Quantum-limited loss sensing: Multiparameter estimation and Bures distance between loss channels
The problem of estimating multiple loss parameters under energy constraints using the most general ancilla-assisted entangled parallel strategy is studied. An upper bound on the quantum Fisher information matrix is derived assuming the environment modes can be accessed. We then present a probe state that achieves this upper bound even in the practical case where the environment modes are inaccessible. This optimal probe can be prepared using single-photon sources and linear optics, and the optimal performance can be attained using on-off detection and classical processing. In the course of our analysis, we calculate explicitly the energy-constrained Bures distance between -fold tensor products of two pure-loss channels.
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Consider the sensing configuration depicted schematically in Fig. 1. optical elements modeled as beam splitters with real-valued transmissivities are probed using a multimode signal-ancilla quantum state with the signal modes being modulated by the loss elements while the ancilla modes are held losslessly. The exact nature of the modes and loss elements need not be specified for our analysis, which is applicable in diverse scenarios. Thus, the loss elements may be actual pixels in an amplitude mask or in a reflectance target in an image sensing scenario Brida et al. (2010), or may represent absorption coefficients of a sample at different frequencies in an absorption spectroscopy setup Whittaker et al. (2017), or a single-photon detector whose quantum efficiency is being calibrated Avella et al. (2011); *QJL+16; *MLS+17; *S-CWJ+17; *MS-CW+17. The probes may also represent temporal modes probing the transmittance of a living cell undergoing a cellular process Taylor and Bowen (2016). The quantum theory of imaging and optical communication Yuen and Shapiro (1978); *Sha09 shows that many natural imaging problems can also be mapped to equivalent transmittance estimation or detection problems, e.g., the estimation of the separation between two point sources Tsang et al. (2016); *NT16prl; *LP16 or deciding if one or two point sources is present Helstrom (1973); *LKN+18arxiv.
While not indicated explicitly in Fig. 1, multiple signal modes may probe each loss element, and the quantum state of all the signal and ancilla modes may be entangled. We assume that the environment modes entering the “unused” ports of the beam splitters are all in the vacuum state, a realistic assumption at optical frequencies if additional background light is absent. The output environment modes are typically inaccessible for measurement, so we assume that only the signal and ancilla modes are measured using an arbitrary quantum measurement in order to estimate the transmissivity values. As a measure of the resources involved in the estimation, we will assume that the energy 111Throughout this paper, we refer to the average photon number of a state of a given set of quasimonochromatic modes simply as its “energy”. allocated to the signal modes probing the -th loss element is specified as . Indeed, not only is arbitrarily precise loss estimation possible if the probe energy is not constrained, but it is necessary under many circumstances to minimize the photon flux through the optical element, e.g., to avoid damage or alteration of processes in live tissue Cole (2014), to calibrate sensitive single-photon detectors Avella et al. (2011), or for covertness.
In this paper, we solve the problem of quantum-optimal estimation of transmittance parameters with a given signal energy budget using the general ancilla-assisted entangled parallel strategy shown in Fig. 1. Specifically, we obtain an upper bound on the quantum Fisher information matrix for the problem, and then present a probe state that achieves the bound. This probe state also achieves the energy-constrained Bures distance between product loss channels, a result of independent interest. Finally, we show that the quantum-optimal performance is achievable using single-photon sources, linear optics, and on-off (single-photon) detectors.
Problem Formulation and Estimation-Theory Review – The action of the -th beam splitter on the -th signal mode annihilation operator and the -th environment mode annihilation operator takes the form
in the Heisenberg picture, where , the total number of signal modes such that . In the Schrödinger picture, the evolution (1) is realized by the system-environment unitary , where the “angle” is defined by . We assume that the joint probe state on the combined signal and ancilla modes is a pure state satisfying the energy constraints for , where is the total photon number operator of the signal modes probing the -th loss element. Note that a mixed probe state can be purified using additional ancilla modes without altering the signal-mode state or worsening the performance. As noted earlier, the environment modes are initially in the multimode vacuum state and the ancilla modes do not suffer any degradation. The quantum channel on the signal mode induced by (1) maps a signal density operator to the output state whose Wigner characteristic function is given by
where is the Wigner characteristic function of (see, e.g., Shapiro (2009)).
We work in the angle parametrization or the equivalent transmittance parameterization according to convenience. Denoting the initial state of the entire system (signal + ancilla + environment modes) by , the evolution (1) results in an output pure state of . Since the output environment modes are inaccessible, the relevant output state is . The state family gives rise to the corresponding multi-parameter quantum Cramér-Rao bound (QCRB), which we now review (see, e.g., refs. Helstrom (1976); *Hol11; *Par09 for details). For each parameter , there exists a Hermitian operator (that depends on in general) called the symmetric logarithmic derivative (SLD) operator satisfying . The quantum Fisher information matrix (QFIM) (also denoted if the parametrization is to be emphasized) is the matrix whose -th matrix element is given by
The operational significance of the QFIM is as follows: Consider any measurement applied to the output modes resulting in an estimate vector for . The error covariance matrix of the estimate has the matrix elements , where denotes expectation over the measurement results. If the estimate is unbiased, i.e., if for all and , the QCRB is the matrix inequality
implying that is a positive semidefinite matrix. One may further choose a positive semidefinite cost matrix in order to define a scalar cost 222For example, if we wish to minimize the sum of the mean squared errors of the parameters, we take . The QCRB then implies that any unbiased estimator has cost .
