

兺

k=1 3 ␹k2 2

冊

S22␳ˆ + ia21 2

兺

_k=1 3 ␹k关␴ˆk,␳ˆ兴+ +S12 ប

兺

k=1 3 ␹k关␴ˆk,␳ˆ兴 − S11 ប2

冉

3␳ˆ −

兺

k=1 3 ␴ˆk␳ˆ␴ˆk

冊

. 共17兲 Comparing this with Eq.共13兲, we see that each detector

con-tributes a term to the equation. Each term comes with the corresponding counting field and ␴matrix.

III. STATISTICS OF QND MEASUREMENT Before turning to the measurements of noncommutative variables, let us first illustrate the formalism with a single-detector setup. Only a single component ␴ˆ₃ will be mea-sured. We will be interested in a quantum QND setup where successive measurements are performed. Such QND mea-surements have been recently realized for superconducting qubits.35 _{To satisfy the nondemolishing condition,}23 _{we set}

Hˆq=⑀␴ˆ3in Eq.共1兲 so that Hqand Hintcommute. In this case,

Hq can be canceled by transformation to the rotating frame,

␳ˆ→eiHˆqt/ប␳_{ˆ e}−iHˆqt/ប_{, and will be disregarded from now on.} Let us perform two measurements that immediately fol-low each other. During the first measurement of duration t₁, the detector output is collected in the time interval共0,t1兲 so the measurement outcome is V1=兰0

t1_dtV共t兲/t1_{. Similarly, for}

the second measurement V2=兰t₁

t₁+t2_dtV共t兲/t2_{. The statistics of}

the two outcomes is computed from Eq. 共13兲 by setting␹共t兲

to a piece-wise constant␹共t兲=␹1共␹2兲 during the first 共second兲 time interval and ␹共t兲=0 otherwise. To solve the equation, we parameterize␳ˆ共␹; t兲 as follows: ␳ˆ共␹;t兲 =1ˆ +␴ˆ3 2 ␳+共␹;t兲 + 1ˆ −␴ˆ₃ 2 ␳−共␹;t兲 +␴ˆ₁␳1共␹;t兲 +␴ˆ₂␳2共␹;t兲, 共18兲

where 1ˆ is the unit 2⫻2 matrix in the qubit space,␳_⫾give diagonal elements of the matrix and␳1,2give the nondiago-nal ones. Equations共18兲 and 共13兲 yield two pairs of

separat-ing equations: ⳵␳+

⳵t = i␹a21␳+−

␹2

⳵␳− ⳵t = − i␹a21␳−− ␹2 2 S22␳−, 共19b兲 and ⳵␳1 ⳵t = − 2iS21␹ ប ␳2−

冉

⌫d+ ␹2 2S22

冊

␳1, 共20a兲 ⳵␳2 ⳵t = 2iS21␹ ប ␳1−

冉

⌫d+ ␹2 2S22

冊

␳2. 共20b兲 We first assume that the initial density matrix of the qubit is diagonal. We solve␳_⫾共␹; t兲 with initial conditions␳_⫾共0兲 at

t = 0. We note that at the initial time t = 0,␳ˆ does not depend

on␹ or Vo and thus we do not write them explicitly for the initial conditions. Solving the equations and Fourier trans-forming the generating function yields a very simple prob-ability distribution of both outcomes:

P共v1,v2兲 =

兺

⫾

冑

␶1␶2 2␲ ␳⫾共0兲e −共␷1⫿1兲 2_␶ 1 2 e− 共␷2⫿1兲2␶2 2 . 共21兲 To keep it simple, we have switched here to the dimension-less durations␶=␶d/␶m, and outcomesv = Vo/a21. The result is in fact classical; it does not depend on the dephasing rate. In allows for an elementary interpretation; initially, the qubit appears to be either in the state + or −, with probabilities ␳+共0兲 and ␳−共0兲, respectively. The state persists during the measurements. The outcome of each measurement is distrib-uted normally around ⫾1 with the standard deviation

冑

␶1,2 set by the duration of the measurement. We note that the persistence of a state is specific for QND measurements, and, as we see in Sec. IV, does not apply to the CWLM of non-commuting variables.

In Fig.1, the solid lines show the distribution of outcome

P versus v for two different durations ␶1= 2 共long, lower curve兲 and ␶1= 0.3共short, upper curve兲. Two obvious peaks located atv =⫾1 for the line␶1= 2 are due to the two states of the qubit that can be distinguished in the course of the long measurement. For the short measurement ␶1= 0.3, the

detector cannot resolve the difference between two eigen-states of the qubit. Thus we see a single peak atv = 0

broad-ened by the noise of the output variable. The dotted lines show the probability distribution P of the outcome of the second measurement of the same duration under condition that the outcome of the first measurement v₁= −1. For long measurement 共␶2= 2兲, the probability distribution is concen-trated near v = −1. For the short measurement 共␶2= 0.3兲, the distribution is similar to that of the first measurement, its average being close to v = 0. This makes a comprehensive

illustration of the fact that the sufficiently long measure-ments are repeatable, that is, the result of the second mea-surement is close to the result of the second one. This is not the case for the short measurement.

