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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]
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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
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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). c Bosonic applications to quantum reading of position-based data and quantum-enhanced direction finding of a remote target. Entangled pairs of signal (red) and idler (blue) are used. In position-based quantum reading, each sub-channel corresponds to a memory cell with reflectivity rB (background) or rT (target); in quantum target finding, each sub-channel corresponds to a sector on a fixed-radius sphere where a target with reflectivity η can be present or absent. If the target is absent, the returning signal is replaced by environmental noise with NB mean thermal photons per mode. being the identity operator. Because the channels are highly Assume that the pattern is probed by a GUS state so that the symmetric, it is natural to input a product state with GUS output ensemble is given by a generally mixed state as in Eq. (2) m ϕ , in which case the output state becomes with target/background state σ(T/B). Then, we show the following k¼1 Sk ðBÞ ðTÞ (see Methods for a proof). ρ ¼ ≠ σ σ ; ð2Þ n k n Sk Sn Theorem 1. where σ(T/B): = Φ(T/B)(ϕ). It is clear that this ensemble of output ρ = n−1ρ †n−1 (Generalized CN receiver) Denote by hn the hypothesis that the states also has GUS, i.e., n S 1S , and it is analogous to Φ(T) the states considered in a pulse-position modulation19,21,22. target channel is encoded in sub-system Sn, so that the global It is known22,26 that the optimal positive-valued operator channel is En. Suppose that there are two partially unambiguous Π ΠðTÞ; ΠðBÞ measure (POVM) { k} minimizing the error probability for POVMs, that we call t-POVM f t t g and b-POVM discriminating an ensemble of GUS states has the same type of fΠðTÞ; ΠðBÞg, such that Π = n−1Π †n−1 b b symmetry, i.e., n S 1S . This POVM has minimum = − ρ Π ΠðTÞσðTÞ ΠðBÞσðBÞ : error probability (Helstrom limit) PH 1 tr( 1 1). For the tr½ t ¼tr½ b ¼1 ð5Þ fi speci c cases where the output states are pure = = = = 2 Then, we design the following receiver. Start with n 1: σT B ¼ ψðT BÞ ψðT BÞ , with overlap ζ ¼ ψðTÞjψðBÞ , we have 1. Check the current hypothesis hn by measuring subsystem Sn with the following expression of the Helstrom limit ΠðTÞ; ΠðBÞ the t-POVM f t t g. hipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ‘ ’ m À 1 2 2. If the outcome from Sn is T , measure all the remaining sub- ; ζ ζ ζ ; m ðTÞ ðBÞ PHðm Þ¼ 1 þðm À 1Þ À 1 À ð3Þ Π ; Π m2 systems fSkgk¼nþ1 in the b-POVM f b b g. If we get out- come ‘T’ for some S then select the hypothesis h . Otherwise, which is achievable by the ‘pretty good’ measurement27–29.In k k select hn. particular, note that for mζ ≪ 1 we have the asymptotic ‘ ’ 3. If the outcome from Sn is B , then discard hn and repeat from expansion point 1 with the replacement n → n + 1. If n + 1 = m, then select 1 P ¼ ðm À 1Þζ2 þ Oðm2ζ3Þ: ð4Þ hypothesis hm. H 4 The error probability of this CN receiver is In general, when Eq. (2) represents an ensemble of mixed 1 ζ PCNðζ ; ζ Þ¼ 2 ðÞmζ þ ðÞ1 À ζ m À 1 ; ð6Þ states, we do not know how to compute the ultimate Helstrom m 1 2 m ζ 1 1 limit. However, we can resort to a sub-optimal detection strategy 1 23 ζ σðBÞΠðTÞ ζ σðTÞΠðBÞ by generalizing the CN receiver . In fact, consider the m-ary CPF where 1 ¼ trð t Þ and 2 ¼ trð b Þ are the two types of problem of Eq. (1) with target/background channel Φ(T/B). error probabilities.
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ζ ≪ Note that, when m 1 1, we have the asymptotic expansion irradiates a total of mM modes and mMNS mean photons over 1 the entire pattern of channels, we show the following lower bound PCN ’ ðm À 1Þζ ζ : ð7Þ (LB) to the mean error probability (see Methods and Supple- m 2 1 2 mentary Note 2 for proof) Also note that the above receiver is a CN receiver because it "#ÀÁffiffiffiffiffi ffiffiffiffiffi pμ pμ 2 exploits partially unambiguous POVMs and a feed-forward m À 1 2M 2MNS B À T P ; ¼ c ; exp À ; ð10Þ 23 H LB EB ET mechanism, similar to the classical CN receiver . However, it 2m 1 þ EB þ ET is a generalized CN receiver because it also involves entanglement pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 À1 with ancillas and may also be applied to mixed-state inputs, while with c ; ½1 þð E ð1 þ E Þ À E ð1 þ E ÞÞ . 23 EB ET B T T B the original CN receiver only applies to pure states with no First note that we can also obtain this bound from Eq. (9)by entanglement. Finally, our receiver only relies on local operations consideringffiffiffiffiffiffi a source that irradiates a single-mode coherent state and classical communication among the different subsystems, an p jiNS for each of the M modes probing subsystem Sk. Then, important feature that makes it practical. = consider no passive signature EB ET, which means that For pure GUS states, one can always devise partially successful discrimination requires signal irradiation, i.e., it cannot unambiguous POVMs and find symmetric error probabilities ζ = ζ = ζ be based on the passive detection of different levels of background 1 2 , in which case the CN receiver asymptotically achieves noise. In this latter case, we find that an energetic single-mode twice the Helstrom limit in Eq. (4). However, for mixed GUS fi coherent state jiMNS on each subsystem is able to produce Eq. states, it is generally dif cult to design such POVMs, and we will (10) from Eq. (9). For this reason, in our next comparisons, we have to give non-trivial constructions in this paper. Also note that will also consider the performance of such a coherent-state feed-forward is crucial for achieving good performance. source. In some cases, the corresponding output ensemble will In fact, suppose that we choose a simple strategy without feed- turn out to be pure, so that we can exactly quantify its forward, e.g., measuring all subsystems in the b-POVM performance via Eq. (3). ΠðTÞ; ΠðBÞ f b b g. In this case, no error occurs when measuring In order to obtain an enhancement by means of entanglement, σ(B) background states . The error only occurs when this POVM is we need to introduce ancillary ‘idler’ systems Ik, for 1 ≤ k ≤ m, applied to the target state σ(T) and gives the erroneous outcome which are directly sent to the measurement apparatus (see ‘ ’ ζ B , which happens with probability 2. When this happens, we Fig. 1b). This means that the generic global channel takes the need to randomly guess (just because all outcomes would be equal