Fundamental Quantum Limit to Waveform Estimation
We derive a quantum Cramér-Rao bound (QCRB) on the error of estimating a time-changing signal. The QCRB provides a fundamental limit to the performance of general quantum sensors, such as gravitational-wave detectors, force sensors, and atomic magnetometers. We apply the QCRB to the problem of force estimation via continuous monitoring of the position of a harmonic oscillator, in which case the QCRB takes the form of a spectral uncertainty principle. The bound on the force-estimation error can be achieved by implementing quantum noise cancellation in the experimental setup and applying smoothing to the observations.
The accuracy of any sensor is limited by noise. To quantify the potential performance of a sensor, it is often useful to compute a lower bound to the error in the estimation of the signal of interest. One of the most widely used bounds is the Cramér-Rao bound (CRB), which limits the mean-square error in parameter estimation vantrees ().
The development of quantum technology highlights the question of how quantum mechanics impacts the performance of sensors. Helstrom formulated a quantum Cramér-Rao bound (QCRB) helstrom (), which stipulates that the minimum estimation error is inversely proportional to a property of the sensor known as the quantum Fisher information. The QCRB is central to quantum sensor design in the burgeoning field of quantum metrology wiseman (); glm () for several reasons. It allows one to determine whether the fundamental sensitivity of a sensor design meets the requirements of an application, provides a criterion against which the optimality of quantum sensing schemes can be tested, and motivates improvements of schemes that are suboptimal. For sensors near the fundamental limit, the QCRB can also be used to quantify the trade-off between sensing accuracy and physical resources of the sensor, so that efficient ways of improving sensitivity can be identified.
Most prior work on the QCRB considered estimation of one or a few fixed parameters. Yet, in most sensing applications, such as force sensing and magnetometry, the signal of interest is changing in time. This time-changing signal, which we call a waveform, is coupled continuously to the sensor, and continuous measurements on the sensor are used to extract information about the waveform braginsky (); berry (); smooth (). Here we derive the QCRB for waveform estimation—the first such derivation to our knowledge—allowing for any quantum measurement protocol, including sequential, discrete or continuous measurements.
Previous work on the QCRB generally did not take into account prior information, but for the task of estimating a waveform, which often depends on an infinite number of unknown parameters, parameter estimation techniques no longer suffice and prior information is required to make the problem well defined vantrees (). The prior information might, for example, restrict the signal to a finite bandwidth, making integrals over frequency finite that otherwise would diverge. Thus a crucial feature of our QCRB is the inclusion of prior waveform information.
Our result provides a rigorous criterion against which the optimality of design, control, and estimation strategies for quantum sensors, such as gravitational-wave detectors, force sensors, and atomic magnetometers, can be tested. As an example, we calculate the QCRB on the error of force estimation via continuous position measurements of a harmonic oscillator, in which case the bound takes the form of a spectral uncertainty principle. We show that the bound can be achieved by implementing quantum noise cancellation (QNC) to remove the backaction noise from the observations qnc () and applying the estimation technique of quantum smoothing smooth () to the observations. This proves the optimality of such control and estimation techniques for force sensing and establishes our QCRB as the fundamental limit to force sensing.
Let denote the classical waveform to be estimated. For simplicity, we assume to be a scalar function; generalization to multiple processes is straightforward. We discretize time as , , and assume that is small enough that we can treat as piecewise-constant, i.e., for . The prior probability density for the vector characterizes what is known or assumed about the waveform prior to the measurements. For a vector of observations made any time during the interval , we define a conditional probability density . The joint probability density is . Finally, we define the estimate of as and the estimate bias, given signal , as , where .
Multiplying both sides of by , differentiating with respect to , and then integrating over all using , we obtain
where the final equality assumes . This assumption, also used in the proof of the classical CRB vantrees (), is satisfied as long as the prior density approaches zero at the infinite endpoints (as it must for any probability density) and the bias there is not infinite.
Quantum mechanics enters this description, which till now is classical, by determining the conditional probability of the observations. Given a quantum system, we can describe any measurement protocol, including sequential measurements and excess decoherence, during the interval by introducing appropriate ancillae, in accord with the Kraus representation theorem wiseman (); kraus (); nielsen (). This also accounts for any feedback during the interval, based on the measurement outcomes, because the principle of deferred measurement nielsen () allows one to put off the measurements on the ancillae till time ; measurement-based feedback is replaced by controlled unitaries prior to the measurements, as schematically shown in Fig. 1.
In this approach, the overall system dynamics is described by unitary evolution of the enlarged system; the conditional probability of observations is given by , where is the density operator of the enlarged system at time , conditioned upon , and is the positive-operator-valued measure (POVM) that describes the (deferred) measurements up to time . We denote expectation values with respect to by angle brackets subscripted by , so that . Continuous measurements can be modeled as the limit of a sequence of infinitesimally weak measurements wiseman ().
We now follow a procedure similar to the one used by Helstrom helstrom () to derive the QCRB. We introduce an operator that satisfies . Unlike Helstrom, we do not require to be Hermitian. Note that the vanishing trace of in the definition of implies that .
It is convenient to incorporate the prior information by working in terms of a density operator in a hybrid quantum-classical space and introducing an operator , which satisfies . In terms of , Eq. (1) takes the form that we use to derive the QCRB:
Multiplying Eq. (2) by , where and are the components of arbitrary real column vectors and , and then summing over all and , we obtain
where , , and denotes transposition. It follows from Eq. (3) that
where the second inequality is the Schwarz inequality.