To illustrate the quantum aspect, let us set the initial den-sity matrix to correspond to a pure state with the wave func-tion that is an equal superposifunc-tion of the base states ⫾: ␳ˆ共0兲=␴ˆ₁. We are interested in the average value of the cor-responding pseudospin projection ␴ˆ₁ after the measurement

provided the outcome v. We evaluate this average if we

know the qubit density matrix␳ˆ共v兲 conditioned on the

out-come v,

␴1共v兲 ⬅ Tr关␴ˆ₁␳ˆ共v兲兴. 共22兲

As discussed in the Sec. II, this qubit density matrix is com-puted from the normalized Fourier transform of ␳ˆ共␹兲, the

latter is obtained by solving Eqs. 共20a兲 and 共20b兲 on time

interval共0,␶兲,␶being duration of the measurement. Since Eqs.共20b兲 do contain the dephasing rate, the result

will depend on actual dephasing. The answer reads

␴1共v;␶兲 =cos共C12v␶兲

cosh共v␶兲 e−C/2␶. 共23兲 Here, we introduce dimensionless constants C⬅4共S11S22 − S₁₂2 兲/共បa21兲2_{− 1 and C}

12= 2S12/បa21. C⬎0 characterizes the quality of the detector, and C = 0 for a quantum-limited one. C₁₂ characterizes the correlations of the noises, for a quantum-limited detector C12= 0 as well.

Generally,␴1共v,␶兲 quickly decays with increasing␶. This manifests the dephasing of the superposition by the measure-ment. Remarkably enough, for a quantum-limited detector 共C=C12= 0兲 and for a special value of the measurement out-come v = 0, the dephasing is absent. The wave function

re-tains its initial value for this particular value of output. This fact has been noted in the work10_{and termed “quantum}

un-demolition measurement.” Let us note that the phase shift between the states ⫾, acquired from the detector, 2兰0␶dtQ共t兲/ប, is zero at this 共rather improbable10兲 value of the outcome. We stress that the strict correspondence between the phase shift and outcome does not hold for a general de-tector, so that ␴1共v=0,␶兲=exp共−C␶/2兲 decreases with the measurement duration␶indicating the nonvanishing dephas-ing of the superposition. We plot ␴1共v;␶= 1兲 versus the de-tector outcome v for the duration of measurement ␶= 1 in Fig. 2. The solid line is for the quantum-limited detector 共C=C12= 0兲, and the dotted line is for the worse detector 共C=C12= 1兲. 0 0.05 0.1 0.15 0.2 -8 -4 0 4 8 0 0.5 1 P (τdur = 0. 3) P( τdur =2) v

FIG. 1. 共Color online兲 Quantum nondemolition setup: Two suc-cessive measurements. In each pair of the curves, the solid one gives the distribution of outcome of the first measurement while the dashed one gives the distribution for the second measurement

pro-vided the first measurement gave v1= −1. Lower 共upper兲 pair of

curves corresponds to long, ␶_1,2= 2 共short, ␶_1,2= 0.3兲 duration of measurement. The long measurement is thus repeatable, the short one is not.

IV. THE STATISTICS OF THE CWLM OF NONCOMMUTING VARIABLES

In Sec. III we provided simple examples to prove the use of the statistical approach. Thus encouraged, we turn to the statistics of the CWLM of noncommuting variables. The in-teraction Hamiltonian is now given by Eq. 共16兲. We do not

want to deal with the qubit Hamiltonian Hq=⑀␴3 and shall assume that it is removed by transforming to the rotating frame. The same transform makes␴_1,2to rotate with angular velocity⑀/ប. To compensate for this, let us presume that the signal from␴1,2is collected at frequencies⑀/ប rather than at zero frequency as the signal from ␴3. Mathematically, we define the outcomes of the detectors 1, 2 as

␶dV1,2=

冕

0 ␶d dt

冋

cos

冉

⑀t ប

冊

V1,2共t兲 ⫿ sin

冉

⑀t ប

冊

V2,1共t兲

册

. 共24兲 This can be practically realized in a very same way as it is done in a radio set. One has to mix a high-frequency signal with a reference signal of the same frequency and detect the low-frequency component of the product.

