Systems and methods for optimized receiver design for entanglement-assisted communication using bpsk

Abstract

Examples of a 2´2 optical hybrid receiver for entanglement-assisted communication are disclosed indicating that the examples of the receiver have roughly 10% lower error probability compared to previously proposed optical parametric amplifier-based receivers for more than 10 modes. Numerical findings indicated that surpassing the Holevo and Homodyne capacities does not require a large number of signal-idler modes. In addition, findings indicated that using unequal priors for Binary Phase-Shift Keying (BPSK) provides roughly three times the information rate advantage over.

Description

Attorney Docket No.: 085067-833105 (UA24-111) PCT SYSTEMS AND METHODS FOR OPTIMIZED RECEIVER DESIGN FOR ENTANGLEMENT-ASSISTED COMMUNICATION USING BPSK CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This is a PCT patent application that claims benefit to U.S. provisional application serial number 63 / 566,709 filed on March 18, 2024, which is incorporated by reference in its entirety. FIELD

[0002] The present disclosure generally relates to quantum technologies; and in particular to an optimized receiver design for entanglement-assisted communication. BACKGROUND

[0003] Quantum Information Processing (QIP) has seen tremendous progress in recent decades, with multiple research directions exploring quantum sensing, covert communication, quantum cryptography, and more. A quantum channel is used to transfer quantum information from one party (known as Alice) to another party (known as Bob). In the case of a perfect channel, the quantum information is transferred intact, but if the channel is noisy, the quantum information undergoes some changes.

[0004] It is with these observations in mind, among others, that various aspects of the present disclosure were conceived and developed. 1 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT BRIEF DESCRIPTION OF THE DRAWINGS

[0005] FIG.1 is an illustration showing the concept of Entanglement- Assisted (EA) communication. EA-assisted communication is enabled by either fiber- optic based distribution of entanglement or entanglement distribution through satellites. The information is transmitted through a lossy and noisy Bosonic channel.

[0006] FIG.2 is an illustration of the Operating Principle of Optical Parametric Amplifier (OPA): at the receiver end, parametric amplification is applied to the return-idler pair with Gain G. Error probability of discrimination is higher at ^̂^, hence the photo detection is made ^^.

[0007] FIG.3 is an illustration of the Operating Principle of Optical Phase Conjugate (OPC) receiver: signals interact with the vacuum followed by mixing with idlers using 50-50 beam splitter and a balanced detection is applied using photodetectors.

[0008] FIG.4 is an illustration of a receiver configuration of a 2 X 2 optical hybrid-based joint balanced detection receiver.

[0009] FIGS.5A-5B are a pair of graphs illustrating symbol-by-symbol (separable) minimum error-probability measurement on each return idler mode pair at the idler output port. It was found that unequal priors have a lower error probability as compared to Binary Phase-Shift Keying (BPSK) symbols with equal prior. At bottom (b), symbol-by-symbol (separable) minimum error-probability measurement on each return- idler mode pair at the return output port of the OPA receiver is shown. As compared to measurement made at the idler output port of the OPA receiver, the error probability is higher at the return output port of the OPA receiver.

[0010] FIG.6 is a graph illustrating optimal threshold for hypothesis testing at the output port of OPA as a function of the number of modes. A higher threshold is required when detection is made at the return output port (red line) compared to the detection made at the idler output port of the OPA receiver (blue line). Unequal prior were p0=0.45, p1=0.55. Plot was generated using Ns=0.01, NB=1, ^=0.01.

[0011] FIGS.7A-7D are a series of graphs illustrating (a) surface plot of ^^^^as a function of prior ^^0and threshold mean photon number ^^^^ℎfrom Eq. (19) for photodetection at idler output port for OPA receiver; and (b)-(d) illustrate a 2D view of 2 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT the surface plot from a different direction. Color map denotes error probability – darkercolor means a lower error probability. The plot was generated using ^^^^ = 0.01, ^^^^ =1, h = 0.01. These plots reveal that for a fixed threshold, error probability is maximum atequal prior while non-equal probable symbol seems to reduce error probability.

[0012] FIG.8 is a report of channel capacity for BPRSK state discrimination using different receiver schemes for single mode. We see that even for a single mode, entanglement assistance offers an advantage in terms of increasing the capacity of the channel and beats the classical capacities such as Holevo capacity andHomodyne Capacity. The plot was generated using ^^^^ = 1, h = 0.01. The red line curvedenotes the theoretical ultimate capacity that can be achieved through the entanglement-assistance. Right plot zoomed-in to clearly illustrate that Holevo and Homodyne capacity are not exactly the same.

[0013] FIG. 9 is a plot showing channel capacity for BPSK state discrimination using different receiver schemes. We see that entanglement assistance offers an advantage in terms of increasing the capacity of the channel. The transmittivityof the Bosonic channel was chosen as ^= 0.01 and the mean background photon as ^^^^ =1. In our numerical study, classical capacity stays below 0.07 bits per channel use formean signal photon ^^^^ < 1.0.

[0014] FIG.10 is a plot of information rate as a function of the number of modes ^^ for OPA and the OPA receiver. It is evident that as we increase the number of modes the R / C for equal and unequal priors attain the same values. Further, we see better performance in terms of information rate when the number of modes is low. The plot was generated using a lower error probability. The plot was generated using ^^^^=0.01, ^^^^ = 1, h = 0.01.

[0015] FIG.11 illustrates reports of (top) capacity error in the case of OPA receiver as a result of approximating negative binomial photodetection statistics to Gaussian. At bottom: capacity of OPA receiver with the number of modes ^^ = 10. We observe that with Gaussian approximation, we overestimate Shannon’s capacity of the OPA receiver for EA communication while discriminating against BPSK states.