The second integral in Eq. (4) is , where
is the estimation-error covariance matrix. The first integral in Eq. (4) is, using the completeness of the POVM, , where is a (real, symmetric) Fisher-information matrix,
Since , separates neatly into a quantum and a classical, prior-information component, i.e., , where
When these results are substituted into Eq. (4), we find that . Setting implies that for arbitrary real vectors . Since is real and symmetric, this implies that is positive-semidefinite; the matrix inequality
is the QCRB in its most general form. To use a CRB in practice, it is customary to define a non-negative, quadratic cost function using a positive-semidefinite (Hermitian) cost matrix suited to the application vantrees (); helstrom (). The matrix QCRB is equivalent to a lower bound, , on all such cost functions.
To calculate the QCRB, we must be more specific about the evolution of the enlarged quantum system. The Hamiltonian that governs overall system dynamics over the interval , of duration , is , with corresponding evolution operator . We have , where . Let denote the evolution operator over the interval . The density operator is related to the initial density operator by , which gives , where
with being the Heisenberg-picture version of . An obvious choice for is the anti-Hermitian , where . The quantum part of the Fisher matrix then becomes
where . Angle brackets with subscript 0 denote an expectation value with respect to . The quantum Fisher information is thus a two-time covariance function, averaged over .
To take the continuous-time limit, we let , , , and . The estimation-error covariance matrix becomes the two-time covariance function of estimation error, , and the Fisher matrix becomes , with
being the functional derivative.
In the continuous-time limit, the matrix QCRB retains the same form as Eq. (9), where the continuous-time inverse is defined by . The bound on a cost function becomes
Equation (14), valid for any cost function, is the most serviceable expression of our chief result. An important special case is the point estimation error,
where angle brackets without a subscript denote an overall quantum-classical average.
To illustrate the use of our QCRB, we consider the estimation of a force on a quantum harmonic oscillator. The Hamiltonian is , with being the position operator, the momentum operator, the mass, and the resonant frequency. In this situation, we have , which leads to a quantum component of the Fisher information,
The further average over in Eq. (11) can be omitted in Eq. (16) because appears linearly in and thus drops out of . If we assume that is a Gaussian process, the classical, prior-information component of the Fisher information is the inverse of the prior two-time covariance function of vantrees ().
We now assume that all noise processes are stationary. For a stationary, zero-mean process , the covariance function depends only on the time difference and can be Fourier-transformed to give the power spectral density . The choice , together with taking and , makes the power spectral density of the estimation error. The QCRB (14) then becomes a spectral uncertainty principle:
In the time-stationary case, the matrix QCRB is equivalent to satisfying this spectral uncertainty principle for all . A bound on the point estimation error now follows from .
To proceed in our approach, we must specify the measurements that extract the force information from the oscillator and include the associated backaction. Thus we now suppose that one performs continuous position measurements, using, for example, a continuous optical probe. The observation process is , and the oscillator equations of motion are and , where is the backaction noise. Here and are like the quadrature components of an optical field, obeying the canonical commutation relation . We assume and have zero mean; their spectra satisfy an uncertainty principle, wiseman (); braginsky ().
If we introduce a small amount of damping, is the inhomogeneous solution for , driven just by , and becomes stationary. In the limit of negligible damping, the spectrum of becomes , where is the oscillator transfer function. The spectral uncertainty principle (17) now takes the form
The corresponding bound on point estimation error is
Notice that a bandwidth constraint on is incorporated in the prior information: goes to zero outside the relevant bandwidth, thus allowing to be zero there and making the integral (19) finite.
We can elucidate the meaning of the QCRB (19) by considering how to estimate the force from the observations in this scenario. In the frequency domain, the observation process reads , being a noise term that depends on and . Using smoothing smooth (); vantrees () to estimate from yields an error
This is the minimum achievable error for a given noise spectrum . It cannot be reached by the more well-known technique of filtering wiseman (), as filtering does not make use of the entire observation record. If and are uncorrelated and quantum limited, we have
where the power spectrum is known as the standard quantum limit (SQL) for force detection braginsky ().
It is now evident that to attain the QCRB (19), it is necessary to beat the SQL. This requires evading or tempering the effects of the backaction . One way to do this is to correlate and , as was proposed for interferometric gravitational-wave detectors by Unruh unruh (). An alternative is to use quantum noise cancellation (QNC) qnc (), which has the advantage of making the QCRB (19) achievable, as we now show. One QNC approach, discussed in qnc (), adds an auxiliary oscillator with position and momentum . One monitors continuously the collective position , giving a process observable ; the backaction force acts on and thus equally, with strength , on each of the two oscillators. Suppose the auxiliary oscillator has the same resonant frequency and equal, but opposite mass (the negative mass can be simulated by an optical mode at the red sideband of the optical probe). The dynamics of the collective position is then determined by and , where . There being no backaction noise in , one easily finds that
with equality for quantum-limited noise. This quantum-noise-cancellation scheme beats the SQL and if the noise is quantum limited, does so optimally: the smoothing error given by Eq. (20) achieves the QCRB (19), which implies that the spectral uncertainty principle (18) is saturated. Our force-sensing QCRB, rigorously proven and demonstrably achievable, thus serves as a fundamental quantum limit, against which the optimality of future force sensing schemes should be tested. More generally, our QCRB for arbitrary cost functions (14) will find application whenever quantum-limited estimation of temporally varying waveforms is attempted.
We acknowledge productive discussions with J. Combes, A. Tacla, Z. Jiang, S. Pandey, M. Lang, and J. Anderson. This work was supported in part by NSF Grants No. PHY-0903953 and No. PHY-1005540, ONR Grant No. N00014-11-1-0082, and ARC Grant CE0348250.
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