Let us evaluate␳ˆ共␹₁,␹2,␹3兲. Without the term Hq, the Eq. 共17兲 is readily solved in a proper basis in the pseudospin

space. In this basis, one of the Pauli matrices is defined as ␴ˆ_␹=共␹1␴ˆ1+␹2␴ˆ2+␹3␴ˆ3兲/␹s, while two others, ␴ˆ␮ and ␴ˆ␯, are chosen to be orthogonal to it. We have introduced ␹s as follows:␹s⬅

冑

␹1

2

+␹₂2+␹₃2. We parameterize␳ˆ as follows:

␳ˆ =␳₀1ˆ +␳_␹␴ˆ_␹+␳_␮␴ˆ_␮+␳_␯␴ˆ_␯. 共25兲 From Eqs.共17兲 and 共25兲 we obtain two equations involving

␳0and␳␹: ⳵␳0 ⳵t = ia21␹s␳␹− ␹s 2 2S22␳0, 共26a兲 ⳵␳␹ ⳵t = ia21␹s␳0−

冉

2⌫d+ ␹s2 2S22

冊

␳␹. 共26b兲 We stress that the CWLM we are about to describe is

hardly a measurement of the initial state of the qubit. In contrast to QND where the dephasing is limited to 1, 2 com-ponents, the input variables Qˆ1,2,3of the detectors randomly rotate the pseudospin in all three directions. The quantum information about initial state is lost rather quickly; at the time scale of 1/⌫d. That is, it is lost before a statistically reliable measurement result can be accumulated. This moti-vates us to choose the unpolarized density matrix as initial condition for the state of the qubit before the measurement,

␳ˆ共0兲 =1

21ˆ. 共27兲

We will see, however, that despite the memory lost the CWLM of noncommutative variables can be rather informa-tive.

Let us first discuss the distribution of the detector outputs. We first solve Eqs.共26a兲 and 共26b兲 with the initial condition

共27兲, and then recall Eq. 共5兲. In the limit of long durations

␶Ⰷ1, the log of the generating function reads: − log Z =␶

冉

Cd−

冑

Cd 2 −␹s 2 +␹s 2 2

冊

, 共28兲

where ␹i has been made dimensionless ␹iS22/a21→␹i as to give the cumulants of dimensionless outputs vi. Here, Cd ⬅⌫d␶m=共C+1+C12

2_兲/2ⱖ1/2.

The cumulants of the outcomes can be evaluated by tak-ing the derivatives of the log Z with respect to␹i. The pres-ence of the qubit enhances output noises of each detector by the factor 1 + 1/Cd: 具具vivj典典 = − ⳵2_{log Z} ⳵␹i⳵ ␹j

冏

␹i,j =0= 1 ␶

冉

1 + 1 Cd

冊

␦ij. 共29兲 There is no correlation of noises between different detectors. Such correlation arise for fourth cumulants

具具vi2vj2典典 = − ⳵4_{log Z} ⳵␹i 2_{⳵ ␹} j 2

冏

␹i,j=0= − 1 + 2␦ij 共Cd␶兲3 . 共30兲

The distribution is isotropic in three outputs depending on

v⬅

冑

v₁2+v₂2+v₃2 only, and in the limit of long durations we can calculate it by the saddle-point method, determining an optimal␹ⴱcorresponding to a given outcomev. We obtain

log P共v兲 = log Z共␹ⴱ共v兲兲;⳵log Z ⳵␹s

兩

␹s=␹

ⴱ= iv␶. 共31兲 ␹ⴱ _{is purely imaginary. We plot log共P兲/␶vsv for three} dif-ferent values of Cd in Fig. 3. The solid line is for the quantum-limited detector 共Cd= 1/2兲, the dashed line is for the worse detector共Cd= 2兲. The dotted line is for the detec-tors not connected to the qubit 共Cd=⬁兲. So it is a parabola corresponding to the Gaussian distribution of outcomes in this case. We see that the distribution is concentrated at zero. Typical values of outcomes v⬃1/

冑

␶ and for these typical values the distribution can be approximated estimated by a Gaussian one P⬃exp关−v2_/2共1+1/C

d兲兴. At larger 共and thus atypical兲 values of outcomes 共v⯝1兲, the distribution is es-sentially non-Gaussian. We see from the plots that the

pres--0.2 0 0.2 0.4 0.6 0.8 1 -6 -3 0 3 6 σ1 v

FIG. 2. 共Color online兲 “Undemolition” measurement. The aver-age value of pseudospin component ␴₁共v;␶兲 conditioned on the detector outcomev characterizes the dephasing of the superposition

after a QND measurement of duration ␶ 共␶=1 for the plots兲. A quantum-limited detector 共C=C₁₂= 0, upper curve兲 allows for the quantum undemolition measurement共␴1= 1 atv = 0兲. This does not

ence of the qubit exponentially enhances probabilities of such outcomes.

Let us discuss the correlation of the detector outputs and the pseudospin after such measurement, thus turning to the joint statistics of the measurement outcomes and the result-ing qubit state.

We characterize the correlation with a fidelity f共v兲, inner product of the normalized vector of the outcomes and aver-aged pseudospin at given outcome v:

f =

兺

i

具␴i典vi

v ;具␴i典 = Trq关␴ˆi␳ˆ共v兲兴, 共32兲

where ␳ˆ共v兲 is defined in Eq. 共15兲. The fidelity is 1 if the

normalized values of the outputs precisely give all three pseudospin components. Analyzing the saddle-point solution for the␳ˆ共v;␶兲, we obtain that f does not depend on␶in the limit ␶Ⰷ1. Importantly, at large values of the outcomes v Ⰷ1, the fidelity reaches the ideal value f ⬇1−Cd/v. This, quite unexpectedly, enables an efficient quantum monitoring of noncommuting variables.

