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Entanglement-Enhanced Testing of Multiple Quantum Hypotheses ARTICLE https://doi.org/10.1038/s42005-020-0369-4 OPEN Entanglement-enhanced testing of multiple quantum hypotheses ✉ ✉ Quntao Zhuang 1,2 & Stefano Pirandola 3,4 1234567890():,; Quantum hypothesis testing has been greatly advanced for the binary discrimination of two states, or two channels. In this setting, we already know that quantum entanglement can be used to enhance the discrimination of two bosonic channels. Here, we remove the restriction of binary hypotheses and show that entangled photons can remarkably boost the dis- crimination of multiple bosonic channels. More precisely, we formulate a general problem of channel-position finding where the goal is to determine the position of a target channel among many background channels. We prove that, using entangled photons at the input and a generalized form of conditional nulling receiver at the output, we may outperform any classical strategy. Our results can be applied to enhance a range of technological tasks, including the optical readout of sparse classical data, the spectroscopic analysis of a fre- quency spectrum, and the determination of the direction of a target at fixed range. 1 Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721, USA. 2 James C. Wyant College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA. 3 Department of Computer Science, University of York, York YO10 5GH, UK. 4 Research Laboratory of ✉ Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. email: [email protected]; [email protected] COMMUNICATIONS PHYSICS | (2020) 3:103 | https://doi.org/10.1038/s42005-020-0369-4 | www.nature.com/commsphys 1 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-0369-4 uantum sensing1 exploits quantum resources and mea- E composed of m sub-channels Φ, each acting on a different = … surements to improve the performance of parameter subsystem Sk (for k 1, , m) and chosen from a binary Q (B) Φ(T) estimation and hypothesis testing, with respect to the best alphabet {Φ , }. Only one of the sub-channels can be the possible classical strategies. One of the fundamental settings of target channel Φ(T), while all the others are copies of a back- quantum hypothesis testing2–5 is quantum channel discrimina- ground channel Φ(B). A quantum pattern is therefore represented 6–10 = ⋯ tion , where the aim is to discriminate between different by a global channel En (for n 1, , m) where the target physical processes, modeled as quantum channels, arbitrarily channel is only applied to subsystem Sn while all the other sub- chosen from some known ensemble. Finding the best strategy for systems undergo background channels (see Fig. 1a for a simple quantum channel discrimination is a non-trivial double optimi- example with m = 3). zation problem which involves the optimization of both input In this scenario, we design entanglement-enhanced protocols, states and output measurements. Furthermore, the optimization based on a two-mode squeezed vacuum source and a generalized is generally performed assuming a certain number of probings entangled version of the conditional-nulling (CN) receiver17,23–25, and it becomes an energy-constrained problem in the dis- that are able to greatly outperform any classical strategy based crimination of bosonic channels, where the available input states on coherent states (see Fig. 1b for a schematic). This quantum have a finite mean number of photons11. advantage is quantified in terms of much lower mean error For the discrimination of bosonic channels, the so-called probability and improved error exponent for its asymptotic ‘classical strategies’ are based on preparing the input signal modes behavior. in (mixtures of) coherent states and then measuring the channel Quantum-enhanced CPF has wide applications (see Fig. 1c). In outputs by means of suitable receivers, e.g., a homodyne detector. quantum reading of classical data, this corresponds to a novel By fixing the input energy to a suitably low number of mean formulation that we call ‘position-based quantum reading’. Here photons per probing, the classical strategies are often beaten by the information is encoded in the position of a target memory cell fl truly quantum sources such as two-mode squeezed vacuum with re ectivity rT which is randomly located among background fl states, where each signal mode (probing the channel) is entangled memory cells with re ectivity rB. This is a particularly suitable with a corresponding idler mode directly sent to the output model for information readout from sparse memory blocks. measurement. This quantum advantage was specifically proven Changing from spatial to frequency modes, it can be mapped into for the readout of data from an optical memory, known as a quantum-enhanced model of photometer or scanner, where the quantum reading12, and the yes/no detection of a remote target, goal is to find an absorbance line within a band of frequencies. known