form ‘ ’ = hi to B ). This gives a conditional error probability ðÞm À 1 m, ðBÞ ðTÞ E I ¼ ≠ ðΦ I Þ ðΦ I Þ: ð11Þ since only one among the m subsystems is correct. The n k n Sk Ik Sn In corresponding error probability for this design is given by For the quantum source, we use the tensor product ϕ mM, where Xm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ME 1 m À 1 P Pt ðζ Þ¼ ´ ζ ´ ¼ðm À 1Þζ =m; ð8Þ ϕ :¼ 1 Nk=ðN þ 1Þkþ1jik; k is a two-mode squeezed m 2 m 2 m 2 ME k¼0 S S k¼1 vacuum state that maximally entangles a signal mode with a where the first 1/m factor is the equal prior. We find that corresponding idler mode, given the mean number of photons NS Pt ðζ Þ ≥ PCNðζ ; ζ Þ, i.e., the CN strategy is always better than the constraining both signal and idler energies. Each subsystem Sk is m 2 m 1 2 ϕ M non-feed-forward strategy and the advantage is particularly large probed by the signal part of ME with a total of MNS photons on when ζ1 is small. average irradiated over Sk. Therefore, the overall GUS ensemble of output states takes the form Classical versus entangled strategy. Given a CPF problem mM ðBÞ ðTÞ ρ ¼ðE IÞϕ ¼ ≠ Ξ Ξ ; ð12Þ expressed by Eq. (1), we aim to minimize the mean error prob- n n ME k n SkIk SnIn ability affecting the discrimination of the corresponding m ΞðT=BÞ ΦðT=BÞ ϕ M m where ¼ð IÞð ME Þ. For generally mixed states, it hypotheses fhngn¼1. The solution of this problem is derived is difficult to calculate the Helstrom limit. One alternative is to assuming that the signal modes irradiated over the subsystems are use the upper bound (UB)30 energetically constrained. More precisely, let us discuss below the ‘ ’ 2 ΞðTÞ; ΞðBÞ : details on how we compare classical strategies (or benchmarks ) PH;UB ¼ðm À 1ÞF ð13Þ with quantum strategies. In a classical strategy (see Fig. 1b), we consider an input source However, far better results can be found by employing the which is described by a state with positive P-representation, so generalized CN receiver of Theorem 1. Note that the formulation that it emits a statistical mixtures of multi-mode coherent states. and proof of this theorem automatically applies to the extended First assume that this classical source has the GUS structure channel En !En I and the corresponding target/background m ϕ σ(T/B) → Ξ(T/B) , so that M modes and MNS mean photons are irradiated state . k¼1 Sk over each subsystem. In this case, we can directly map Eq. (1) into In the following we explicitly compare classical and quantum Eq. (2) and write the following lower bound based on ref. 30 (see performance for the paradigmatic cases mentioned in our Methods for more details) introduction, i.e., position-based quantum reading and quantum target finding, including their frequency-based spectroscopic m À 1 4 σðTÞ; σðBÞ ; formulations. In all cases we exactly quantify the quantum PH;LB ¼ F ð9Þ 2m advantage that is achievable by the use of entanglement. where F is the quantum fidelity. For the problem of CPF with arbitrary single-mode phase- Position-based quantum reading and frequency scanner.As insensitive bosonic Gaussian channels11,31 (see Methods for a depicted in Fig. 1, a possible specification of the problem is for the detailed definition), we prove a general classical benchmark. quantum readout of classical data from optical memories. In Suppose the target and background channels have transmissivity/ quantum reading12, the bosonic channels are used to model the μ μ fl gain T, B and output noises ET, EB. Given the most general re ection of light from the surfaces of an optical cell with dif- fl classical source at the input, i.e., a multimode mixture of coherent ferent re ectivities, whose two possible values rT and rB are used states not necessarily with GUS structure, and assuming it to encode a classical bit. In the absence of other noise, the readout
4 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 process is therefore equivalent to discriminating the value probability using thehi formula in Eq. (13), where the fidelity term ∈ ÀÁ r {rT, rB} of the loss parameter of a pure-loss bosonic channel F2 ΞðTÞ; ΞðBÞ ¼ F2M ðL IÞϕ ; ðL IÞϕ can be rT ME rB ME Lr. In our position-based formulation of the protocol, the clas- sical information is encoded in the position of a target cell (with exactly calculated (see Supplementary Note 1 for details). The fl μ = QR re ectivity T rT) within a pattern of m cells, where all the exact expression of the bound PH;UB is too long to display, but fl μ = ‘ ’ remaining are background cells (with re ectivity B rB). In will be used in our numerical comparisons (here QR stands for general, we probe each cell with M bosonic modes, so that we quantum reading). ðTÞ M ≪ ≫ fi have target channel Φ ¼L and background channel For NS 1 and M 1at xed MNS per cell, we have the rT ΦðBÞ M simple asymptotic expansion ¼Lr . In the following, we develop our theory of position- ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B p pffiffiffiffiffiffiffi QR based quantum and classical reading in this pure-loss setting, P ’ðm À 1ÞeÀ2MNSð1À ð1ÀrBÞð1ÀrT ÞÀ rBrT Þ: ð17Þ = = H;UB where EB ET 0. Our analysis can be extended to the presence + of extra noise (thermal-loss channels) as discussed in Methods. Comparing Eqs. (16) and (17), we can already see that, for rT ≥ As previously mentioned, we can map the model from spatial rB 1, the error exponent of the quantum case is better than the to frequency modes. This means that the problem may be exact error exponent of the classical case. In particular, this translated into a spectroscopic one where the goal is to find a faint advantage becomes large when both rT and rB are close to unity. absorbance line rT < 1 within a range W of transparent We can improve this result and show a greater quantum frequencies (rB ~ 1). This can be resolved into a discrete advantage by employing the generalized CN receiver of Theorem ensemble of m = W/δW modes, where δW is the bandwidth of 1. An important preliminary observation is that the output state ϕ the detector. The corresponding quantum-advantage can then be ðLr IÞ ME, from each probing of a generic cell, can be directly re-stated in terms of better identifying an absorbance line transformed into a tensor product form, where the signal mode is in a frequency spectrum, where we are constrained to use a white in the vacuum state and the idler mode is in a thermal state with − power spectral density over W for a certain time duration, so that mean photon number (1 r)NS. This is possible by applying a the total irradiated energy is equal to mMNS. This model can be two-mode squeezing operation S2[s(r, NS)], with strength considered both in transmission (e.g., in a spectro-photometer pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi þ À setup) and in reflection (e.g., in a scanner-like setup). ; 1 NffiffiffiffiffiffiffiffiffiffiffiffiffiffiS 1 ffiffiffiffiffiffiffiffirNS : sðr NSÞ¼ ln p p ð18Þ 2 NS þ 1 þ rNS Position-based reading with classical light. We can easily specify This allows us to design a CN receiver for the cell output state the lower bound in Eq. (10) to the reading problem, so that we get Ξ(ℓ), which consists of two-mode squeezing operations followed the following lower bound for position-based classical reading of by photon counting on the signal modes. By applying S2[s(rB, NS)] a block of m cells irradiated by mMNs mean photons to each pair of the 2M signal-idler modes, we have that Ξ(B) is pffiffiffi pffiffiffiffi 2 ~ðBÞ CR m À 1 À2MN r À r transformed into a state Ξ with vacuum signal modes; while ¼ SðÞB T ; ð14Þ PH;LB e ðTÞ 2m Ξ(T) becomes a state Ξ~ where the signal modes are in a product where ‘CR’ stands for classical reading. As discussed before, we of M thermal states, each with mean photon number can also obtain this bound from Eq. (9) by irradiating energetic pffiffiffiffiffi pffiffiffiffiffi 2 single-mode coherent states on each subsystem, i.e., m jiα ; ; NSðNS þ 1Þð rB À rT Þ : ffiffiffiffiffiffiffiffiffiffi k¼1 Sk nðN r r Þ¼ ð19Þ α p S B T 1 þ N ð1 À r Þ with ¼ MNS. S B Assuming the input source m jiα , the output states k¼1 Sk ffiffiffiffi Let us now measure the number of photons on the M signal ρ m σð‘Þ p α ℓ f ngn¼1 are pure, expressed by Eq. (2) with ¼ r‘ for modes. The outcomes are interpreted as follows: If we count any = T, B. Thus we can use Eq. (3) to calculate the Helstrom limit at photon then return ‘T’, otherwise return ‘B’. Assuming this rule, ðBÞ the output the background state Ξ~ does not lead to any photon count and, CR ; ; ; ; ζCR ; therefore, to any error. An error occurs only if, in the presence of P ðrB rT M NSÞ¼PHðm Þ ð15Þ H ðTÞ ffiffiffi ffiffiffiffi a target state Ξ~ , we get zero count on all M signal modes, which pffiffiffiffiffi pffiffiffiffiffi p p 2 ζCR α α 2 ÀMNSðÞrBÀ rT where ¼j rB j rT j ¼ e . In the limit of happens with probability small overlap ζ ≪ 1, we have ζQR ; ; ÀM: ð20Þ ffiffiffi ffiffiffiffi 1 ¼½1 þ nðNS rB rT Þ 1 p p 2 PCR ’ ðm À 1ÞeÀ2MNSðÞrBÀ rT ; ð16Þ H 4 This measurement implements the b-POVM of our CN receiver (unambiguous over background cells). which is only m/2 times larger than the lower bound in Eq. (14). Let us now realize the t-POVM, which is unambiguous on This also means that the lower bound is tight in the error target cells. In this case, we apply the operator S [s(r , N )] with fi 2 T S exponent. Although it is extremely dif cult to minimize the Ξ~ðTÞ Helstrom limit by varying the input among general non- different squeezing, so that has vacuum signal modes, while ~ðBÞ Rsymmetric classical states, we can show that mixtures of the type Ξ has thermal signal modes, each with mean photon number 2 m d αPðαÞ jiα hjα or increasing the modes in each sub- n(NS, rT, rB). By performing photon counting on the signal modes k¼1 Sk system do not improve the value of PCR (see details in Methods). and using the same rule above, we have that an error occurs only H ðBÞ if a background state Ξ~ gets zero counts on all M modes, which Position-based reading with entangled light. To get a quantum happens with probability advantage in terms of a lower error probability and, therefore, a ζQR ¼ ½1 þ nðN ; r ; r Þ ÀM: ð21Þ higher rate of data retrieval from the pattern, we interrogate each 2 S T B cell with the signal-part of an M-pair two-mode squeezed vacuum We can now study the performance of the CN receiver from state ϕ M. At the output of each cell, we get the state Ξð‘Þ ¼ Theorem 1, where we use the formula of Eq. (6) computed over hiME ζQR ζQR M the two types of error probabilities 1 and 2 . For position- ϕ ℓ = fi ðLr‘ IÞ ME for B, T. We can upper-bound the error based quantum reading of a block of m cells, we nd the
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Fig. 2 Position-based quantum reading. Quantum advantage shown for a block of m = 100 cells and NS = 5 mean photons per mode. a We consider the PQR =PCR PQR log ratio of the error probabilities (log 10½ CN H ), between quantum reading with conditional-nulling receiver CN and classical reading in the Helstrom CR limit PH . This ratio is plotted as a function of the background and target reflectivities, rB and rT, for M = 10 modes per cell. Note that since Eq. (6) is not r r PQR PCR M symmetric in B and T, we observe asymmetric patterns. b Error probabilities CN (black solid) and H (black dashed) versus number of modes , for fl r = r = PCR re ectivities B 0.95 and T 0.9. We also include the ultimate classical benchmark given by the lower bound for classical reading H;LB (gray dashed). c As in b but with rB = 1 and rT = 0.4. achievable error probability which shows a large advantage in the error exponent with respect to the classical strategy of Eq. (16). In Fig. 3 we show the quantum PQR ¼ PCNðζQR; ζQRÞ: ð22Þ CN m 1 2 advantage both in terms of error exponent and actual values of ≪ At low photon numbers NS 1 while keeping the total the error probabilities. This further quantum enhancement is irradiated energy MNS as a finite value, we have that particularly relevant to spectroscopy, where the background is QR P ’ 2PCRðr ; r ; M; N Þ, i.e., a factor of two worse than the indeed highly transparent with rB very close to unity. CN H B T S = classical performance in Eq. (16). However, for larger values of N Finally, let us note that the other case of rT 1 and rB < 1 can pffiffiffiffiffi pffiffiffiffiffi S and assuming the condition N ð r À r Þ2 1, we find that be improved in the same way, leading to an improved type-II S B T error probability ffiffiffi ffiffiffiffi p p 2 1 1 QR m À 1 ÀMðNSþ1ÞðÞrBÀ rT 1Àr þ1Àr ÂÃQR T B ; ð23Þ pffiffiffiffiffi À2M ζ PCN ’ e ζQR 2 ; 2 2Ã ¼ 1 þ NSð1 À rBÞ ¼ M ð26Þ ½1 þ NSð1 À rBÞ which has a large advantage in the error exponent when rB and rT are close to 1, as also evident from Fig. 2. and the overall error probability QR P ’ PCRðr ; 1; M; N Þ ´ 2eÀMNSð1ÀrBÞ: ð27Þ Further quantum enhancement. Let us consider an ideal sce- CNÃ H B S nario for position-based quantum reading, where the target cell with rT < 1 has to be found among many background cells with Quantum target finding. In general, target detection involves a fl = fi perfect re ectivity rB 1. This con guration allows us to show an search in multiple space-time-frequency bins. Time bins are even higher quantum advantage. In fact, for ideal background associated with ranging, frequency bins can be used for speed = (rB 1), the application of S2[s(rB, NS)] generates a background detection via Doppler effect, while space bins are associated with ðBÞ fi state Ξ~ which is vacuum in all signal and idler modes, and a direction nding. Let us study the latter problem here, i.e., dis- Ξ~ðTÞ covering the position of a single target in terms of polar and target state which is non-vacuum on all these modes. We can azimuthal angles, while we assume it is at some fixed range R and therefore apply the b-POVM of the CN receiver to the entire set does not create large Doppler shifts. Let us divide the R-radius of 2M signal and idler modes. horizon sphere into m non-overlapping sectors, one of which The type-I error probability is obtained by calculating the contains the reflective target. For large m, each sector S is ~ðTÞ k fidelity between Ξ and the vacuum state (see Supplementary approximately subtended by a corresponding small solid angle Note 1 for details). This leads to (see Fig. 1). We simultaneously probe all m sectors, while using M bosonic ÂÃpffiffiffiffiffi ζQR ζQR À2M 1 ; modes for each of them (e.g., a train of temporal pulses or a single 1Ã ¼ 1 þ NSð1 À rT Þ ¼ M ð24Þ ½1 þ NSð1 À rT Þ broadband pulse). Each signal mode will shine NS mean number N ζQR of photons. Let us denote by Lμ a thermal-loss channel with loss with a clear improvement with respect to the previous case 1 . Consider now the t-POVM. The application of the other parameter μ and mean number of thermal photons N, so that its = − μ Ξ~ðTÞ output noise is E (1 )N. When the target is present in a squeezing operator S2[s(rT, NS)] generates a target state with sector, the M signal modes go through the target channel vacuum signals but non-vacuum idlers, so that we must again M ðTÞ N =ð1ÀηÞ restrict photon counting to the signal modes, implying that we Φ ¼Lη B , so that each mode is affected by loss achieve the same type-II error probability as before, i.e., parameter μ = η and output noise E = N . By contrast, if the ζQR ζQR T T B 2Ã ¼ 2 . target is absent in a sector, then the M signal modes are lost and Using Eq. (6), we derive the overall error probability replaced by environmental modes, each having NB mean thermal QR CN ζQR; ζQR ≪ PCNÃ ¼ Pm ð 1Ã 2Ã Þ. At low photon numbers NS 1 while photons. For target absent, we therefore have the background fi fi M keeping the total energy MNS as nite, we nd ΦðBÞ NB μ = = channel ¼L0 , with B 0 and EB NB (no passive QR CR ; ; ; ´ ÀMNSð1ÀrT Þ; PCNÃ ’ PH ð1 rT M NSÞ 2e ð25Þ signature).
6 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. 3 Position-based quantum reading with ideal background. Quantum advantage for ideal background reflectivity (rB = 1) and considering m = 100 PQR =PCR PQR cells. a We consider the log ratio of the error probabilities (log 10½ CNÃ H ), between quantum reading with improved conditional-nulling receiver CNÃ CR and classical reading in the Helstrom limit PH . This ratio is plotted as a function of the target reflectivity rT and mean photon number per mode NS for fixed MNS = 12, where M is the number of modes. b We show the various error probabilities, i.e., quantum reading with the improved conditional nulling receiver PQR PQR CNÃ (including measurements of the idlers, gray solid), quantum reading with the conditional nulling receiver CN (based on the measurement of the PCR PCR signals only, black solid), the classical performance H (black dashed), and the ultimate classical benchmark H;LB (gray dashed). These are plotted versus the number of modes M, for rT = 0.95 and NS = 5. c As in b but choosing parameters rT = 0.4 and NS = 5.
We consider the region of quantum illumination13, where where v = N /(N + 1) and Ck is the binomial coefficient ≫ B B m bright thermal noise NB 1 is present in the environment, as it (number of combinations of k items out of m). would be the case at the microwave wavelengths15. We then In the high-noise N ≫ 1 and large number of modes M ≫ 1 ≪ B consider low energy signals (NS 1) so that the probing is non- limit, this error probability is dominated by the smallest error revealing and/or non-destructive for the target. In these exponent in the sum, and it becomes conditions, the considered quantum channels are clearly CTF m À 1 η = : entanglement-breaking. Before we present the corresponding PDD ’ expðÞÀM NS 2NB ð32Þ results, let us note that the model for target finding can also be 2m mapped to a model of quantum-enhanced frequency scanner, This is only a factor 2 worse than the bound in Eq. (28). In these now in the presence of bright environmental noise. See Methods limits, we expect that classical target finding via a DD scheme is for more details on this mapping and also for a discussion on close to the optimum. target ranging. Target finding with entangled light. Let us now assume a tensor fi ϕ mM Target nding with classical light. The general lower bound in product of two-mode squeezed vacuum states ME at the input. Eq. (10) can be specified to classical target finding, by setting In each M-mode probing of a sector, the ensemble of possible E = E = N and μ = η, μ = 0, so that we have output states takes the form of Eq. (12) with the following T B B T B η background and target states CTF m À 1 2M NS hi P ; ¼ exp À ; ð28Þ M H LB ΞðBÞ NB ϕ ; 2m 2NB þ 1 ¼ðL0 IÞ ME ð33Þ where ‘CTF’ stands for classical target finding. This expression hi = η M bounds the best performance achievable by classical sources of ΞðTÞ NB ð1À Þ ϕ : ¼ðLη IÞ ME ð34Þ light that globally irradiate mMNS mean photons over the entire sphere. In particular, we can also obtain this bound fromffiffiffiffiffiffiffiffiffiffi Eq. (9) Let us compute an upper bound based on Eq. (13). Its exact m p by considering m single-mode coherent states jiMNS , expression is too long to display, even though it is used in our k¼1 Sk ≪ ≫ each shining MN mean photons on a sector. numerical evaluation. In the limits of NS 1 and M 1 while S fi fi Let us compute the classical performance with a specific keeping the total energy per sector MNS as xed, we nd the following asymptotic bound for quantum target finding receiver.pffiffiffiffiffiffiffiffiffiffi When we use the uniform coherent source m η k¼1jiMNS S at the input, the ensemble of output states of QTF M NS k P ; ðη; N ; M; N Þ’ðm À 1Þ exp À ; ð35Þ Eq. (2)isdefined on the following background and target states H UB B S 1 þ N ÀÁ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi B σðBÞ NB ; where ‘QTF’ stands for quantum target finding. This has no ¼L0 MNS MNS ð29Þ advantage with respect to Eq. (28), but both bounds are likely to ÀÁ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi = η be non-tight. It has instead a factor of 2 