[0016] FIG.12 is an example process associated with the inventive concepts described herein. 3 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT

[0017] Corresponding reference characters indicate corresponding elements among the view of the drawings. The headings used in the figures do not limit the scope of the claims. 4 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT DETAILED DESCRIPTION

[0018] The present disclosure relates to systems and methods for optimized receivers for entanglement-assisted communication using Binary Phase-Shift Keying (BPSK). The use of pre-shared entanglement in entanglement-assisted communication offers a superior alternative to classical communication, especially in the photon-starved regime and highly noisy environments. By the inventive concept described herein, the performance of several low-complexity receivers that use optical parametric amplifiers is examined. The simulations demonstrate that receivers employing an entanglement- assisted scheme with phase-shift-keying modulation can outperform classical capacities. A 2x2 optical hybrid receiver is presented for entanglement-assisted communication and show that it has a roughly 10% lower error probability compared to previously proposed optical parametric amplifier-based receivers for more than 10 modes. However, the capacity of the optical parametric amplifier-based receiver exceeds the Holevo capacity and the capacities of the optical phase conjugate receiver and 2x2 optical hybrid receiver in the case of a single mode. The numerical findings indicate that surpassing the Holevo and Homodyne capacities does not require a large number of signal-idler modes. Furthermore, it was found that using unequal priors for BPSK provides roughly three times the information rate advantage over equal priors.

[0019] 1. Introduction

[0020] Quantum Information Processing (QIP) has seen tremendous progress in recent decades, with multiple research directions exploring quantum sensing, covert communication, quantum cryptography, and more. A quantum channel is used to transfer quantum information from one party (known as Alice) to another party (known as Bob). In the case of a perfect channel, the quantum information is transferred intact, but if the channel is noisy, the quantum information undergoes some changes. Quantum channels can also be used to transmit classical information. Additionally, if the channel is noisy within certain limitations, the quantum channel can be used to share entanglement between Alice and Bob. The use of pre-shared entanglement can enhance classical capacity and protect against an adversary, commonly referred to as Eve. Recent experiments have shown that even in entanglement-breaking scenarios, the rate of entanglement-assisted (EA) communication can be much higher than 5 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT ^^ communication without entanglement. The ratio^^^^^^ , where ^^^^^^is the entanglement-assisted capacity and ^^ is the Holevo- Westmoreland (HSW) capacity inthe classical regime, diverges logarithmically the inverse of the signal power over a lossy and noisy bosonic channel. Recent efforts have been made to design receivers for EA communication, where authors have utilized the Gaussian approximation of the cumulative distribution function to calculate the Bit Error Rate (BER).

[0021] Technical Solutions

[0022] In the present disclosure, we analyze the receiver design for entanglement-assisted (EA) communication using Optical Parametric Amplifiers (OPAs) and expand upon previous results to determine the optimality of the receiver design. We show that EA communication does not need to occupy the entire C-band. Additionally, we analyze a 2x2 optical hybrid-based receiver for EA communication that is suitable for implementation in integrated optics and quantum nanophotonics. In our scheme, optical phase conjugation is performed on the transmitter side when signal photons are brighter, rather than on the receiver side where the signal photons are buried in noise and highly attenuated.

[0023] We further propose an optimized hypothesis testing scheme and demonstrate numerically that the optimized receiver design provides a superior communication capacity compared to capacity without entanglement assistance. When using the BPSK modulation format to represent digital information, we find that non-equal priors perform at least three times better in terms of information rate compared to an equal prior encoding scheme.

[0024] The rest of the disclosure is organized as follows. In Section 2, we provide a brief review of entanglement-assistance with mathematical formalism necessary for the rest of the paper. In Section 3, we present an overview of the receiver design schemes for entanglement-assisted communication, including the optical parametric amplifier-based receiver design with threshold detection, the optical phase conjugation receiver, and the 2x2 optical hybrid-based joint receiver. These receiver designs are then evaluated in Section 4.

[0025] 1.1 Notations 6 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT

[0026] |∙^ is used for ket-notation in quantum mechanics, equivalent to a vector notation in linear algebra. The Hermitian conjugate of the vector, ^∙|, is referred toas bra-notation. The scalar product of two vectors |^^1^ and |^^2^ is denoted by ^^^1‖^^2^.Additionally, the ket-notation |^^^ represents a coherent state of amplitude ^^. The imaginary unit or a complex number√−1 is represented by ^^. Random variables ^^ and ^^ denote the input and detected respectively. The measurement operator isrepresented by Π. Shannon’s entropy is denoted by ^^(∙)and mutual information is represented by ^^(∙,∙). Probabilities are written as ^^, while conditional probabilities are represented as ^^^^|^^and conditioned on ^^ given ^^. The binomial coefficient is represented by ( ^^). The tensor product is represented by ^ and the cumulative distribution function ^^ of a statistical distribution is represented by ℱ.

[0027] 2. Entanglementclassical communication concept

[0028] Quantum entanglement is a phenomenon where two particles are strongly correlated, such that the state of one particle immediately provides information about the state of the other particle, no matter how far apart they are. These particles, such as photons or electrons, are individual systems, but they remain connected even when separated by vast distances, forming a composite system. As an example, giventwo basis vectors {|0^^^, |1^^^} in Hilbert space ℋ^^ and {|0^^^, |1^^^} in Hilbert space ^^^^,then the following is an entangled state:1 ^^ − 1^^^^ 0^^^) (1)

[0029] When a composite system is in the state (1), it is impossible to attribute either system A or B a definite pure state. Although the von Neumann entropy of the whole state is zero, the entropy of the subsystem is greater than zero, indicating the systems are entangled.