The monitoring procedure is as follows. Starting from some initial state, one performs a series of repetitive mea-surements of duration␶. The three outcomesviof each mea-surement are written down. For most meamea-surements, the val-ues of the outcomes are typical, that is, and do not exceed the results of such measurements which correspond to low fidel-ity and are therefore discarded. One specifically waits for a measurement that gives sufficiently big values of outcomes. To decide if the outcomes are sufficiently big, one estimates the fidelity of each measurement given the values of out-comes and the relation f共v兲. If one wants to achieve the desired accuracy ades, one thus waits for the outcomes satis-fying f共v兲⬎1−ades. The big values of outcomes guarantee that the fidelity f共v兲 is sufficiently high. Sooner or later, a measurement gives the sufficiently big outcomes. The quan-tum monitoring takes place. At this moment, the state of the qubit is known with the accuracy desired and it is given by the values of outcomes␳ˆ =␴ivi/v. Since␳ˆ2= 1ˆ, this is a pure state.

We stress that the monitoring does not constitute a single-shot measurement of all components of the initial unknown quantum state. This would be forbidden by the basic laws of

quantum mechanics. Indeed, the time required for an accu-rate monitoring exceeds by far the measurement time of the detectors. By this time, the initial state is completely forgot-ten. However, the monitoring gives the observer complete information about the final quantum state.

One also could argue that any pure state of the qubit can be obtained in a simpler fashion. One would just choose a proper Hq and wait for dissipation to bring the qubit to the state of lowest energy. One could also try a projective mea-surement in a certain basis: after several tries, such measure-ment would give the state desired. We note, however, that for both approaches the resulting pure state is a priori known to the observer. This is not the case of quantum monitoring: here we let the quantum system to make “its own choice” of the final pure state and do not enforce this choice by any means.

The better the accuracy desired ades⬅1− f Ⰶ1, the bigger outputs are required, v⬎Cd/ades. The typical waiting time grows exponentially. To estimate it, we assume the duration of each measurement␶is in the order of ␶m. This is because the longer durations are not favorable due to decoherence. The waiting time is inversely proportional to the probability to have sufficiently high outcomes␶w⬃␶/ P共v;␶兲. Since the success probability is exponentially small: P共v;␶兲⬃exp共 −v2␶/2兲 from the saddle-point solution, we then estimate

␶w⬃exp共v2兲. Therefore, we shall expect log共␶w兲 ⬃ ades

−2 .

This estimation of the waiting time of successful quantum monitoring sounds pessimistic or at least causes a doubt con-cerning the practical feasibility of the monitoring. To prove that the monitoring is practical, we are going to show that a reasonably high fidelity can be achieved in a reasonably short time.

The above arguments are based on the analytical saddle-point solution valid at large␶. Now we investigate the mea-surements of moderate duration,␶⬃1. This can only be done by numerical calculation. Solving Eqs. 共26a兲 and 共26b兲 and

making the Fourier transformation according to Eq.共15兲, we

evaluate and plot f共v兲 of a single measurement of duration␶ versus ␶for the quantum-limited detector and the worse de-tector 共Cd= 2兲 in Fig.4and Fig. 5, respectively. Each curve gives f共␶兲 at a given outcome v. At each curve, we put a “bullet” to indicate the duration of measurement ␶at which the probability to obtain the outcomes larger than the given one is 10%. Similarly, we put a “triangle” to indicate ␶ at which the probability is 50%. To find the positions of the symbols, for eachv we solve numerically for the values of␶

that satisfy

冕

v +⬁ Tr␳ˆ共v

⬘

,␶兲v

⬘

2dv

⬘

冕

0 +⬁ Tr␳ˆ共v

⬘

,␶兲v

⬘

2dv

⬘

= 10%共50%兲, 共33兲

respectively. We see from the plots that f = 0.95 is achieved for a quantum-limited detector at v = 4 and ␶= 0.7. At these parameters, 10% of the measurements are successful, i.e.,

-10 -8 -6 -4 -2 0 0 1 2 3 4 ln(P)/ τ v

FIG. 3. 共Color online兲 Logarithm of the outcome distribution. The curves from the top to the bottom: quantum-limited detector 共Cd= 1/2兲, worse detector 共Cd= 2兲, detector not connected to the

give the outputv⬎4. To achieve a success, the measurement

has to be repeated typically ten times given the success prob-ability of 10%. We conclude that the 5% accuracy is typi-cally achieved in a time interval ⯝10⫻0.7␶m= 7␶m. This is not much slower than the QND measurement of the same accuracy.