advantange in the error σðTÞ ¼LNB ð1À Þ MN MN : ð30Þ η S S exponent with respect to the DD result in Eq. (32) for large noise. This is identicalffiffiffiffiffiffiffiffiffiffiffiffiffi to classical pulse-position modulation decoding To better evaluate the performance of the entangled case, we need pη 22 to analyze an explicit receiver design. with signal MNS and thermal noise NB . We can therefore consider the direct detection (DD) scheme based on photon We adapt the quantum illumination receiver based on sum- 32 counting (see refs. 2,22), giving the error probability frequency-generation (SFG) process to the CN approach in Theorem 1. Consider the problem of binary hypothesis testing Xm 1 ð1 À vÞð1 À vkÀ1ÞηMN between the states Ξ(B) and Ξ(T). An SFG receiver converts the PCTF ¼ ðÀ1ÞkCk ´ exp À S ; DD m m 1 À vk signal-idler cross correlations into photon number counts, k¼2 through the combination of multiple cycles of SFG process and ð31Þ ≪ ≫ interference. In the limit of NS 1 and NB 1 with feed-forward
COMMUNICATIONS PHYSICS | (2020) 3:103 | https://doi.org/10.1038/s42005-020-0369-4 | www.nature.com/commsphys 7 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-0369-4 disabled, the photon counting statistics of Ξ(T) is equivalent to a position modulation18, so that it clearly departs from other η + coherent state with mean photon number M NS(NS 1)/NB, and approaches that exploit pulse position modulation for state-based Ξ(B) is equivalent to a vacuum state. After this conversion, encoding (e.g.33). In this scenario, we showed that the use of an suppose we perform the photon-counting stage of the SFG entangled source and a suitably constructed conditional-nulling measurement on the background state Ξ(B), then there is always receiver can outperform any classical strategy in finding the zero count and therefore no ambiguity. For Ξ(T), there is instead unknown position of the channel. This quantum advantage, QTF η = ζ ÀM NSðNSþ1Þ NB which is quantified in terms of improved error probability and some type-II probability 2 ¼ e of getting zero count and therefore selecting the wrong hypothesis ‘B’. This error exponent, has been demonstrated for paradigmatic exam- corresponds to the b-POVM of the generalized CN receiver. On ples of position-based quantum reading and quantum target the other hand, for the t-POVM, suppose we apply a two-mode finding, besides their spectroscopic formulations as quantum- QTF squeezer S2(r ) before performing the previous SFG measure- enhanced frequency scanners. As further theoretical directions, it ment, where would be interesting to exactly establish the optimal performance "#pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for discriminating quantum channels with geometrical uniform 1 À2 ηN ðN þ 1Þ symmetry. Finally, although our analysis relies on symmetry, we rQTF ¼À arctan S S ð36Þ 2 1 þ NS þ NB expect that a similar quantum advantage exists in problems with completely arbitrary channel patterns. QTF ΞðTÞ y QTF is chosen such that S2ðr Þ S2ðr Þ has zero cross correla- tions. Then we decide ‘T’ when no photon is counted, making no Methods error. However, when the input is Ξ(pB),ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the squeezer will create Phase-insensitive bosonic Gaussian channels. The action of a single-mode x^ ^; ^ T phase sensitive cross correlations ’ ηN ðN þ 1Þ. When no (covariant) phase-insensitive Gaussian channel overffiffiffi inputpffiffiffiffiffiffiffiffiffiffiffiffiffiffi quadratures ¼ðq pÞ S S x^ pμx^ μ x^ ξ μ ‘ ’ can be represented by the transformation ! þ j1 À j E þ , where is counts are registered, we select the wrong hypothesis T , with ≤ μ ≤ μ ≥ x^ a transmissivity (0 1) or a gain ( 1), E are the quadratures of an envir- ζQTF ζQTF ω = + type-I error probability 1 ¼ 2 . onmental mode in a thermal state with noise variance 2N 1 with N being the According to Theorem 1, the performance of the generalized mean number of photons, and ξ is additive classical noise, i.e., a random 2D CN receiver (here applied to signals and idlers) corresponds to Gaussian distributed vector with covariance matrix waddI. Here we assume vacuum shot noise equal to 1. the following mean error probability Note that, for a coherent state at the input, the output state of the channel is = μ + ∣ − μ∣ω + ω ω = 1 η = generally thermal with covariance matrix V ( 1 add)I. Setting QTF CN ζQTF; ζQTF À2M NS NB : + − ω − μ ∣ − μ∣ + I PCN ¼ Pm ð 1 2 Þ’ ðm À 1Þe ð37Þ (1 2E add )/ 1 , this matrix simply becomes (2E 1) . Therefore, 2 conditionally on a coherent state input, the channel can be described by the two μ ≤ μ ≤ Comparing with Eq. (28), we see that the achievable performance parameters and E. In particular, for a thermal-loss channel, we have 1 1, fi and E = (ω − 1)(1 − μ)/2 = (1 − μ)N; for a noisy amplifier, we have μ ≥ 1, and E = of quantum target nding clearly outperforms the bound on (ω + 1)(μ − 1)/2 = (μ − 1)(N + 1); and finally, for an additive Gaussian noise fi μ = = ω classical target nding. In particular, we see that the error channel, we have 1 and E add/2. exponent is increased by a factor 2. We explicitly compare these results in Fig. 4. Optimal receiver design for standard quantum reading. The novel CN receiver design also provides a new insight into the original quantum reading model, related to the binary discrimination between the two lossy channels L and L .Withnoloss Discussion rT rB of generality, let us assume rB > rT. When the two-mode squeezed vacuum state is In this work we showed that the use of quantum entanglement used at the input, the corresponding outputs for the two channels are Ξ(T) and Ξ(B). can remarkably enhance the discrimination of multiple quantum Therefore, the t-POVM and b-POVM can be directly used to perform their dis- ζQR = hypotheses, represented by different quantum channels. More crimination, leading to the error probability 1 2 for equal prior probabilities, where ζQR = precisely, we considered a basic problem of quantum pattern 1 is given in Eq. (20) (see orange line in Fig. 5). In the ideal case of rB 1, the recognition that we called CPF. This model can also be regarded further improved detection, given by the application of the CN receiver to both signals ζQR = ζQR fi as a quantum channel formulation of the classical notion of pulse and idlers, leads to the error probability 1à 2, where 1à is de ned in Eq. (24)(see ζQR = reddottedlineinFig.5). We see that the improved performance 1à 2 saturates the 34,35 ζQR = quantum Chernoff bound , while the general applicable performance 1 2isable to beat the best known Bell-measurement receiver12,whenM is sufficiently large (Fig. 5a) or NS is large (Fig. 5b).