[0030] Compared to classical communication, entanglement enhances communication by increasing the number of messages that can be sent perfectly over the channels, resulting in higher one-shot zero-error capacity and increased security as 7 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT described in Miller (J. Miller, “Entanglement enhances classical communication,” Phys. Today 64(4), 15–16 (2011).). However, Miller doesn’t explain what kind of measurement device and receiver scheme the experimentalists used. Theoretical proofs and discussions of entanglement-assisted communication can be found in various literature.

[0031] In entanglement-assisted classical communication, entangled states can be distributed through either optical fibers or satellites and stored in quantum memories. The classical data is transmitted by Alice using the signal photon of the entangled pair, which is affected by noise and loss in the quantum channel. On the receiver side, Bob uses the idler photon of the entangled pair to determine what was transmitted by employing an optimal quantum receiver. The overall design is illustrated in Fig.1. Error correction can be applied to the quantum states to restore the transmitted information and mitigate the effects of decoherence.

[0032] 3. Receiver design for EA communication

[0033] In entanglement-assisted communication, two-mode Gaussian states are generated through spontaneous parametric down-conversion (SPDC) of entangled-photon pairs [19,20]. The SPDC source is a broadband source with a numberof modes ^^ = ^^^^^^, where ^^ is the phase-matching bandwidth and ^^^^ is themeasurement interval and generates ^^ independent pairs of signal-idler photons in spaceand time denoted by their annihilation operators ^̂^(^^), ^̂^(^^)^^ ^^ , with ^^^^[1, ^^]. These pairs areprepared in identical entangled two-mode(TMSV) states, which can be represented in a Fock state basis as in Eq. (2) ∞ ^^ ^^ ^^(2)

[0034] where ^^^^is the mean photon number in the signal mode. The meanphoton number for the idler mode is ^^^^ = ^^^^. TMSV belongs to a class of Gaussian states,where an ^^-mode Gaussian state ^̂^ consisting of modes ^̂^(^^), ^^ ∈ [1, ^^], is characterizedby the mean and variance of their respective quadrature field operators such that ^̂^(^^)=^̂^^^ + ^^^̂^(^^). The covariance matrix for a TMSV state is given by8 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT 2^^^^ + 1 0 ^^ 0^^ = 0 2^^^^ + 1 0 −^^^^^^^^^^ ^^(3)

[0035] where ^^ = 2√^^^^(^^^^ + 1), I and Z are 2 × 2 Pauli matrices. Othertwo Pauli matrices are X and Y.

[0036] If we consider the Phase-Shift Keying (PSK) modulation scheme forcommunication, then mathematically, we can use the unitary operator ^^ ^ = ^^^^^^^̂^†^^̂^to denote the rotation of the base annihilation operator ^̂^. In transmitting information using entangled photons generated from SPDC, the signal photon of the signal-idler pair is used while the idler is pre-shared before transmission occurs. In order to transmit information, Alice modulates the signal ^̂^^^′using a phase modulator to apply a rotation of ^^. The signal then passes through a thermal, lossy Bosonic quantum channel. The received photon mode (after passing through the communication channel) at Bob’s end is denoted by ^̂^^^= ^̂^^^′exp(^^q) where ^̂^^^′is the base photon mode at the receiving end. Bob uses the undisturbed idler part of the pre-shared entangled photon pair and an optimal quantum detector to perform hypothesis testing and determine which symbol was transmitted. For simplicity, we will drop the mode notation from the annihilation operator. Under the phase-encoding scheme, the covariance matrix of the return-idler pair ^̂^^^ , ^̂^^^ is given by(2^^^^ + 1)^^ ^^ ^ ^^^^^^ ^^^[^^ (^^ − ^^^^)](4)

[0037] where ^^^^ = ^^^^^^ + ^^^^, ^^^^ = 2√^^^^^^(^^^^ + 1), ^^ is the transmittivity ofthe Bosonicmode. In the case of a pre-shared entangled state, the idler is assumed to be undisturbed as it has 9 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT been shared through fiber optics or satellite and is stored in quantum memory. In such a case, attenuation experienced by the idler is negligible. Hence, at the receiver side, theidler mean photon number ^^^^ = ^^^^ = ^^^^. As the signal mode passes through a thermallossy bosonic channel, the mode is altered and referred to as the return mode onthe receiver side with mean number ^^^^.

[0038] 3.1 OPA-based receiver with threshold detection

[0039] A joint detection receiver for state discrimination of EA communication consists of an optical parametric amplifier (OPA). On the receiver side, an optical parametric amplifier (OPA) is used to combine the return-idler pair, as shown in Fig.2. The return and idler modes are evolved as given by Heisenberg’s picture: ^̂^ = √^^^̂^^^ + √^^ − 1^̂^† ^^^^(5) ^− †^

[0040] whereOPA receiver can be used to combine and amplify the return-idler pair using a strong local pump. This gives rise to Eq. (5). At the output ports, a photodetector is used for photon counting, and a threshold detection rule is applied to make state discrimination.We further assume an ideal OPA, where the gain ^^ is fixed.

[0041] The photodetector outputs are designated as ^̂^ and ^^ at two ports, referred to as the return and idler outputs, respectively. For each output, the meanphoton number is given by the expectation 〈^̂^†^̂^〉 or 〈^^†^^〉, depending on whetherthreshold detection is made at the signal output port or idler output port. The photocurrent operators and their expectations are given by Eq. (6) and Eq. (7). ^̅^ †1^^ = 〈^̂^ ^̂^〉 = ^^(^^^^^^ + ^^^^) + (^^ − 1)(1 + ^^^^) + 2 ^^^^^^ ^^√^^(^^ − 1)√^^^^^^(^^^^ + 1) (6)(7)

[0042] The derivation is provided in Appendix 8.