The estimation of a single loss parameter has been studied before Sarovar and Milburn (2006); Venzl and Freyberger (2007); Monras and Paris (2007); Adesso et al. (2009); Monras and Illuminati (2010); *MI11, but not in the generality considered here. Ref. Sarovar and Milburn (2006) focuses on measurement optimization and did not consider probe optimization, Venzl and Freyberger (2007) focuses on entangled coherent state probes with a fixed measurement, Monras and Paris (2007) focuses on single-mode Gaussian-state probes, and Adesso et al. (2009) considers optimization of a probe state of a single signal mode. Ref. Monras and Illuminati (2011) considers joint Gaussian probes of a signal and ancilla mode. Thus, none of the above works addressed the fully general multimode ancilla-assisted parallel strategy or the multi-parameter case solved here.
Upper bound on the QFIM – We now obtain an upper bound (in the matrix-inequality sense) on the QFIM for estimating , extending the approach of Monras and Paris Monras and Paris (2007) for the single-parameter case. We do this by evaluating the QFIM under the assumption that the output environment modes are also accessible. By the monotonicity of the QFIM under partial trace [; Theorem10.3.]Pet08qits, this upper bounds the QFIM for the actual situation where the environment modes are inaccessible.
With the angle parametrization, we can write the output state of the joint evolution of as for
In writing the above, we have assumed for notational simplicity that exactly one signal mode is used to probe each of the beam splitters – this assumption will be relaxed shortly. Since is pure, differentiating implies that an SLD operator for is . Using the purity of and the fact that the environment modes are in vacuum, a direct calculation of the -th matrix element of the QFIM (where the tilde denotes that this matrix is calculated assuming access to the environment modes) gives
where is the energy of in the -th signal mode. If multiple signal modes are used to probe each loss element, a similar calculation gives the same result with now the total energy in the signal modes used to probe the -th beam splitter. The monotonicity of the QFIM then implies that the true QFIM matrix satisfies
Note that this bound is valid for any probe state with the given signal energy distribution. We refer to (6) as the generalized Monras-Paris (MP) limit.
Energy-constrained Bures distance between loss channels – Consider the situation of Fig. 1 with , a probe with signal modes with total energy , an arbitrary ancilla , and two possible values of transmittance (angle) and , i.e., as an ancilla-assisted channel discrimination problem between product channels and of the form of Eqs. (1-2) with a total signal energy constraint. This perspective arises naturally in the quantum reading of a digital optical memory Pirandola (2011) in which the loss channels represent the two possible values of a bit stored in the memory. Several measures of general channel distinguishability under an energy constraint have been proposed recently, e.g., the energy-constrained diamond distance Pirandola et al. (2017); *Shi17arxiv; *Win17arxiv, the energy-constrained Bures distance (ecb-distance) Shirokov (2016) and general energy-constrained channel divergences Leditzky et al. (2018); *SWA+17arxiv. We focus here on the ecb-distance defined for any bosonic channels and on the signal modes as 333Our definition (7) differs slightly – purely for convenience of analysis – from that in Shirokov (2016) in that it uses an equality energy constraint, and is normalized to lie between 0 and 1. As we show, these are inessential differences for the problem at hand:
where is the fidelity between states, is an arbitrary ancilla system, id is the identity channel on , is a joint state of , is the total photon number operator on , and the optimization is over all pure states of with signal energy .
We now evaluate . An arbitrary probe state can be written as
where is an -mode number state of , are normalized states of , and is the probability distribution of . The energy constraint takes the form
i.e., the probability mass function of the total photon number in the signal modes. It was shown in (Nair (2011), Sec. II) that for any probe (8) with given , the fidelity between the outputs of the channels is bounded from below as , where
Moreover, it was shown (Nair (2011), Sec. IV.A) that if the are orthonormal, the lower bound is achieved regardless of the way in which the photon number is distributed among the signal modes. This orthonormality condition is equivalent to the reduced density operator of being diagonal in the multimode number state basis, so such probe states are called Number-Diagonal Signal (NDS) states Nair (2011).
Thus, in order to calculate the ecb-distance (7), we need to minimize the NDS probe output fidelity under the energy constraint (9). Consider an arbitrary satisfying the energy constraint and let , and . For and , we have , , and . Since the function is convex, we have . Considering the graph of the function , convexity implies that the chord joining and lies above that joining and in the interval . Since the energy constraint Eq. (9) can be satisfied if and , i.e., if is concentrated at the two points and with and 444The strict convexity of implies that this is the only signal photon number distribution that achieves the minimum fidelity. Thus, Gaussian probes can be seen to be strictly suboptimal., the energy-constrained minimum fidelity is given by:
where is the fractional part of .
