Ultimate limits of thermal pattern recognition

Quantum channel discrimination presents a fundamental task in quantum information theory, with critical applications in quantum reading, illumination, data readout, and more. The extension to multiple quantum channel discrimination has seen a recent focus to characterize potential quantum advantage associated with quantum-enhanced discriminatory protocols. In this paper, we study thermal imaging as an environment localization task, in which thermal images are modeled as ensembles of Gaussian phase insensitive channels with identical transmissivity, and pixels possess properties according to background (cold) or target (warm) thermal channels. Via the teleportation stretching of adaptive quantum protocols, we derive ultimate limits on the precision of pattern classification of abstract, binary thermal image spaces, and show that quantum-enhanced strategies may be used to provide significant quantum advantage over known optimal classical strategies. The environmental conditions and necessary resources for which advantage may be obtained are studied and discussed. We then numerically investigate the use of quantum-enhanced statistical classifiers, where quantum sensors are used in conjunction with machine-learning image classification methods. Proving definitive advantage in the low-loss regime, this work motivates the use of quantum-enhanced sources for short-range thermal imaging and detection techniques for future quantum technologies.

Figure 1

(a)–(c) Depict the teleportation stretching of a general adaptive protocol, described by a quantum comb $\mathcal{P}$ that contains teleportation covariant channels. Via teleportation stretching, the general adaptive protocol is stretched to (b) a block-assisted protocol with the generation of $M$-copy Choi states, followed by a premeasurement adaptive operation $(\mathcal{B},\tilde{\mathrm{\Lambda }})$. The performance of (a) and (b) can then be bounded by the block-assisted protocol (c) in the limit of infinitely squeezed TMSV states in a probe and idler configuration, which achieves the ultimate performance limit.

Figure 2

Minimum relative probe copy number ${\overline{M}}_{\text{adv}}$ required for guaranteed quantum advantage, with respect to (a) thermal-loss channels under low-loss and different thermal noise parameters, and (b) additive-noise channels under different noise conditions. In (a) we model images using thermal target and background channels with pixel temperature variance $\mathrm{\Delta }{}^{\circ }\text{C}$ (in a microwave setting with wavelength $\sim 1\text{mm}$, see Appendix pp3). The darker plots describe images with this variance at higher temperatures within $[0,50]{}^{\circ }\text{C}$, while the brighter plots describe inherently colder environments within $[-10,40]{}^{\circ }\text{C}$. In (b), the background channel noise ${\nu }_B$ is varied against the fixed target noise for a number of examples.

Figure 3

Panels (a) and (b) consider ultimate bounds on thermal image classification based on a uniform distribution of $m$ target channels, $\mathit{i}\in {\mathcal{U}}_m$. (a) Considers $m=9$ pixel images such that each channel is a thermal-loss channel under conditions $\{{\epsilon }_B,{\epsilon }_T\}=\{18.5,20.2\}$ and different values of transmissivity $\tau$ in the low-loss regime. In (b) this is shown for additive-noise channel patterns ${\nu }_T=0.01$ for $m=9$ pixels and different values of background noisy channels ${\nu }_B>{\nu }_T$. Panels (c) and (d) focus on $k\text{−}\mathrm{BCPF}$ image spaces $\mathit{i}\in {\bigcup }_{k\in \mathit{k}}{\mathcal{U}}_{\text{CPF}}^k$. (c) Considers $m=9$ pixel images such that each channel is a thermal-loss channel under conditions $\{\tau ,{\epsilon }_B,{\epsilon }_T\}=\{0.99,18.5,20.2\}$. In (d) this is shown for additive-noise channel patterns $\{{\nu }_B,{\nu }_T\}=\{0.02,0.01\}$ for $m=50$ pixels.

Figure 4

Image classification via supervised machine-learning postprocessing. (a) Depicts a (potentially quantum) sensor probing a thermal image, which is then classified via a convolutional neural network (CNN) which has been trained on similar data. The result of the classification is encoded in the final layer of the network. (b) Illustrates the use of a nearest-neighbor classifier, which infers the class of a detected image by locating its closest (in terms of some distance measure) image in a training set $\mathcal{T}$.

Figure 5

Thermal image classification performance on a thermally rendered MNIST handwritten digit data set, using a NN classifier. Each pixel is modeled as a thermal-loss channel with $\{{\epsilon }_B,{\epsilon }_T\}=\{18.5,20.2\}$. The error region for the quantum-enhanced classifier at $M=5000$ is extremely narrow, and essentially takes the form of the lower bound for $M=2500$. Dashed lines indicate lower bound of the correspondingly colored error region.

Figure 6

Quantum-enhanced thermal imaging on the MNIST data set, using NN and CNN pattern classifiers, for (a) thermal-loss channel patterns with $\{{\epsilon }_B,{\epsilon }_T\}=\{18.5,20.2\}$ and (b) additive-noise channels with $\{{\nu }_B,{\nu }_T\}=\{0.02,0.01\}$. (a) Depicts quantum advantage bounds for thermal-loss channels for variable $\tau$, using a CNN classifier. Change in color of the regions for $M=500$ is used to indicate a transition from potential advantage to guaranteed advantage at $\tau \sim 0.99$ using realistic quantum resources. (b) Illustrates the error-classification bounds for additive-noise channels under variable probe copy number $M$, using both statistical classifier models. In both cases $T=10000$, dashed lines indicate lower bounds, and performances have been averaged over a number of experiments.

Figure 7

Finite-energy output fidelity behavior for (a) thermal-loss channels with background and target parameters $\{{\epsilon }_B,{\epsilon }_T\}=\{18.5,20.2\}$, and (b) additive-noise channels with target noise ${\nu }_T=0.01$. Beginning in a vacuum state (optimal classical probe) the finite-energy Choi-state approximations quickly approach the optimal quantum probe-state fidelities in the limit of infinite squeezing, for realistic energetic resources, shown here for $a\in \{2.5,10,100\}$.