[0043] For practical communication, consider that information is encoded using repetition codewords that employ binary phase-shift keying (BPSK) modulation 10 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCTwith phases ^^^^0, ^^. Decoding BPSK can be modeled as hypothesis testing: if hypothesis^^0 is true, then the BPSK symbol with ^^ = 0 was transmitted, and if hypothesis ^^1 istrue, then the symbol with ^^ = ^^ was transmitted. In this paper, we do not discussoptimal encoding, which is beyond the scope of this paper. However, BPSK is a suitable choice for weak signals, as it is power-efficient.

[0044] To allow for efficient error correction, repeated PSK codewordsconsisting of ^^ signal-idler pairs are used in EA communication. In a joint-detectionscheme, the receiver mixes all ^^ received modes and counts the total number ofphotons at the output ports. The joint detection state in this case becomes an ^^-fold tensor product ^^^^^, with identical zero-mean thermal states, and the per-mode meanphoton number is given by ^̅^1(^^) or ^̅^2(^^), depending on which output port of the OPAwe use. An optimum joint for state discrimination requires photoncounting at an output port and thus deciding between two hypotheses using the totalphoton number ^^ over ^^ modes. Under such a scenario, the probability mass function(pmf) is negative binomial with mean ^^^̅^(^^) and standard deviation ^^(^^) =√^^^̅^(^^)(^̅^(^^) + 1), given by:(8)

[0045] where ^^ ∈ 1,2, and ( −) is the binomial coefficient. ^^

[0046] Equation (8) can beas a Gaussian distribution withmean ^^^^^̅^(^^) and standard deviation ^^(^^) = √^^^̅^(^^)(^̅^(^^) + 1) for sufficiently large ^^(see Appendix 7). At the detectorand decide in favor of^^0 if the total number of photons detected is ^^ > ^^^^ℎ(^^), otherwise we choose ^^1 for^^ ≤ ^^^^ℎ(^^) where ^^^^ℎ(^^) is the threshold number of photons, which is aof thephase ^^. A suitable value for the threshold number of photons is chosen according tothe scheme described later in Section 4.

[0047] 3.2 Optical phase conjugation receiver with threshold detection 11 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT

[0048] OPA can also be used differently, where the return ^̂^^^modeinteracts with the vacuum mode ^̂^^^ to produce √^^^̂^^^ + √^^ − 1^̂^^^ , which becomes ^̂^^^ =√2^̂^ + ^̂^†^^ ^^ for ^^ = 2. By mixing the idler with ^̂^^^ using a 50-50 beamsplitter, we get two1 modes √2(^̂^^^ ± ^̂^^^). The outputs from the two 2 arms are fed to a balanced detector, andtheir difference is measured as a photocurrent. We call this the Optical Phase Conjugate Receiver (OPC receiver). Consider the schematic of the OPC receiver shownin Fig. 3.

[0049] For the case of BPSK, the mean photon operators of two output arms of beamsplitters are given by 1† † †^̂^^†^ / ^^^̂^^^ / ^^=[(^^ − 1)2^̂^^^^̂^^^ ± √^^ − 1^̂^^^^̂^^^ ± √^^ − 1^̂^†^^^̂^^^ + ^̂^^^ ^̂^^^ ] (9)– , doesn’t appear as it denotes the vacuum mode.

[0051] We adopt a joint-detection scheme similar to the one adopted forthe OPA receiver discussed in Section 3.1, containing ^^ modes for error correction.The difference in the mean photon number detected at the two photodetectors of the OPC is converted to a photocurrent with a photocurrent operator ^^ given by Eq. (10),setting ^^ = 2.††^^ = ^̂^^^^̂^^^ − ^̂^^^ ^̂^^^ = √^^ − 1^̂^^^^̂^^^ + √^^ − 1^̂^†^^^̂^^^^^^^^^ ^^as, ^̂^ =(10^^) 〈^̂^ ′^̂^ 〉 = √^^^^ (^^ + 1), ^̂^ ^̂^† = ^̂^†^^ ^^ ^^ ^^ ^^ ^^ ^^ ^̂^^^ + ^^2is given by Eq. (11), setting ^^ = 2.

[0052] The variance ^^^^^^^^2 2 2 〈 〉 〈 〉 ( )^^ = ^^ − ^^^^^^^^^^12 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT

[0053] At the detector end, the decision scheme uses threshold detection, similar to the OPA-based receiver design discussed in Section 3.1.

[0054] 3.3.2x2 optical hybrid-based joint receiver with threshold detection

[0055] In this section, we describe a practical receiver design using a 2x2 optical hybrid for EA communication. An optical hybrid-based joint detection scheme is suitable for EA communication as it can be directly implemented in integrated optics and quantum nanophotonics. For a two-dimensional constellation, a 2x2 optical hybrid receiver can be used, as shown in Fig.4. The scattering matrix ^^ of the 2x2 optical hybrid is described by Eq. (12). ^^ ^^^^1√1−^^^^ = [ √1 − ^^^^^] √1 − ^^ ^^ ^2√^^(12)

[0056] where ^^ is the power-splitting ratio of Y-junction in a 2x2 optical hybrid; and ^^1, and ^^2are phase shift parameters. Return and idler at the receiver are transformed based on the scattering matrix. ^^^̂^ [^^ ] = ^^ [ ^^^^^̂^] .^^ ^^

[0057] We consider equal power splitting set by K = 0.5 and write the scattering matrix as ^^^^1 ]= [^^ ^^^^11]^̂^ [[ ^^^^] ^̂^