Quantum-enhanced frequency scanner in noisy conditions. The previous result on quantum-enhanced target finding can be mapped into the model of quantum- enhanced frequency scanner, now in the presence of bright environmental noise. Here we assume a target at some fixed linear distance which only reflects radiation at a narrow bandwidth δν around some carrier frequency. The target is assumed to be still (or slowly moving) and it completely diffracts the other frequencies. This limited reflection could also be the effect of meta-materials employed in a cloak. The previous m sectors now become m different non-overlapping frequency windows with bandwidth δν, each of them probed by pulses with the same bandwidth. One choice is to use a single δν-pulse per window containing M ≃ δν−1 effective frequencies, each with NS mean number of photons. Alternatively, we may use δν Fig. 4 Target direction finding with classical and entangled light. We plot M -pulses per window which are irradiated as a train of independent temporal modes, each with N mean photons. In our basic model, reflection occurs in only the error probabilities in terms of number of modes M, considering m = S one of these frequency windows, while background thermal noise is detected for all −3 50 sectors, NS = 10 photons per mode, NB = 20 thermal photons per the other windows. The previous results (see Fig. 4) automatically imply that the environmental mode, and η = 0.1 round-trip loss. We consider the use of an entangled source outperforms any classical strategies in the regime of few performance of classical target finding via direct detection from Eq. (31) photon numbers per mode. (CTF-DD, solid black line) and assuming the lower bound of Eq. (28) (CTF- LB, black dashed line). We then consider the performance of quantum About target ranging. In quantum target finding, if we consider time bins instead fi target finding assuming the upper bound of Eq. (35) (QTF-UB, red dashed of spatial bins, we can map the problem of direction nding into that of ranging. However, at fixed direction but unknown distance, there is a crucial problem which line) and via the generalized CN receiver from Eq. (37) (QTF-CN, solid makes the entangled strategy problematic. We must in fact ensure that the red line). returning signal (if any) is combined with the corresponding idler. Since we do not
8 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. 5 Error probability versus number of modes M for binary quantum reading. Background and target reflectivities are respectively rB = 1 and rT = 0.4. 12 Comparisons are done for a number of photons per mode NS = 0.1 in a and NS = 10 in b. We plot the performance of the original Bell receiver (solid black ζQR= line), the asymptotically tight quantum Chernoff bound (QCB, solid blue line), the generalized conditional nulling receiver with performance 1 2 (CN, ζQR= solid orange line), and the generalized conditional nulling receiver with improved performance 1Ã 2 (CN*, red dashed line).
know, a priori, the round-trip time from the target, we cannot synchronize signal Suppose that, from the first measurement, we instead get the correct outcome ‘ ’ − ζ and idler in a joint detection. A potential way around this issue is to generate a B (with probability 1 1). Then, the receiver would correctly discard the false train of m signal-idler pulses with well-separated carrier frequencies (e.g., with a hypothesis h1 and would check the next one h2. Denote by Pm−1 the total error bandwidth larger than the maximum Doppler shift from the target). Signal-idler probability of the receiver in distinguishing the remaining m − 1 hypotheses. Then, pulses with different carrier frequencies are then jointly detected at the different m the overall (conditional) probability of error is given by the product of Pm−1, and ‘ ’ time bins. In principle this procedure can make the quantum measurement work the joint probability of outcome B for h1 being false. Therefore, we have but it opens another issue. The best classical strategy does not need to employ this PB ¼ðm À 1ÞmÀ1ð1 À ζ ÞP .Ifm = 2, then in this case there is only one h6 1 1 mÀ1 time slicing approach. In fact, one could just send a single coherent pulse and wait = hypothesis left, and we have the initial condition P1 0. for its potential return. From an energetic point of view, the classical source would CN ζ ; ζ Overall, the error probability of the receiver Pm Pm ð 1 2Þ will be equal to only irradiate MNS photons (assuming M modes per pulse) while the quantum case T B the sum of Ph6 and Ph6 , so that we have the recursive formula needs to irradiate mMNS photons on the target. Taking into account of this dif- 1 1 fi m À 1 ference, we cannot directly apply our previous ndings and derive a conclusive P ¼ ½ð1 À ζ ÞP þ ζ ζ : ð39Þ result for target ranging. m m 1 mÀ1 1 2 = = ζ ζ The initial conditions of the recursion is that P1 0 and P2 1 2/2. To solve the = − − ζ − = − ζ ζ Optimality of pure states. Here we state two lemmas to summarize the results recursion, let us set Pm gm/m so that we have (1 1)gm−1 gm (m 1) 1 2 = = − ζ ζ fi (See Supplementary Note 2 for their proofs). with initial conditions g1 0 and g2 1 2.We nd the solution XmÀ2 g ¼Àζ ζ ðm À nÞðÞ1 À ζ nÀ1 ¼Àζ ζ ðmζ þ ðÞ1 À ζ m À 1Þ=ζ2; ð40Þ Lemma 2. m 1 2 1 1 2 1 1 1 n¼1 Consider the discrimination of N channels fEng with prior probabilities {pn}. Inputting pure states minimizes the mean error probability. which leads to Note that if there is a constraint on the Hilbert space (e.g., an energy constraint ζ 1 m for an infinite-dimensional space), then the previous lemma might not hold. P ¼ 2 ½mζ þð1 À ζ Þ À 1 ; ð41Þ m m ζ 1 1 However, this result may still hold in the presence of convexity properties, as in the 1 proof of the following lemma. completing the proof. Note that, when the receiver outcomes are all ‘B’, this automatically means that the true hypothesis is the last one hm, which is compatible = Lemma 3. with the initial condition P1 0. Consider position-based quantum reading, with a