Using Eq. (7), we get the ecb-distance between the loss channels and – see Fig. 2. Since the ecb-distance is an increasing function of , it equals (up to normalization) that defined with an inequality constraint in Shirokov (2016).
Note that the optimal probe states achieving the ecb-distance are independent of the loss values and , while the ecb-distance between and is independent of . Thus, the probe state can be chosen to have a single signal and ancilla mode, but we may also use the state
with signal modes and at most one ancilla mode, which can be prepared using single-photon states and linear optics.
Optimal Multiparameter Loss Estimation – Let us now apply this result to the original estimation scenario of Fig. 1. Consider first the estimation of a single angle parameter using a signal energy . If the probe state (12) is used, we can calculate the resulting QFI using Eq. (11) and the relation between fidelity and the QFI Hayashi (2006); *BC94:
so that the MP limit is saturated by this probe at all values of and . For integer , we have in the optimal state, so that the single-mode number state of photons is also optimal, agreeing with the result of Adesso et al. (2009). For non-integer however, and in particular for , the unentangled states proposed in Adesso et al. (2009) are suboptimal if ancilla entanglement is allowed.
Consider now the multiparameter case with the given energy budget . Let us use the probe state for with the implied number of signal and ancilla modes. Denoting as before the output state as , it easy to see that the -th SLD operator for satisfying . The SLDs are commuting and the -th element of the QFIM is:
where we have used for all and the single-parameter result . Thus, the product probe achieves the generalized MP limit (6) and is quantum-optimal. Since the SLDs commute and the QFIM is diagonal, there is no obstacle to the simultaneous achievement of the QCR bounds for the parameters Helstrom (1976); Ragy et al. (2016).
The SLD operators corresponding to the parametrization in terms of are given by resulting in the optimal QFIM
in the transmittance parametrization. In comparison, the QFIM for a product coherent-state input with the given energies is
so that a large advantage is available for transmittance values close to unity.
We now show that the optimal performance is obtainable using on-off detection. For any of the parameters, and for general non-integer , the probe has signal modes and one ancilla mode. Performing on-off detection in each of these modes after probing the channel gives a vector observation , where each component of is a bit indicating whether or not the detector in that mode fired. The classical Fisher information of this measurement is then, using the chain rule Zamir (1998) (the last term is the conditional Fisher information for the detection of the entangled signal mode given the result of detection of its ancilla mode):
which is the QFI. Thus, per-shot on-off detection suffices to attain the QFI and more involved adaptive measurements over multiple shots are unnecessary Barndorff-Nielsen and Gill (2000); *Fuj06. Finally, by using a large number of copies of the state of (12), the maximum-likelihood estimator on the multi-shot measurement record can be used to approach the QCRB (15) by increasing for any finite Kay (1993).
Discussion – We obtained the optimal probe state and measurement for the simultaneous estimation of transmittance parameters using the most general ancilla-assisted entangled parallel strategy. For arbitrary values of , the probe states can be prepared using on-demand or heralded single-photon sources – now a mature technology Eisaman et al. (2011) – and linear optics, circumventing the difficulties of preparing nonclassical states of large energy. Further, the optimal estimation of all parameters is simultaneously achievable without entanglement between the signal modes and using only on-off detection and classical processing. Interestingly, our results also show the near-optimality of the so-called absolute calibration method for measuring transmittances Klyshko (1980); *JR86; *HJS99; *WM04 which has been of recent experimental interest Avella et al. (2011); *QJL+16; *MLS+17; *S-CWJ+17; *MS-CW+17 provided only the energy of the postselected signal modes is counted. It is remarkable that by using at most 1 ebit of ancilla entanglement, one can achieve the exact same performance that access to the output modes would give. This is in contrast to the case of estimating Hamiltonian shift parameters in the presence of noise, for which the performance is strictly worse than the noiseless case even with ancilla entanglement Escher et al. (2011); *DKG12; *Tsa13. We have focused on ancilla-assisted parallel strategies in this paper – it remains to be seen if sequential adaptive estimation strategies Demkowicz-Dobrzański and Maccone (2014); *CMW16; *Yua16; *PL17; *TW16arxiv; *PLL18arxiv can yield further improvements over the performance obtained here.
Enroute to our estimation theory results, we derived the ecb-distance between -fold products of loss channels. It is remarkable that the ecb-distance for loss channels can be calculated exactly while, to the best of our knowledge, the available results for unitary channels, e.g., the phase shift channel, are in the form of quantum speed limit bounds (see, e.g., Frey (2016) for a review).The results of Nair and Yen (2011) show that NDS probes also achieve the energy-constrained diamond distance. Independently, Sharma et al. have shown Sharma et al. (2017); Wilde (2018) that NDS input probes optimize general energy-constrained channel divergences between any phase-covariant bosonic channels. It may thus be hoped that other energy-constrained channel divergences may be calculated using similar techniques, and their connections to interesting problems in optical quantum metrology may be elucidated.
Acknowledgements – I thank Cosmo Lupo, Stefano Pirandola, and Mark M. Wilde for useful discussions. This work is supported by the Singapore Ministry of Education Academic Research Fund Tier 1 Project R-263-000-C06-112.
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