[0058] For BPSK, ^̂^^^ = ^̂^^^′^^ ^^^^ with ^^^^{0, ^^} The photocurrent operator isgiven by 13 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT 1 1 ^^ ̂ = ^^−^^^^(^^−^^^^1 − ^^−^^^^2)^̂^†^^^−^^^^(^^^^^^1 − ^^−^^^^^^^^ 2)^̂^2^^′̂^^^+ 2^†^^̂^^^′(15)

[0059] 1 ^^ −^^^^ −^^^^^^^^ = 〈^^ ̂^^^^〉 = ^^ √^^^^^^(^^2^^ + 1)(^^ 1 − ^^^^^^21 )+ ^^^^^^√^^^^^^(^^^^ + 1)(^^^^^^12− ^^−^^^^2)(16)

[0060] In this disclosure, we consider a special case of 2x2 optical hybridreceiver where ^^1 = 0 and ^^2 = ^^, for which, ^^^^^^ = √^^^^^^(^^^^ + 1) cos ^^. The variance ofthe photocurrent operator is given by ^^2 2^^^^ = 〈^^ ̂^^^^ 〉 − 〈^^ ̂^^^^〉2 1 −^^^^^^ ^^^^ ^^^^

[0061] For equal prior BPSK symbols, 〈^^±2^^^^〉 = (^^±2^^∙^^ + ^^±2^^∙^^ ) / 2 =1. For non-equal prior symbols with priors ^^0and+ ^^1^^±2^^∙^^ which is still 1. Further, regardless of phase value ^^ ∈ {0, ^^} for BPSKsymbols, ^^±2^^^^ = cos 2^^ . Putting these values in Eq. (17), and considering special caseof ^^1 = 0 and ^^2 = ^^, the variance for BPSK is^^^2^^^(^^) = (2^^^^^^1 + ^^^^ + ^^1) + 2^^^^^^(^^^^ + 1)(1 − ^^^^^^ 2^^) − 2^^^^^^(^^^^ + 1)(18)

[0062] where ^^^^ = ^^^^^^ + ^^^^ and ^^^^ = ^^^^. Similar to OPA and OPCreceiver design, the decision scheme uses threshold detection for state discrimination. 14 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT Since the target of this paper is a highly noisy and lossy environment, we choose ^^^^=1, ^^^^ = 0.01, ^^ = 0.01, and ^^ = 1.1 as a representative of such a condition.

[0063] 4. Evaluation of entanglement-assisted communication receivers

[0064] 4.1 Error probability calculation

[0065] The probability of error of state discrimination for the case of BPSK using OPA is given by ^^^^ = ^^0^^^^^^^^(^^ < ^^^^ℎ|^^ = 0; ^^, ^^) + ^^1[1 − ^^^^^^^^(^^ < ^^^^ℎ|^^ = ^^; ^^, ^^)](19)can errorterms in Eq. (19) which, for case of symbols with equal priors gives us ^^^^ℎ(^^) =^^(^^(^^)^̅^(0) + ^^(0̅)̅̅^̅^ (^^))(^^(^^) + ^^(0)). Derivation of the optimum threshold for OPAinGaussian approximation.

[0067] However, for unequal priors, the optimum threshold ^^^^ℎis the one that satisfies the condition ^^0^^^^^^^^(^^ < ^^^^ℎ|^^ = 0; ^^, ^^) = ^^1[1 − ^^^^^^^^(^^ < ^^^^ℎ|^^ = ^^; ^^, ^^)](20)

[0068] We solve Eq. (20) for ^^^^ℎusing grid search procedure and plug into Eq. (19) to calculate the error probability. In such a case, there is no closed-form solution.

[0069] The joint detection can be made either at the idler output port or the signal output port. The error probability of discrimination is higher at the return port compared to detection made at the idler port, as shown in Fig.5(b). As a result, our further analysis focuses solely on making joint detection at the idler port and we drop the index ^^ from the probability notation moving forward. We find that for the case of non-equal priors, the mean threshold photon number ^^^^ℎfor BPSK discrimination is higher for any detection made at the return output port than at the idler output port (see Fig.6). Note that even though the error probability ^^^^in Eq. (19) is a convex function of ^^^^ℎ, it is a monotonic 15 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT function of the prior ^^0, as shown in Fig.7. Hence, there does not exist an optimum prior that minimizes the probability of error for state discrimination.

[0070] For the OPC receiver, we calculate the error probability by taking a Gaussian approximation of photodetection statistics similar to Eq. (19). This is because we measure the difference of photocurrents obtained at the two arms of the beamsplitter at the detection side (as shown in Fig.3). The Gaussian approximation yields the probability of error formula given in Eq. (21). ^^^^ = ^^0ℱ^^^^^^ (^^^^ℎ, ^^ ∙ ^^^^^^^^(0), √^^ ∙ ^^^^^^^^(0)) + ^^1 [1 − ℱ^^^^^^ (^^^^ℎ, ^^ ∙ ^^^^^^^^(^^), √^^ ∙ ^^^^^^^^(^^))](21)

[0071] In Eq. (21), ^^^^^^^^(0) and ^^^^^^^^ are given by Eqs. (10) and (11)respectively. ℱ^^^^^^is cumulative distribution function of a Gaussian distribution withmean ^^ ∙ ^^^^^^^^(0) and standard deviation √^^ ∙ ^^^^^^^^.

[0072] Equation (21) is similar to Eq. (19) but written explicitly using the cumulative distribution function (CDF) notation. Like the OPA receiver, we can calculate the optimum ^^^^ℎby equating the two terms of Eq. (21). From Fig.5(a), we see that the OPC receiver’s performance in terms of error probability in discriminating BPSK symbols is better than that of the OPA receiver. However, for a low number of modes ^^, OPA receivers with non-equal priors still perform better than OPC receivers with equal priors and perform similarly to OPC receivers with non-equal priors. Our evaluation suggests that lower-complexity receivers like OPA receivers with fewer optical components can provide superior information retrieval with a suitable choice of prior.