constraint of MNS mean photon General bounds. Here we present various general bounds that apply to m-ary state numbers per cell. Any statisticalp mixtureffiffiffiffiffiffiffiffiffiffi of GUS coherent states can be reduced to – m α α discrimination (in the setting of symmetric hypothesis testing)30,36 38. These k¼1jiS with amplitude ¼ MNS. The minimum error probability is k bounds apply to the mean error probability and can be computed from the CR ; ; ; ; ζCR ; quantum fidelity (which has a closed formula for arbitrary multimode Gaussian PH ðrB rT M NSÞ¼PH ðm Þ ð38Þ 39 ; ρ m ’ ffiffiffiffi ffiffiffiffi states ). In particular, for any ensemble of m mixed states fp g , where pk s CR pffiffiffiffiffi pffiffiffiffiffi 2 p p 2 k k k¼1 ζ α α ÀMNS ð rBÀ rT Þ ρ ’ where ¼jh rB j rT ij ¼ e and the function PH is given in Eq. are the prior probabilities and k s are the states, we may write the following upper 30 (3) of the main text. bound on the minimum error probability or Helstrom limit PH X ffiffiffiffiffiffiffiffiffiffi ≤ p ρ ; ρ ; PH PH;UB 2 pk0 pkFð k0 kÞ ð42Þ Generalized CN Receiver (proof of theorem 1). Let us describe the measurement 0 = k >k process starting from n 1, i.e., by checking the hypothesis h1 that the target state σ(T) is in subsystem S .Ifh is true, then the receiver will not make any error, due where F is the Bures’ fidelity 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΠðTÞσðTÞ fi ΠðBÞσðBÞ pffiffiffipffiffiffi pffiffiffi pffiffiffi to trð t Þ¼1 on the rst subsystem S1 and trð t Þ¼1 on all the other ρ; σ : ρ σ ρσ ρ: ð43Þ m m Fð Þ ¼ 1 ¼ tr subsystems fSkgk¼2. There is an error only if the true hypothesis is one of fhkgk¼2. σ(B) ΠðTÞ; ΠðBÞ ‘ ’ In this case, S1 would be in the background state and the t-POVM f t t g The result of Eq. (42) is a bound on the performance of a pretty good mea- ‘ ’ ζ ‘ ’ 27–29 fi would return the incorrect outcome T with probability 1 and correct outcome B surement and is tight up to constan t factors in the exponent. A delity-based − ζ 40 with probability 1 1. lower bound is instead given by ref. , ‘ ’ ζ X Suppose that we get T (with type-I false-positive probability 1) while the ≥ 2 ρ ; ρ : ~ PH PH;LB p 0 p F ð 0 Þ correct hypothesis is h~ for some k>1. In measuring the remaining subsystems k k k k ð44Þ k k0 >k m ðTÞ ðBÞ fS g in the b-POVM fΠ ; Π g, the outcomes will be certainly equal to ‘B’ for = −1 k k¼2 b b Assume equi-probable hypotheses, so that pk m for any k, and the ≠ ~ σ(B) ρ ; ρ ; ≠ 0 fi all systems with k k since they will all be in a background state . However, the symmetry Fð k k0 Þ¼F 8k k . We then have the simpli ed bounds application of b-POVM over the target state σ(T) of subsystem S~ could give the k ; ‘ ’ ζ PH;UB ðm À 1ÞF ð45Þ wrong outcome B with type-II (false-negative) probability 2. If this happens the receiver would select the false hypothesis h1. In this case, the overall (conditional) ζ ζ m À 1 2 probability of error is given by the product of the two incorrect outcomes 1 2 times P ; F : ð46Þ − −1 H LB the probability that h1 is false, i.e., (m 1)m . Therefore, we get 2m PT ¼ðm À 1ÞmÀ1ζ ζ . These bounds appear in our main text with the following expressions for the h6 1 1 2
COMMUNICATIONS PHYSICS | (2020) 3:103 | https://doi.org/10.1038/s42005-020-0369-4 | www.nature.com/commsphys 9 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-0369-4
fi delity or rT in such a way that the block of m cells has GUS. The block is probed by bosonic modes for a total of mMNS mean photons irradiated. In the classical case, ρ ; ρ 2 ΞðTÞ; ΞðBÞ ; Fð n n0≠nÞ¼F ð47Þ we compute a lower bound to the performance of all possible classical states (globally irradiating mMNS mean photons over the m block of cells), while for the for the entangled case and quantum case, we consider a tensor-product of two-mode squeezed vacuum states, 2 ðTÞ ðBÞ so that M signal modes probe each cell, with each mode irradiating NS mean ðρ ; ρ 0 Þ¼ σ ; σ ð Þ F n n ≠n F 48 photons. for the classical case. As before, this problem is mapped into the discrimination of an ensemble of GUS bosonic channels fE g with equal priors, which are expressed by n ðBÞ ðTÞ Classical benchmarks. Let us now introduce a general bound to the ultimate E ¼ ≠ Φ Φ ; ð54Þ n k n Sk Sn performances achievable by classical states in CPF, with direct application to the fi ΦðB=TÞ problems of position-based reading and target nding. Recall that the general with acting on cell Sk. For M-mode probing of the cell, we have the target Sk problem of CPF consists of discriminating an ensemble of GUS bosonic channels M M ΦðTÞ NB ΦðBÞ NB fE g with equal priors. These are expressed by channel ¼Lr and the background channel ¼Lr , where n T B NB fl ðBÞ ðTÞ Lr is a single-mode thermal-loss channel with re ectivity r and thermal noise NB. E ¼ ≠ Φ Φ ; ð49Þ n k n Sk Sn In general, the protocol of position-based quantum reading can be formulated = with two generic thermal-loss channels as discussed above. In such a case, the where ΦðB TÞ is the background/target channel acting on subsystem S (e.g., a cell Sk k classical benchmark can be easily derived from Eq. (52). Then, we may introduce a or a sector). Each of these channels is generally meant to be a multi-mode channel. finer classification of the protocol in two types: one with active and the other with In the bosonic setting, single-mode phase-insensitive Gaussian channels model passive signature. In the first type of protocol, the parameters of the channels are various physical processes. This channel Gμ;E can be parameterized by a such that the noise variance at the output of the two channels is different assuming transmissivity/gain parameter μ > 0 and a noise parameter E >011,31. In particular, the vacuum state at the input. In other words, their statistical discrimination is E accounts for the thermal photons at the output of the channel, when the input possible without sending a probing signal. In the second type, the parameters are state is a vacuum or coherent state. Besides the single-mode phase-insensitive such that there are no different levels of noise at the output. Here we analyze this fl (covariant) bosonic Gaussian channels discussed above, we can also include the second type, so that the channels have re ectivity rl and mean number of thermal − = contravariant conjugate thermal-amplifier channel, whose action on an input photons NB/(1 rl) for l B, T. The corresponding classical benchmark can be annihilation operator is described by computed from Eq. (53) and takes the form ffiffiffi pffiffiffiffiffiffiffiffiffiffiffi "#ffiffiffiffiffi ffiffiffiffiffi ^ pμ^y μ ^; p p 2 a ! a þ þ 1e ð50Þ CR;N ; ; ; m À 1 À2Mð rB À rT Þ NS : PH;LB ðrB rT M NSÞ¼ exp ð55Þ where μ > 0 and ^e is in a thermal state with mean photon number (E − μ)/(μ + 1). 