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arXiv:1701.06206

Fundamental limits of quantum-secure covert optical sensing

Boulat A. Bash, Christos N. Gagatsos, Animesh Datta, Saikat Guha

Published Jan 22, 2017• 1 citation
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Abstract

We present a square root law for active sensing of phase $θ$ of a single pixel using optical probes that pass through a single-mode lossy thermal-noise bosonic channel. Specifically, we show that, when the sensor uses an $n$-mode covert optical probe, the mean squared error (MSE) of the resulting estimator $\hatθ_n$ scales as $\langle (θ-\hatθ_n)^2\rangle=\mathcal{O}(1/\sqrt{n})$; improving the scaling necessarily leads to detection by the adversary with high probability. We fully characterize this limit and show that it is achievable using laser light illumination and a heterodyne receiver, even when the adversary captures every photon that does not return to the sensor and performs arbitrarily complex measurement as permitted by the laws of quantum mechanics.

DOI
10.48550/arxiv.1701.06206

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We investigate the fundamental limit of quantumsecure covert communication over the lossy thermal noise bosonic channel, the quantum-mechanical mode underlying many practical channels. We assume that the adversary has unlimited quantum information processing capabilities as well as access to all transmitted photons that do not reach the legitimate receiver. Given existence of noise that is uncontrolled by the adversary, the square root law (SRL) governs covert communication: up to c√(n) covert bits can be transmitted reliably in n channel uses. Attempting to surpass this limit results in detection with unity probability as n → ∞. Here we present the expression for c, characterizing the SRL for the bosonic channel. We also prove that discrete-valued coherent state quadrature phase shift keying (QPSK) constellation achieves the optimal c, which is the same as that achieved by a circularly-symmetric complex-valued Gaussian prior on coherent state amplitude. Finally, while binary phase shift keying (BPSK) achieves the Holevo capacity for non-covert bosonic channels in the low received signal-to-noise ratio regime, we show that it is strictly sub-optimal for covert communication.
We consider the problem of estimating an unknown but constant carrier phase modulation $\ensuremath{\theta}$ using a general, possibly entangled, nnn-mode optical probe through nnn independent and identical uses of a lossy bosonic channel with additive thermal noise. We find an upper bound to the quantum Fisher information (QFI) of estimating $\ensuremath{\theta}$ as a function of nnn, the mean and variance of the total number of photons NS{N}_{\mathrm{S}}NS​ in the nnn-mode probe, the transmissivity $\ensuremath{\eta}$, and mean thermal photon number per mode n‾B{\overline{n}}_{\mathrm{B}}nB​ of the bosonic channel. Since the inverse of QFI provides a lower bound to the mean-square error (MSE) of an unbiased estimator $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\theta}}$ of $\ensuremath{\theta}$, our upper bound to the QFI provides a lower bound to the MSE. It already has found use in proving fundamental limits of covert sensing and could find other applications requiring bounding the fundamental limits of sensing an unknown parameter embedded in a correlated field.
We explore covert communication of qubits over the lossy thermal-noise bosonic channel, which is a quantum-mechanical model of many practical channels, including optical. Covert communication ensures that an adversary is unable to detect the presence of transmissions, which are concealed in channel noise. We show a square root law (SRL) for quantum covert communication similar to that for classical:∝n\propto\sqrt{n}∝n​qubits can be transmitted covertly and reliably overnnnuses of an optical channel. Our achievability proof uses photonic dual-rail qubit encoding, which has been proposed for long-range repeater-based quantum communication and entanglement distribution. Our converse employs prior covert signal power limit results and adapts well-known methods to upper bound quantum capacity of optical channels. Finally, we believe that the gap between our lower and upper bounds for the number of reliable covert qubits can be mitigated by improving the quantum error correction codes and quantum channel capacity bounds.
Quantum-secure covert communication is an emerging technology crucial for secure transmission in hostile environments. However, existing studies primarily focus on single-transmitter scenarios, leaving a gap in the research on multi-user settings. This paper investigates the fundamental limits of quantum-secure covert communications over lossy bosonic random access channels with Gaussian thermal noise in view of realistic communication systems. The square root law is first established in the scenario where only randomly activated covert users exist. We then demonstrate that the square root law can be surpassed with the assistance of randomly activated overt users in quantum communication system, enabling covert users to transmitO(N)\mathcal {O}(N)O(N)bits of information overNNNchannel uses, provided that the mean photon numbers of all users reaching the detecting end are equal. To prove this, a recursive upper bound is developed on the trace distance (TD) to measure the detection probability of the warden, which is tighter than the traditional quantum relative entropy measurement. However, if the mean photon number of any covert user deviates from the above condition, its covert capacity will degrade toO(N)\mathcal {O}(\sqrt{N})O(N​)bits, whereas the capacities of other users will remain atO(N)\mathcal {O}(N)O(N)bits. This result is derived by exploiting the iterative relationship of the upper bound of TD and introducing a two-step detection strategy for the warden to handle random user activation uncertainty. Numerical experiments validate the theoretical findings, confirming the performance of covert transmission in both scenarios.
We propose a scheme for covert active sensing using floodlight illumination from a terahertz-bandwidth amplified spontaneous emission (ASE) source and heterodyne detection. We evaluate the quantum-estimation-theoretic performance limit of covert sensing, wherein a transmitter's attempt to sense a target phase is kept undetectable to a quantum-equipped passive adversary, by hiding the signal photons under the thermal noise floor. Despite the quantum state of each mode of the ASE source being mixed (thermal), and hence inferior compared to the pure coherent state of a laser mode, the thousand-times-higher optical bandwidth of the ASE source results in achieving a substantially superior performance compared to a narrow-band laser source by allowing the probe light to be spread over many more orthogonal temporal modes within a given integration time. Even though our analysis is restricted to single-mode phase sensing, this system could be applicable or extendible for various practical optical sensing applications.
We explore covert communication of qubits over the lossy thermal-noise bosonic channel, which is a quantum-mechanical model of many practical channels, including optical. Covert communication ensures that an adversary is unable to detect the presence of transmissions, which are concealed in channel noise. We show a \emph{square root law} (SRL) for quantum covert communication similar to that for classical: ∝n\propto\sqrt{n}∝n​ qubits can be transmitted covertly and reliably over nnn uses of an optical channel. Our achievability proof uses photonic dual-rail qubit encoding, which has been proposed for long-range repeater-based quantum communication and entanglement distribution. Our converse employs prior covert signal power limit results and adapts well-known methods to upper bound quantum capacity of optical channels. Finally, we believe that the gap between our lower and upper bounds for the number of reliable covert qubits can be mitigated by improving the quantum error correction codes and quantum channel capacity bounds.