[0073] The error probability of a 2x2 optical hybrid can be calculated using a formula similar to the one in Eq. (21) with means and variances from Eqs. (16) and (18), respectively. From the error probability plot in Fig.5(a), we see that the 2x2 optical hybrid offers a roughly 10% improvement in BPSK state discrimination compared to the OPC receiver.

[0074] 4.2 Mutual information calculation 16 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT

[0075] The Holevo capacity quantifies the maximum amount of information, in bits per channel use, that can be sent over a quantum channel when the use of entangled states at the input and arbitrary measurements at the output are permitted. In the situation under consideration in this work, the Holevo capacity evaluates to Eq. (22): ^^ = ^^(^^^^^^ + ^^^^) − ^^(^^^^)(22) where ^^(^^) = (^^ + 1) ^^^^^^2(^^ + 1) − ^^ ^^^^^^2(^^)(23)

[0076] is the entropy of the thermal state with mean photon number ^^. To write the mutual information and, in turn, the capacity for entanglement-assisted classical communication that requires symbol-by-symbol joint detection, we are required to calculate the conditional probability distribution.

[0077] Assuming that the random variable ^^ denotes the transmittedsymbols and ^^ denotes the detected symbols, we calculate the mutual information asfollows: we first calculate the conditional probabilities to complete the transition matrix. Using the conditional probabilities, we can calculate the posteriors, which are then used to calculate the conditional entropies, followed by the calculation of the mutual information. The steps to calculate the mutual information are provided in Eq. (32) in Appendix 9.

[0078] The Shannon’s capacity for transmitting classical information withour EA receivers can be calculated by taking the maximum of mutual information I(^^; ^^)over prior ^^ and threshold mean photon number ^^^^^^ , i.e.^^^^^^ = ^^^,^^^^^^^^^ℎ^^(^^; ^^)(24)

[0079] We find that symbols with equal priors maximize the mutual information, as expected. We conducted a simulation study with a varying number ofmodes ^^ to optimize the mutual information as a function of the signal mean photonnumber transmitted over a noisy Bosonic channel with ^^^^ = 1 and transmittivity ^^ =0.01. In Fig.8 and Fig.9, we present a comparison of the capacities of various receiver 17 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT designs proposed for the BPSK constellation using Eq. (24). At the same time, we also plot the capacity of the Homodyne receiver, where the average number of photonsreceived is 4^^^^^^ and the average number of noisy photons is 2^^^^ + 1. The capacity ofa Homodyne receiver is given by ^^^^ = 0.5 log2[1 + 4^^^^^^ / (2^^^^ + 1)]. As a reference,we also plot the Holevo capacity, given by Eq. (22). Note that the Holevo capacity requires coherent states with Gaussian modulation. For EA communication with the BPSK constellation, we find that joint receivers based on OPA and OH proposed in this paper outperform the Holevo capacity, even for a single mode, as shown in Fig. 8. We also conducted simulation studies for a large number of modes. From our analysis and results, shown in Fig.9, we conclude that for the proposed EA receiver design employing a joint-detection scheme, a large number of signal-idler modes is not required. For ^^ = 1, theOPA receiver’s performance is better than the OPC receiver and 2x2 optical hybridreceiver. C-band width 35nm, hence ^^^^ = 35^^^^. The central wavelength, ^^ = 1550^^^^.The typical observation interval of symbols is 1µs. Phase matching bandwidth of the C- band is calculated as: ^^3 × 108^^ = ^^^^ =^^2(1550 × 10−9)2⋅ 35 × 10−9 = 4.3704 × 1012(25)

[0080] Then the number of bosonic mode, ^^ for C-band is4.3704 × 1012 · 1µ^^ = 4.3704 × 106. Hence if we were to use the number of modesin the order of 106, we would end up using C-band. However, as we have shown above,achieving a superior performance requires as little as ^^ = 1 if we choose optimalvalues of prior and threshold mean photon.

[0081] We also compared our capacities with ultimate bound, i.e., entanglement-assisted classical capacity ^^^^^^^^^^^^^^^^^^. To calculate the ^^^^^^^^^^^^^^^^^^, we adopted Eq. (26): ^^ − 1 ^^ − 1^^^^^^^^^^^^^^ = ^^ + ^^ − ++ ^^ −^^^^ ^^ ^^18 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT

[0082] With ^^ = 2^^^^ + 1, ^^ = 2^^^^ + 1, ^^^^ = 2√^^^^^^(^^^^ + 1) , ^^± =Fig. 8, it is.

[0083] For a large number of modes, the 2x2 optical hybrid receiverperforms better in terms of capacity as shown in Fig. 9. Previous work in this directionhave not considered the use of OH receivers. An OH receiver uses a balanced detector, similar to OPA to distinguish the modulation which is more practical and efficient. Further, OH is known to suppress intensity noise. It could be argued that the capacity should be divided by the number of modes for a scheme using multiple modes. However, in this paper, we are talking about the overall receiver design’s capacity, rather than bits per mode. In addition, we find that a number of modes greater than 1 may not be needed to outperform the Holevo capacity, as seen from Fig.8. Thus, for a well-designed receiver, repetition coding may not always be useful. This claim is further corroborated by Fig.10.