2m 2NB þ 1 α All these channels Gμ;E map a coherent state jito a displaced thermal state with Similarly, for the quantum case, we can easily repeat the calculations to find the pffiffiffi pffiffiffi ? μα μα fi QR;N QR ≪ amplitude ( for the conjugate thermal-ampli er channel) and corresponding noisy expression PH;UB of the upper bound PH;UB. For NS 1 and + I ≫ fi covariance matrix (2E 1) . M 1at xed MNS, we may generalize Eq. (17) of our main text into the following Therefore, let us consider the problem of CPF where target and background form "#ffiffiffiffi channels are tensor products of a phase-insensitive bosonic Gaussian channel Gμ;E. p pffiffiffiffiffiffiffiffiffi μ ; À2MN ð1 þ N À H À r r Þ Denote the transmissivity/gain and noise of the target channel as T and ET, while QR N ; ; ; S B B T ; μ PH;UB ðrB rT M NSÞ’ðm À 1Þ exp ð56Þ those of the background channel as B and EB. For the entangled case, we assume 1 þ NB that each subsystem is exactly probed by M signal modes, each irradiating N mean S = + − + − photons, for a total of mMN mean photons. For the classical case, we can relax this where H (1 NB rB)(1 NB rT). S ϵ structure and include the more general case of different energies irradiated by the Denote the error exponent in Eq. (55)as CR and the error exponent in Eq. (56) ϵ fi M modes over each subsystem S . More generally, for the classical case with no as QR.We nd that the quantum case is always better than the classical case, i.e., k ϵ ϵ ϵ ϵ ≃ + = QR > CR. For rT and rB close to 1, we have QR/ CR 1 1/2NB. In this regime, we passive signature (EB ET), we can also allow for arbitrary number of modes Mk see that the advantage becomes huge when N ≪ 1, which agrees with our ΦðlÞ Mk B per subsystem Sk so that S ¼Gμ ;E . In other words, for classical CPF with no ≫ k l l observation in Eqs. (16) and (17). However, when NB 1, the advantage decays, in passive signature, the only surviving constraint is the mMNS mean photons globally agreement with the observation related to Eqs. (28) and (35). Note that this irradiated. More precisely, we can state the following result (See Supplementary conclusion is based on a quantum lower bound and a classical upper bound, and Note 2 for proof). we expect them to be not tight when noise NB is large.
Lemma 4. Data availability Consider the problem of CPF where target and background channels are tensor The data supporting the findings of this study are available as Supplementary Data, more products of a single-mode phase-insensitive bosonic Gaussian channel with para- details can be obtained from the first author upon reasonable request. μ μ meters T,ET (for target) and B,EB (for background). Assume a global energetic constraint of mMNS mean photons with M modes irradiated over each of the m subsystems Sk. The optimal classical state (with positive P-representation) mini- Code availability mizing the lower bound PH,LB of Eq. (44) is any tensor product of coherent states The theoretical results of the manuscript are reproducible from the analytical formulas qffiffiffiffiffiffiffiffiffiffi and derivations presented therein. Additional code is available from the first author upon 0 0 m M iθðk Þ ðk Þ jiα ¼ 0 e k N ; reasonable request. k¼1 k ¼1 Sk ð51Þ S P k ðk0Þ M ðk0Þ where the phases θ are arbitrary and 0 N ¼ MN for any k, so that each k k ¼1 Sk S Received: 10 February 2020; Accepted: 1 May 2020; subsystem is irradiated by the same mean number of photons. The corresponding minimum lower bound is given by "#ffiffiffiffiffi ffiffiffiffiffiffi pμ pμ 2 m À 1 2M 2MNSð B À T Þ P ; c ; ´ exp À ; ð52Þ H LB EB ET 2m 1 þ EB þ ET ÀÁpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 À1 References with cE ;E ¼½1 þ EBð1 þ ET Þ À ET ð1 þ EBÞ . In particular, for no pas- B T = ≡ fi sive signature (ET EB E), we have the simpli cation 1. Pirandola, S., Bardhan, B. R., Gehring, T., Weedbrook, C. & Lloyd, S. "#ffiffiffiffiffi ffiffiffiffiffiffi Advances in photonic quantum sensing. Nat. Photon. 12, 724–733 (2018). pμ pμ 2 m À 1 2MNSð B À T Þ 2. Helstrom, C. Quantum Detection and Estimation Theory, Mathematics in P ; exp À ; ð53Þ H LB 2m 1 þ 2E Science and Engineering: A Series of Monographs and Textbooks. (Academic Press, New York, 1976). and bound holds under the general energetic constraint of mMNS mean photons, 3. Chefles, A. & Barnett, S. M. Quantum state separation, unambiguous with no restriction on the number of modespffiffiffiffiffiffiffiffiffiffi irradiated per subsystem. In this case, an discrimination and exact cloning. J. Phys. A: Math. Gen. 31, 10097 optimal state is the tensor-product m jiMN . k¼1 S Sk (1998). 4. Chefles, A. Quantum state discrimination. Contemp. Phys. 41, 401–404 (2000). Position-based quantum reading with thermal noise. Let us now generalize the 5. Hirota, O. Properties of quantum communication with received quantum state study of position-based quantum reading to the case where thermal noise is present control. Opt. Commun. 67, 204–208 (1988). in the environment. This means that the environmental input of each cell Sk is not 6. Kitaev, A. Y. Quantum computations: algorithms and error correction. fl – the vacuum but a thermal state with NB mean photons. Each cell has re ectivity rB Russian Math. Surv. 52, 1191 1249 (1997).
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