Let us explain the relation between the monitoring dis-cussed and the quantum algorithms that use the results of partial measurement. These algorithms have been introduced in the context of two-qubit systems36,37_{but also may be}

ap-plied to a general quantum state.38_{Speaking very generally,}

such algorithms start with a quantum state, pure or mixed, and aim at producing another state 共pure and/or highly en-tangled兲. They proceed in steps. Each step involves interac-tion with ancilla qubits that results in an entanglement of the qubit and the ancillae. Importantly, the projective measure-ment of ancilla qubit共s兲 is performed at each step. The result of this measurement is used to decide upon next step: pos-sible decisions include to request the initial state, to stop since the fidelity desired is reached, and to apply a certain quantum gate.

The quantum monitoring proceeds similarly. It starts with an almost isotropic initial state␳ˆ = 1ˆ. The qubit is being

mea-sured by our three independent quantum limit detectors dur-ing a time interval␶. The outcome of the detectors is used to

make a decision. If the sum of the square of the three

detec-tors outputviof are small:

冑

v1 2

+v₂2+v₃2⬍4, the measurement

is disregarded; this is an analog of requesting the initial state. The measurement is repeated until the values of the outputs are sufficiently high v =

冑

v₁2+v₂2+v₃2⬎4. In this case, the

monitoring may stop since its goal is reached: the qubit is purified to a state given by the detector outputs v1,v2, and

v3. Since thevi共i=1,2,3兲 are random, the purified state of the qubit is also random. Thus one could say that the CWLM monitoring implements a quantum algorithm of the sort de-scribed, in the same sense as an analogous computer may implement discrete computer algorithms.

V. SUMMARY

In conclusion, we have shown how to evaluate the full statistics of the outcomes of a CWLM on a qubit. We are also able to evaluate joint probability distribution of the outcomes and the qubit variables after the measurement. For a single detector, we have illustrated the QND measurements and un-derstood the recent proposal of quantum undemolition mea-surement. Most interesting results concern the simultaneous CWLM of three noncommuting variables by three detectors. Such “measurement” is obtrusive and typically scrambles the initial qubit wave function. However, we have demonstrated a high degree of correspondence between the wave function

after the measurement and the outcomes of the three

detec-tors. Therefore, such CWLM may be used for high-fidelity quantum monitoring of the qubit. The monitoring in fact amounts to a purification of the qubit state in a random di-rectionvជ=共v1,v2,v3兲/兩v兩 at Bloch sphere, v1−3being random outputs of the detectors. We have drawn analogy with quan-tum algorithms that use the outputs of ancilla measurements to decide on the purification degree reached and the quantum gates to be applied.

The interpretation we give to the results is of course not the only possible one. The communications with several col-leagues have convinced us that “interesting” and “important” defy an unambiguous definition as far as theory of quantum measurement is concerned. In any case, we have developed calculational tools to access the joint probability of the qubit degrees of freedom and the outcomes of linear detectors measuring the qubit. We have also derived the representative results. It is up to the reader to conclude.

ACKNOWLEDGMENTS

H.W. acknowledges the financial support in the frame-work of NanoNed initiative共project DSC.7023兲. Y.N. appre-ciates the participation in 2006 Aspen Summer Program where he got the impetus for this work.

APPENDIX A: DERIVATION OF THE DETECTOR ACTION

The derivation of the path-integral representation for a set of variables linear in boson creation/annihilation is a

0 0.25 0.5 0.75 1 0 1 2 3 4 f τ

FIG. 4. 共Color online兲 Fidelity of quantum monitoring f vs the measurement duration␶ for quantum-limited detector. From upper to lower curvev are, respectively, 4, 3.5, 3, 2.5, 2, and 1.5. Bullets

at each curve indicate the value of␶ at which the probability to get the outcome larger than the correspondingv for each curve is 10%;

triangles at each curve indicate the value of ␶ at which the prob-ability to get the outcome larger than the correspondingv for each

curve is 50%. 0 0.25 0.5 0.75 0 1 2 3 4 f τ

FIG. 5. 共Color online兲 Fidelity of quantum monitoring f versus the measurement duration␶ for worse detector 共Cd= 2兲. From upper

to lower curvev are, respectively, 4, 3.5, 3, 2.5, 2, and 1.5. Bullets

at each curve indicate the value of␶ at which the probability to get the outcome larger than the correspondingv for each curve is 10%;

triangles at each curve indicate the value of ␶ at which the prob-ability to get the outcome larger than the correspondingv for each

straightforward task. It is instrumental in dissipative quan-tum mechanics and therefore is to be found in basic literature on the subject. References24are usually cited in this respect. Owing to the simplicity of the problem, there are many other derivations of the kind that are tailored to specific models 共e.g., Ref.31兲 and usually assume thermal equilibrium of the

boson bath. To avoid any confusion, we present this part of the derivation here. We do the derivation in the most general terms possible and specify to the concrete model in use at the later stage of the calculation.

Let Xj be a set of the variables linear in boson creation/ annihilation operators, Xˆj共t兲 being the Heisenberg time-dependent operators of these variables. Let us first disregard the coupling with the qubit, so that the time dependence of

Xˆj共t兲 is governed by the detector Hamiltonian Hd only. Ex-plicitly, the Heisenberg equation reads

dXˆj共t兲

dt = i

ប关Hd,Xˆj共t兲兴.