[0084] Additionally, we also plot the per-mode communication rate ^^, normalized by the Holevo capacity for classical communication ^^, in Fig. 10. The communication rate R is given by Eq. (27). 1+ Ρ ( ) ( ) (^ ℯ log2( Ρℯ + 1 − Ρℯ log2 1 − Ρ )^ = ℯ(27) ^^

[0085] Equation (27) is based on symmetric hypothesis testing. We find that in terms of the normalized communication rate, the OPA and OPC receivers perform almost three times better in the photon-starved regime when BPSK symbols with non- equal priors are used, compared to when BPSK symbols are equally likely. At the same time, the 2x2 optical hybrid receiver for non-equal priors performs roughly 2.5 times better, compared to BPSK with equal priors in the photon-starved regime. Furthermore, the 2x2 optical hybrid receiver can outperform an OPA-based receiver by as much as 30% in terms of information rate. Finally, it’s worth noting that a large number of modes does not necessarily equate to superior performance.

[0086] 4.3 Discussion 19 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT

[0087] It is expected that the order of error probability of two receiver designs should be the reverse of the order of capacity when compared. We find that this is not the case for OPA, OPC, and OH as evident from Fig.5 as well as Fig.8 and Fig.9. The ordering in the error-probability plot and capacity plot is only related when mutual information is optimized with respect to the prior only. However, we optimize mutual information with respect to the prior as well as threshold mean photon number as shown in Eq. (24). Hence, the usual ordering relationship is no longer applicable. Further, as the number of modes increases, the probability distribution seen in Eq. (8) resembles more and more Gaussian. Thus the ordering of capacity is altered as we move from the numberof modes ^^ = 1 to higher modes. Thus, with the higher number of modes, the OHreceiver design has the best performance, followed by OPC and then the OPA receiver.However, with ^^ = 1, the best performance is obtained by using the OPA receiver,followed by the OH receiver and then the OPC receiver.

[0088] 5. Concluding remarks and possible extensions

[0089] Entanglement is a unique phenomenon in quantum information science that can be leveraged to design new types of sensors, allowing computing devices to solve problems that are intractable for conventional computers. In communication systems, the use of entanglement assistance offers a unique advantage in terms of providing a better communication rate in low-photon number regimes. Pre- shared entanglement can be used to surpass the performance of classical capacities and the Holevo capacity in highly noisy and low-brightness conditions. However, there are several challenges in terms of the practical realization of entanglement, such as: (i) transmitting entanglement over long distances is challenging, and (ii) the optimum quantum receiver to achieve entanglement-assisted channel capacity has not yet been derived. Nevertheless, simulation results indicate that even when entanglement is not perfect, entanglement-assisted (EA) communication based on signal-idler pairs outperforms the Holevo capacity and the capacities of classical channels.

[0090] In this disclosure, we analyze several low-complexity receiver designs employing optical hybrids and balanced detectors. We demonstrate that for BPSK modulation, a 2x2 optical hybrid-based joint detection can outperform the OPA and 20 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT optical phase-conjugation receivers. Numerical results demonstrate that we do not need a large number of signal-idler modes to outperform the Holevo and Homodyne capacities.

[0091] Appendix

[0092] 7. Gaussian approximation to negative binomial photon statistics

[0093] Although the photodetection statistics given by OPA receivers in Eqs. (8) are of negative binomial nature, they can be computationally expensive to calculate for large values of M. By recognizing that Eq. (19) contains cumulative distributions and approximating them as Gaussian distributions, we can rewrite the equation as the error function (erf), which is commonly used to write the cumulative distribution function of a Gaussian distribution: 1^^ − ^^ ⋅ ^^(0)1^^ − ^^ ⋅ ^^(^^)(1 + ^^^^^^ ( ^^ℎ)) = 1 − (1 + ^^^^^^ ( ^^ℎ)) 2 ^^^^ 2 ^^^^ ^^

[0094] Considering that erf(−^^) = −erf(^^) and equating the arguments oferf: ^^(^^(^^)^^(0) + ^^(0)^^(^^))^^^^ℎ(^^) =^^

[0095] However, we should be aware of how we may misinterpret the true performance of receivers due to approximation.

[0096] In Fig.11, we plot the difference between ^^Gaussianand ^^NB. ^^Gaussianrepresents the capacity of the OPA receiver as discussed in Section 3.1, where the photodetection statistics are approximated as Gaussian. ^^NBrepresents the capacity using the exact negative binomial distribution from Eq. (8). Our calculations have led us to the following observations: (i) the error of the approximation increases as the signal 21 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT mean photon number, ^^^^, increases; (ii) with the Gaussian approximation, the capacity of the channel is overestimated compared to its true value; (iii) as the number of modes increases, the error of the approximation decreases. These observations are depicted in Fig.11. Although the Gaussian approximation overestimates the capacity, the error is of the order of 10−3, which is small compared to the value of the capacity and enables faster numerical calculations.

[0097] 8. Derivation of mean photon number for optical parametric amplifier

[0098] In this section, we derive in detail, the mean photon number for OPA using Eq. (3.1). ^̂^†^̂^ = (√^^^̂^†^^ + √^^ − 1^̂^^^)(√^^ − 1^̂^^†^) =^^^̂^†^^ ^̂^^^ + √^^(^^ − 1)^̂^^†^ ^̂^† + √^^(^^ − 1)^̂^ ^̂^ + (^^ − 1)^̂^ ^̂^† ^^ ^^ ^^ ^^ ^^= with mean thermal photon number ^^^^^^^^ = ^^^^, As, idler per shared^^1(^^) = 〈^̂^†^̂^〉 = ^^(^^^^^^ + ^^^^) + (^^ − 1)(1 + ^^^^)+2 ^^^^^^ ^^ √^^(^^ − 1)√^^^^^^(^^^^ + 1)Similarly, ^̂^†^̂^ = (√^^^̂^†^^ + √^^ − 1^̂^^^)(√^^^̂^^^ + √^^ − 1^̂^^†^) ^^^^^^ ^^

