We demonstrate the achievability of a square root limit on the amount of information transmitted reliably and with low probability of detection (LPD) over the single-mode lossy bosonic channel if either the eavesdropper's measurements or the channel itself is subject to the slightest amount of excess noise. Specifically, Alice can transmit $\mathcal{O}(\sqrt{n})$ bits to Bob over $n$ channel uses such that Bob's average codeword error probability is upper-bounded by an arbitrarily small $δ>0$ while a passive eavesdropper, Warden Willie, who is assumed to be able to collect all the transmitted photons that do not reach Bob, has an average probability of detection error that is lower-bounded by $1/2-ε$ for an arbitrarily small $ε>0$ . We analyze the thermal noise and pure loss channels. The square root law holds for the thermal noise channel even if Willie employs a quantum-optimal measurement, while Bob is equipped with a standard coherent detection receiver. We also show that LPD communication is not possible on the pure loss channel. However, this result assumes Willie to possess an ideal receiver that is not subject to excess noise. If Willie is restricted to a practical receiver with a non-zero dark current, the square root law is achievable on the pure loss channel.We demonstrate the achievability of a square root limit on the amount of information transmitted reliably and with low probability of detection (LPD) over the single-mode lossy bosonic channel if either the eavesdropper's measurements or the channel itself is subject to the slightest amount of excess noise. Specifically, Alice can transmit $\mathcal{O}(\sqrt{n})$ bits to Bob over $n$ channel uses such that Bob's average codeword error probability is upper-bounded by an arbitrarily small $δ>0$ while a passive eavesdropper, Warden Willie, who is assumed to be able to collect all the transmitted photons that do not reach Bob, has an average probability of detection error that is lower-bounded by $1/2-ε$ for an arbitrarily small $ε>0$. We analyze the thermal noise and pure loss channels. The square root law holds for the thermal noise channel even if Willie employs a quantum-optimal measurement, while Bob is equipped with a standard coherent detection receiver. We also show that LPD communication is not possible on the pure loss channel. However, this result assumes Willie to possess an ideal receiver that is not subject to excess noise. If Willie is restricted to a practical receiver with a non-zero dark current, the square root law is achievable on the pure loss channel.

arXiv:1403.5616Mar 22, 20142 citations

Fundamental Limits of Quantum-Secure Covert Communication Over Bosonic Channels

Michael S. Bullock, Christos N. Gagatsos, Saikat Guha +1 more

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 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.

arXiv:1907.04228Allerton Conference on Communication, Control, and ComputingSep 1, 201962 citations

Limits of Reliable Communication with Low Probability of Detection on AWGN Channels

Boulat A. Bash, Dennis Goeckel, Don Towsley

We present a square root limit on the amount of information transmitted reliably and with low probability of detection (LPD) over additive white Gaussian noise (AWGN) channels. Specifically, if the transmitter has AWGN channels to an intended receiver and a warden, both with non-zero noise power, we prove that o(√n) bits can be sent from the transmitter to the receiver in n channel uses while lower-bounding α + β ≥ 1-ε for any ε > 0, where α and β respectively denote the warden's probabilities of a false alarm when the sender is not transmitting and a missed detection when the sender is transmitting. Moreover, in most practical scenarios, a lower bound on the noise power on the channel between the transmitter and the warden is known and O(√n) bits can be sent in n LPD channel uses. Conversely, attempting to transmit more than O(√n) bits either results in detection by the warden with probability one or a non-zero probability of decoding error at the receiver as n→∞.We present a square root limit on the amount of information transmitted reliably and with low probability of detection (LPD) over additive white Gaussian noise (AWGN) channels. Specifically, if the transmitter has AWGN channels to an intended receiver and a warden, both with non-zero noise power, we prove that o(√n) bits can be sent from the transmitter to the receiver in n channel uses while lower-bounding α + β ≥ 1-ε for any ε > 0, where α and β respectively denote the warden's probabilities of a false alarm when the sender is not transmitting and a missed detection when the sender is transmitting. Moreover, in most practical scenarios, a lower bound on the noise power on the channel between the transmitter and the warden is known and O(√n) bits can be sent in n LPD channel uses. Conversely, attempting to transmit more than O(√n) bits either results in detection by the warden with probability one or a non-zero probability of decoding error at the receiver as n→∞.

arXiv:1202.6423IEEE Journal on Selected Areas in CommunicationsAug 23, 20131,078 citations

Square Root Law for Covert Quantum Communication over Optical Channels

Evan J. D. Anderson, Christopher K. Eyre, Isabel M. Dailey +2 more

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: $\propto\sqrt{n}$ qubits can be transmitted covertly and reliably over $n$ 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.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:$\propto\sqrt{n}$qubits can be transmitted covertly and reliably over$n$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.

IEEE International Conference on Quantum Computing and Engineering (QCE)Sep 15, 20246 citations

Quantum-Secure Covert Communications Over Lossy Bosonic Random Access Channels

Bichen Kang, Neng Ye, Yue Zhang +3 more

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 transmit $\mathcal {O}(N)$ bits of information over $N$ channel 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 to $\mathcal {O}(\sqrt{N})$ bits, whereas the capacities of other users will remain at $\mathcal {O}(N)$ 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.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 transmit$\mathcal {O}(N)$bits of information over$N$channel 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 to$\mathcal {O}(\sqrt{N})$bits, whereas the capacities of other users will remain at$\mathcal {O}(N)$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.

IEEE Transactions on Vehicular TechnologyFeb 13, 20260 citations

Covert Quantum Communication Over Optical Channels

Evan J. D. Anderson, Christopher K. Eyre, Isabel M. Dailey +2 more

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: $\propto\sqrt{n}$ qubits can be transmitted covertly and reliably over $n$ 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.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: $\propto\sqrt{n}$ qubits can be transmitted covertly and reliably over $n$ 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.

arXiv:2401.06764Jan 12, 20241 citation

\mathcal{O}(\sqrt{n})

bits to Bob over

n

channel uses such that Bob's average codeword error probability is upper-bounded by an arbitrarily small

δ>0

while a passive eavesdropper, Warden Willie, who is assumed to be able to collect all the transmitted photons that do not reach Bob, has an average probability of detection error that is lower-bounded by

1/2-ε

for an arbitrarily small

ε>0

. We analyze the thermal noise and pure loss channels. The square root law holds for the thermal noise channel even if Willie employs a quantum-optimal measurement, while Bob is equipped with a standard coherent detection receiver. We also show that LPD communication is not possible on the pure loss channel. However, this result assumes Willie to possess an ideal receiver that is not subject to excess noise. If Willie is restricted to a practical receiver with a non-zero dark current, the square root law is achievable on the pure loss channel.

\propto\sqrt{n}

qubits can be transmitted covertly and reliably over

n

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.

\mathcal {O}(N)

bits of information over

N

channel 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 to

\mathcal {O}(\sqrt{N})

bits, whereas the capacities of other users will remain at

\mathcal {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.

\propto\sqrt{n}

qubits can be transmitted covertly and reliably over

n