We are interested in the generating function of the vari-ables which we present in the following form关c.f. Eq. 共4兲兴

Z兵关␹j共t兲兴其 = Tr关Tជei/2兰dtXˆj共t兲␹j共t兲␳ˆd共0兲Tឈei/2兰dtXˆj共t兲␹j共t兲兴. We assume summation over repeating indices j and skip time indices of Xˆ 共t兲, ␹共t兲 for brevity. Differentiating with respect to the parameters ␹j⫾共t兲 of the generating function, one re-produces all possible products of the operators Xˆj共t兲.

Let us introduce a path integral over the variables X⫾_j共t兲 associated with the operators Xˆj共t兲 by means of the following identity: Z兵关␹j共t兲兴其 =

冕

DX¯+DX¯−Tr兵Tជ

兿

t,j ␦关Xj −_{共t兲 − Xˆ} j共t兲兴 ⫻ei_/2兰dt共Xj−+X+_j兲␹j_␳ˆ d共0兲Tឈ

兿

t,j ␦关Xj +_{共t兲 − Xˆ} j共t兲兴其. 共A1兲 Here we insert␦functions that replace the operators Xˆj共t兲 by the fluctuating fields X⫾j共t兲, separately for two parts of the Keldysh contour.

The use of this representation is that we can treat the coupling between the detector and an arbitrary quantum sys-tem in the form of the influence functional. If the coupling between the detector and the system has the form

H_int=

兺

j

Xˆj共t兲Sˆj;

Sˆj being operators defined in the subspace of the quantum system, we can formally substitute Xˆj共t兲→Xj⫾共t兲. The result-ing influence functional thus reads

ZIq兵关Xj −_{共t兲兴,关X} j +_{共t兲兴其 = Tr} S关T ជ_e−i/ប

冕

dtX−_jSˆ_j␳ˆ S共0兲Tឈei/ប兰dtXj +_Sˆ j兴, 共A2兲 where the trace is over the subspace of the quantum system, ␳ˆS共0兲 is its initial density matrix and the time dependence of

Sˆjis governed by the separate Hamiltonian on the subspace of the quantum system.

Let us turn to the evaluation of the representation共A1兲. To

facilitate the operations with ␦functions, we represent them by means of extra integration over the auxiliary variables

k⫾共t兲/ប. At each time moment,

␦共Xj⫾− Xˆj兲 =

冕

dkj⫾ 2␲e

ik+_j共X_j⫾−Xˆj兲_.

With these extra variables, the integral becomes

Z兵关␹j共t兲兴其 =

冕

DX¯+DX¯−Dk¯+Dk¯−ei兰dtXj −_共k j −_+␹j/2兲 ei兰dtXj +_共k j +_+␹j/2兲 ⫻Tr关Tជe−i兰dtXˆjkj − ␳ˆd共0兲Tឈe−i兰dtXˆjkj + 兴. 共A3兲

Let us now take the trace over the boson degrees of free-dom. To do this, we use the widely known relation

具eAˆ

eBˆ典 = e

具

A

ˆ2_+Bˆ2

2 +Aˆ Bˆ

典

that holds for Aˆ , Bˆ that are linear in boson operators under condition of Wick’s theorem. This allows us to express the trace in terms of the two-point correlators of boson variables, 具Xˆi共t兲Xˆj共t

⬘

兲典. We may assume 具Xˆj典⬅0 without compromising generality. The resulting expression reads

Z兵关␹j共t兲兴其 =

冕

DX¯+DX¯−Dk¯+Dk¯−ei兰dtXj −_共k j −_+␹j/2兲 ei兰dtXj +_共k j +_+␹j/2兲 eA˜d_; A ˜ d= −

冕

dtdt

⬘

冋

1 2ki +_共t兲k j +_共t

_⬘

_兲具Tឈ_共Xˆ i共t兲Xˆj共t

⬘

兲兲典 +1 2ki −_共t兲k j −_共t

_⬘

_兲具Tជ_共Xˆ i共t兲Xˆj共t

⬘

兲兲典 + ki −_共t兲k j +_共t

_⬘

_兲具Xˆ i共t兲Xˆj共t

⬘

兲典

册

. 共A4兲 It is instructive to introduce at this stage “classical” and “quantum” variables defining Xj⫾= Xj⫾បxj/2, k⫾j = Kj⫾kj/2ប. In these variables, the two-point correlators are naturally collected to symmetrized noises Sij共t,t

⬘

兲 and Kubo-like response functions Aij共t,t

⬘

兲,

Sij共t,t

⬘

兲 = 1 2具Xˆi共t兲Xˆj共t

⬘

兲 + Xˆj共t

⬘

兲Xˆi共t兲典; Aij共t,t

⬘

兲 = − i ប⌰共t − t

⬘

兲具关Xˆi共t兲,Xˆj共t

⬘

兲兴典. and the action becomes

A ˜

d= −

冕

dtdt

⬘

关2Ki共t兲Kj共t

⬘

兲Sij共t,t

⬘

兲 + iki共t兲Kj共t

⬘

兲Aij共t,t

⬘

兲兴, and does not containប. One can now integrate over the aux-iliary variables Kj共t兲, kj共t兲 to get the action in terms of Xj共t兲,

xj共t兲. This Gaussian integral can be readily taken by the saddle-point method. For our model, it is convenient to make the time-local approximation first. We replace the kernels Sij,

Aij with their time-local expressions 关Eqs. 共7兲 and 共8兲 and perform integration over K, k to arrive at Eq.共9兲.