as quantum illumination13–16. The advantage can therefore be interpreted as a quantum- While quantum advantage with entangled-assisted protocols enhanced tool for non-invasive spectroscopy. has been proven in problems of binary quantum channel dis- Another potential application of CPF is quantum target crimination with bosonic channels, the potential advantage of finding, where we simultaneously probe multiple space cells that quantum entanglement over the best classical strategies still needs are now represented by sectors of a sphere with some fixed radius. to be explored and fully quantified in the more general setting of Only a single sector has a target with reflectivity η while all the discrimination between multiple quantum channels. As a matter other sectors are empty. Moreover, each sector is characterized by of fact, this problem is very relevant because real physical bright noise so that NB mean thermal photons per bosonic mode applications often involves multiple hypotheses, and their treat- are irradiated back to the receiver. Of course the problem is not ment lead to non-trivial mathematical complications. In fact, limited to a spherical geometry. For instance, it can be seen in the naively decomposing a multi-hypothesis quantum channel dis- context of defected device detection. Suppose there is an assembly crimination into multiple rounds of binary cases does not line for producing a device that implements a channel, and with necessarily preserve the quantum advantages from the low probability, the assembly line produces a defective device that binary case. implements a different channel. Similarly, the problem can In this work, we formulate a basic problem of multiple channel equivalently be mapped from spatial to frequency modes, so as to discrimination that we call “channel-position finding”. Here the realize a quantum-enhanced scanner now working in very noisy goal is to determine the position of a target channel among many conditions. copies of a background channel. We prove that, using entangled Besides these potential applications, we expect that our results photons at the input and a generalized form of conditional nul- will have other implications beyond the model of CPF. For ling receiver at the output, we may outperform any classical instance, as a by-product, we also found that our generalized CN strategy in finding the position of the target channel, with a clear receiver beats the best known receiver for the original binary advantage in terms of mean error probability and its error problem of quantum reading12 (see Methods for more details). exponent. In particular, our receiver design only relies on state- of-the-art technology in quantum optics, i.e., direct photo- Generalized conditional nulling receiver. From a mathematical detection (not requiring number-resolution), two-mode squeez- point of view, the model of CPF exploits a relevant symmetry ing (which can be realized by standard optical parametric property that enables us to perform analytical calculations. For- amplifiers) and feed-forward control (which has been demon- mally, we consider the discrimination of m possible global strated17). Our results can be applied to various applications, m channels fEngn¼1, each with equal prior probability and expres- including position-based quantum reading, spectroscopy and sed by target finding. ðBÞ ðTÞ E ¼ ≠ Φ Φ ; ð1Þ n k n Sk Sn Results ðB=TÞ fi where Φ is the background/target channel acting on sub- General setting and main ndings. We study the discrimination Sk of multiple quantum channels by introducing and studying the system Sk. In general, each subsystem may represent a collection problem of channel-position finding (CPF). This is a basic model of M bosonic modes. m of pattern recognition involving quantum channels, which has It is easy to see that the ensemble of global channels fEngn¼1 19–22 22 nÀ1 ynÀ1 relations with the notion of pulse-position modulation .In has the geometric uniform symmetry (GUS) En ¼ S E1S , CPF, a pattern is represented by a multi-mode quantum channel where the unitary S is a cyclic permutation and Sm = I, with I 2 COMMUNICATIONS PHYSICS | (2020) 3:103 | https://doi.org/10.1038/s42005-020-0369-4 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-0369-4 ARTICLE Fig. 1 Channel-position finding (CPF) schematics. CPF represents a fundamental model of pattern recognition with quantum channels. a Example for m = ; ; Φ S S S 3 subsystems. Global channels E1 E2 E3 consist of sub-channels on subsystems 1, 2, 3. Each sub-channel can be chosen to be a background (B) (T) channel Φ or a target channel Φ . Channel En (for n = 1, ⋯ , m) means that the target channel is applied to subsystem Sn while all the other subsystems undergo background channels. b The classical strategy sends coherent-state signals (red, Sk), while the entangled strategy sends signals (red, Sk) entangled with locally stored idlers (blue, Ik).
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