[0099] 9. Mutual information calculation

[0100] In this section, at a very high level, we provide a calculation of how mutual information can be calculated. The values ofPOPA can be plugged from Eq. (8). We can first write the conditional probabilities, assuming that the random variable ^^denotes the transmitted symbols and ^^ denotes the detected symbols:22 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT ^^^^|^^(^^ = 0|^^ = 0) = 1 − ^^^^^^^^(^^ < ^^^^ℎ|^^ = 0; ^^)^^^^|^^(^^ = 1|^^ = 1) = ^^^^^^^^(^^ < ^^^^ℎ|^^ = ^^; ^^)= = = − ^^^^^^^^ =

[0101] Using conditional probabilities, we can obtain mutual information as follows. ^^(^^|^^ = 0) = −^^^^|^^(^^ = 0|^^ = 0)^^^^^^2(^^^^|^^(^^ = 0|^^ = 0))can Eq. (24).

[0103] It should be understood from the foregoing that, while particular embodiments have been illustrated and described, various modifications can be made thereto without departing from the spirit and scope of the invention as will be apparent to those skilled in the art. Such changes and modifications are within the scope and teachings of this invention as defined in the claims appended hereto. 23 100787335.3

Claims

Attorney Docket No.: 085067-833105 (UA24-111) PCT CLAIMS What is claimed is:

1. A method, comprising: accessing signal-idler photon pairs at an optical parametric amplifier (OPA)- based receiver for entanglement-assisted detection; mixing the signal-idler photon pairs by the OPA; determining, based on photon counting statistics, an optimum threshold value that a first hypothesis associated with a first phase value of a binary phase-shift keying modulation operation is true or that a second hypothesis associated with a second phase value of the binary phase-shift keying modulation operation is true, the first hypothesis being associated with a first prior and the second hypothesis being associated with a second prior, the first prior and the second prior being unequal; and detecting the transmitted phase by the threshold receiver.

2. The method of claim 1, further comprising: obtaining the optimum threshold by minimizing bit error probability subject to the threshold value and the prior probabilities while employing the exact photon counting statistics.

3. The method of claim 1, further comprising: obtaining the optimum threshold by minimizing bit error probability subject to the threshold value and the prior probabilities while employing the Gaussian approximation of photon counting statistics with the goal of reducing the system cost.

4. The method of claim 1, further comprising: 24 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT obtaining the optimum threshold by maximizing the mutual information subject to the threshold value and the prior probabilities while employing the exact photon counting statistics.

5. The method of claim 1, further comprising: obtaining the optimum threshold by maximizing the mutual information subject to the threshold value and the prior probabilities while employing the Gaussian approximation of photon counting statistics with the goal of reducing the system cost.

6. A method, comprising: accessing signal-idler photon pairs at an optical phase-conjugation (OPC) receiver for entanglement-assisted detection; processing the received signal photons by the OPA for gain 2, mixing the signal photons as processed with idler photons on a beam splitter followed by a balanced detector; determining, based on the balanced detector output statistics, the optimum threshold value that a first hypothesis associated with a first phase value of a binary phase-shift keying modulation operation is true or that a second hypothesis associated with a second phase value of the binary phase-shift keying modulation operation is true; the first hypothesis being associated with a first prior and the second hypothesis being associated with a second prior, the first prior and the second prior being unequal; and detecting the transmitted phase by the threshold receiver. 25 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT 7. The method of claim 6, further comprising: obtaining the optimum threshold by minimizing bit error probability subject to the threshold value and the prior probabilities while employing the exact balanced detector output statistics.

8. The method of claim 6, further comprising: obtaining the optimum threshold by minimizing bit error probability subject to the threshold value and the prior probabilities while employing the Gaussian approximation of the balanced detector output statistics with the goal of reducing the system cost.

9. The method of claim 6, further comprising: obtaining the optimum threshold by maximizing the mutual information subject to the threshold value and the prior probabilities while employing the exact the balanced detector output statistics.

10. The method of claim 6, further comprising: obtaining the optimum threshold by maximizing the mutual information subject to the threshold value and the prior probabilities while employing the Gaussian approximation of the balanced detector output statistics with the goal of reducing the system cost.

11. A method, comprising: accessing signal-idler photon pairs at optical hybrid (OH)-based receiver for entanglement-assisted detection; mixing the received signal photons with idler photons on the proposed OH, followed by the balanced detector; determining, based on the balanced detector output statistics the optimum threshold value, that a first hypothesis associated with a first phase value of a binary phase-shift keying modulation operation is true or that a second hypothesis associated with a second phase value of the binary phase-shift keying modulation operation is true; 26 100787335.3Attorney Docket No.: 085067-833105 (UA24-111) PCT the first hypothesis being associated with a first prior and the second hypothesis being associated with a second prior, the first prior and the second prior being unequal; and detecting the transmitted phase by the threshold receiver.

11. The method of claim 11, further comprising: obtaining the optimum threshold by minimizing bit error probability subject to the threshold value and the prior probabilities while employing the exact balanced detector output statistics.

12. The method of claim 11, further comprising: obtaining the optimum threshold by minimizing bit error probability subject to the threshold value and the prior probabilities while employing the Gaussian approximation of the balanced detector output statistics with the goal of reducing the system cost.

13. The method of claim 11, further comprising: obtaining the optimum threshold by maximizing the mutual information subject to the threshold value and the prior probabilities while employing the exact the balanced detector output statistics.

14. The method of claim 11, further comprising: obtaining the optimum threshold by maximizing the mutual information subject to the threshold value and the prior probabilities while employing the Gaussian approximation of the balanced detector output statistics with the goal of reducing the system cost. 27 100787335.3

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