APPENDIX B: AUGMENTED BLOCH-REDFIELD EQUATION AND LINDBLAD FORM

In this appendix we cast the Eq. 共13兲 to the Lindblad

form33_{to illustrate the entanglement of the qubit and detector}

and to prove the positivity of the density matrix encompass-ing the qubit and output variable of the detector.

To specify this underlying density matrix, it is useful to introduce a quantum variable pˆ defined as

pˆ =

冕

0 t

Vˆ 共t

⬘

兲dt

⬘

.

This variable represents the integrated detector output over the interval 共0,t兲. It differs only by a factor t from the out-come Voof the measurement at the same time interval共0,t兲 as we have defined in the main text.

We denote by p and兩p典 the eigenvalue and eigenvector of

pˆ, respectively: pˆ兩p典=p兩p典. The subject of our interest is the

time-dependent reduced density matrix in the space 兩spin典

丢兩p典,␳ˆ共p,p

⬘

; t兲, where “hat” denotes the matrix in the

pseu-dospin space. Since p is related to Vo upon a factor, this matrix is equivalent to Rˆ 共Vo, Vo

⬘

兲 used in the main text. The quantity ␳ˆ共␹兲 in the augmented Bloch-Redfield equation is

related to the diagonal part of this density matrix by Fourier transform

␳ˆ共p,p;t兲 =

冕

d␹e−ip␹␳ˆ共␹,t兲. 共B1兲

Making the inverse Fourier transform, we deduce from Eqs. 共13兲 and 共B1兲 the equation for␳ˆ共p,p;t兲:

⳵␳ˆ共p,p;t兲 ⳵t = − i ប关Hq,␳ˆ共p,p;t兲兴 + S₂₂ 2 ⳵2_{␳ˆ共p,p;t兲} ⳵p2 −a21 2

冋

⳵␳ˆ共p,p;t兲 ⳵p ␴ˆ3+␴ˆ3 ⳵␳ˆ共p,p;t兲 ⳵p

册

−iS12 ប

冋

⳵␳ˆ共p,p;t兲 ⳵p ␴ˆ3−␴ˆ3 ⳵␳ˆ共p,p;t兲 ⳵p

册

−S11 ប2关␳ˆ共p,p;t兲 −␴ˆ3␳ˆ共p,p;t兲␴ˆ3兴. 共B2兲 This is an evolution equation in partial derivatives.

Next, we demonstrate that Eq. 共B2兲 is indeed of a

Lind-blad type. This also proves that the density matrix satisfying the equation has positive diagonal elements. We work in the space兩spin典丢兩p典. We can check that the operator oˆ=i_⳵p⳵ has the following properties:

具p兩␳ˆ oˆ兩p

⬘

典 = − i⳵␳ˆ共p,p

⬘

;t兲

⳵p

⬘

, 共B3a兲

具p兩oˆ␳ˆ兩p

⬘

典 = i⳵␳ˆ共p,p

⬘

;t兲

⳵p . 共B3b兲

The Lindblad form of an evolution equation reads33,34

⳵␳ˆ ⳵t = − i ប关H,␳ˆ兴 + 1 2

兺

_␯ 共关Lˆ␯␳ˆ ,Lˆ␯ †_{兴 + 关Lˆ} ␯,␳ˆ Lˆ␯†兴兲, 共B4兲 where H is a Hermitian operator, generally not coinciding with the qubit Hamiltonian, and Lˆ_␯共␯= 1 , ¯兲 are arbitrary operators. We need to prove that a proper choice of the Lind-blad operators Lˆ_␯ and the Hamiltonian H reproduces Eq. 共B2兲.

To do so, we introduce two Lindblad operators as follows:

Lˆ1=

冑

4S11S22− 4S12 2 −ប2a₂₁2 4ប2_S 22 ␴ˆ₃, 共B5a兲 Lˆ2=

冑

S22

冉

oˆ − S12 បS22␴ˆ3− ia21 2S22 ␴ˆ₃

冊

, 共B5b兲 here 4S11S22− 4S122 −ប2a212 ⱖ0 is guaranteed by the Cauchy-Schwartz inequality 共see Sec. II兲. The Hermitian operator reads:

H = Hq− បa21

2 oˆ␴ˆ3, 共B6兲

where Hqis the qubit Hamiltonian. We substitute the opera-tors to Eq. 共B4兲 to obtain the following:

⳵␳ˆ共p,p

⬘

;t兲 ⳵t = − i ប关H,␳ˆ共p,p

⬘

;t兲兴 + S₂₂ 2

冉

2 ⳵2_␳ˆ共p,p

_⬘

_;t兲 ⳵p⳵p

⬘

+⳵ 2_␳ˆ共p,p

_⬘

_;t兲 ⳵p2 + ⳵2_␳ˆ共p,p

_⬘

_;t兲 ⳵p

⬘

2

冊

−a21 2

冉

⳵␳ˆ共p,p

⬘

;t兲 ⳵p

⬘

␴ˆ3+ ⳵␳ˆ共p,p

⬘

;t兲 ⳵p ␴ˆ3 +␴ˆ₃⳵␳ˆ共p,p

⬘

;t兲 ⳵p +␴ˆ3 ⳵␳ˆ共p,p

⬘

;t兲 ⳵p

⬘

冊

−iS12 ប

冉

⳵␳ˆ共p,p

⬘

;t兲 ⳵p ␴ˆ3+ ⳵␳ˆ共p,p

⬘

;t兲 ⳵p

⬘

␴ˆ3 −␴ˆ₃⳵␳ˆ共p,p

⬘

;t兲 ⳵p −␴ˆ3 ⳵␳ˆ共p,p

⬘

;t兲 ⳵p

⬘

冊

+S11 ប2共␴ˆ3␳ˆ共p,p

⬘

;t兲␴ˆ3−␳ˆ共p,p

⬘

;t兲兲. 共B7兲 We made use of the properties共B3a兲 and 共B3b兲. We have

not done yet since the above equation is for a two-indexed density matrix␳ˆ共p,p

⬘

兲 and not for its diagonal part. We still

have to prove that the above equation does not mix the di-agonal and nondidi-agonal elements. So that, to relate Eq.共B2兲

Ps= p + p

⬘

2 , 共B8a兲 Pd= p − p

⬘

2 . 共B8b兲 Thus, ⳵ ⳵p= 1 2

冉

⳵ ⳵Ps + ⳵ ⳵Pd

冊

, 共B9a兲 ⳵ ⳵p

⬘

= 1 2

冉

⳵ ⳵Ps − ⳵ ⳵Pd

冊

. 共B9b兲

From Eqs.共B7兲, 共B9a兲, and 共B9b兲 we obtain:

⳵␳ˆ共Ps, Pd;t兲 ⳵t = − i ប关H,␳ˆ共Ps, Pd;t兲兴 + S22 2 ⳵2_␳ˆ共P s, Pd;t兲 ⳵Ps 2 −a21 2

冋

⳵␳ˆ共Ps, Pd;t兲 ⳵Ps ␴ˆ₃+␴ˆ₃⳵␳ˆ共Ps, Pd;t兲 ⳵Ps

册

−iS12 ប

冋

⳵␳ˆ共Ps, Pd;t兲 ⳵Ps ␴ˆ₃−␴ˆ₃⳵␳ˆ共Ps, Pd;t兲 ⳵Ps

册

+S11 ប2␴ˆ3␳ˆ共Ps, Pd;t兲␴ˆ3. 共B10兲 Equations共B2兲 and 共B10兲 has the same form. Therefore, we

have proved that ␳ˆ共p,p;t兲 satisfies the Lindblad equation in

the 兩spin典丢兩p典 space matrix form and thus is positive.

We can straightforwardly extend the above scheme to Eq. 共17兲 of Sec. II that is valid for the case of three detectors. In

this case, we introduce six Lindblad operators

Lˆ1i=

冑

4S11S22− 4S122 −ប2a212 4ប2_S 22 ␴ˆi, 共B11a兲 Lˆ_2i=

冑

S₂₂

冉

oˆi− S₁₂ បS22␴ˆi− ia₂₁ 2S₂₂␴ˆi

冊

, 共B11b兲 where i = 1 , 2 , 3. The Hermitian operator reads:

H = Hq− បa21

2

兺

_i=1 3

oˆi, 共B12兲

where oˆi= i_⳵p⳵_i and pi are the eigenvalues of the operator pˆi =兰₀tVˆi共t

⬘

兲dt

⬘

共i=1,2,3兲 corresponding to each detector.

To conclude, we have shown that the density matrix ␳ˆ共p,p

⬘

; t兲 in the space 兩spin典丢兩p典 satisfies the Lindblad

equation, and thus its diagonal part is positive. The diagonal part of the matrix is related to the quantity␳ˆ共␹, t兲 in Sec. II

by the Fourier transform. We note that Lindblad operators Eqs. 共B5a兲, 共B5b兲, 共B11a兲, and 共B11b兲 include both the

de-gree of qubit and the dede-gree of the detector共s兲. This signals the entanglement of the qubit and detector共s兲 In this sense, we measure both the qubit and detector共s兲 in the context of FCS theory.

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