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arXiv:1601.06826 · IEEE Transactions on Information Theory

Fundamental Limits of Covert Communication Over Classical-Quantum Channels

Michael S. Bullock, Azadeh Sheikholeslami, Mehrdad Tahmasbi, Robert C. Macdonald, Saikat Guha, Boulat A. Bash

Published Feb 13, 2025• 10 citations
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Abstract

We investigate covert communication over general memoryless classical-quantum channels with fixed finite-size input alphabets. We show that the square root law (SRL) governs covert communication in this setting when product a ofninput states is used:$L_{\mathrm { SRL}}\sqrt {n}+o(\sqrt {n})$covert bits (but no more) can be reliably transmitted innuses of classical-quantum channel, where$L_{\mathrm { SRL}}\gt 0$is a channel-dependent constant that we callcovert capacity. We also show that ensuring covertness requires$J_{\mathrm { SRL}}\sqrt {n}+o(\sqrt {n})$bits secret key shared by the communicating parties prior to transmission, where$J_{\mathrm { SRL}}\geq 0$is a channel-dependent constant. We assume a quantum-powerful adversary that can perform an arbitrary joint (entangling) measurement on allnchannel uses. We determine the single-letter expressions for$L_{\mathrm { SRL}}$and$J_{\mathrm { SRL}}$, and establish conditions when$J_{\mathrm { SRL}}=0$(i.e., no pre-shared secret key is needed). Finally, we evaluate scenarios where covert communication is not governed by the SRL.

DOI
10.1109/tit.2025.3537970

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We investigate covert communication over general memoryless classical-quantum channels with fixed finite-size input alphabets. We show that the square root law (SRL) governs covert communication in this setting when product a ofninput states is used:LSRLn+o(n)L_{\mathrm { SRL}}\sqrt {n}+o(\sqrt {n})LSRL​n​+o(n​)covert bits (but no more) can be reliably transmitted innuses of classical-quantum channel, whereLSRL>0L_{\mathrm { SRL}}\gt 0LSRL​>0is a channel-dependent constant that we callcovert capacity. We also show that ensuring covertness requiresJSRLn+o(n)J_{\mathrm { SRL}}\sqrt {n}+o(\sqrt {n})JSRL​n​+o(n​)bits secret key shared by the communicating parties prior to transmission, whereJSRL≥0J_{\mathrm { SRL}}\geq 0JSRL​≥0is a channel-dependent constant. We assume a quantum-powerful adversary that can perform an arbitrary joint (entangling) measurement on allnchannel uses. We determine the single-letter expressions forLSRLL_{\mathrm { SRL}}LSRL​andJSRLJ_{\mathrm { SRL}}JSRL​, and establish conditions whenJSRL=0J_{\mathrm { SRL}}=0JSRL​=0(i.e., no pre-shared secret key is needed). Finally, we evaluate scenarios where covert communication is not governed by the SRL.
Recently, the fundamental limits of covert, i.e., reliable-yet-undetectable, communication have been established for general memoryless channels and for lossy-noisy bosonic (quantum) channels with a quantum-limited adversary. The key import of these results was the square-root law (SRL) for covert communication, which states that O(√n) covert bits, but no more, can be reliably transmitted over n channel uses with O(√n) bits of secret pre-shared between communicating parties. Here we prove the achievability of the SRL for a general memoryless classical-quantum channel, showing that SRL covert communication is achievable over any quantum communication channel with a product-state transmission strategy. We leave open the converse, which, if proven, would show that even using entangled transmissions and entangling measurements, the SRL for covert communication cannot be surpassed over an arbitrary quantum channel.
We establish the fundamental limits of covert communication over quantum channels, focusing on both finite-dimensional classical-quantum (CQ) channels and the infinite-dimen\-sional lossy thermal-noise bosonic channel. For general memoryless CQ channels, we characterize when covert communication is possible and prove a square root law (SRL): at most Ln+o(n)L\sqrt{n} + o(\sqrt{n})Ln​+o(n​) bits can be reliably transmitted covertly over nnn uses of the channel where LLL is the channel-dependent covert capacity. We derive a single-letter expression for LLL and show that the required pre-shared secret key also scales as n\sqrt{n}n​ under mild conditions. We also identify and characterize corner cases where either constant-rate or zero-rate covert communication is possible. For the bosonic channel, we prove that the SRL holds and derive exact covert capacities under different resource assumptions. In particular, we show that the covert secrecy and covert entanglement generation capacities both equal the covert classical capacity without entanglement-assistance, Lno-EAL_{\text{no-EA}}Lno-EA​, and show that the required pre-shared classical secret key scales as n\sqrt{n}n​. We also find that quadrature phase-shift keying (QPSK) achieves the optimal covert capacity, outperforming binary modulation in contrast to the classical channel setting. When Alice and Bob instead share entanglement, the covert classical capacity improves to LEAnlog⁡n+o(nlog⁡n)L_{\text{EA}} \sqrt{n} \log n + o(\sqrt{n} \log n)LEA​n​logn+o(n​logn), though this scaling is sensitive to noise in the entangled resource. Finally, we propose a practical covert transceiver and simulate its performance compared to recently experimentally-tested transceivers under realistic conditions.
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 explore covert communication of qubits over an arbitrary quantum channel. Covert communication conceals the transmissions in the channel noise, ensuring that an adversary is unable to detect their presence. We show the achievability of a square root law (SRL) for quantum covert communication similar to that for classical:M(n)nM(n) \sqrt{n}M(n)n​qubits can be transmitted covertly and reliably overnnnuses of a general quantum channel. We lower boundM(n)M (n)M(n)with and without assistance from a two-way covert classical channel. In the former case, we quantify the number of classical covert bits sufficient for our protocol.
We determine the covert capacity for entanglement generation over a noisy quantum channel. While secrecy guarantees that the transmitted information remains inaccessible to an adversary, covert communication ensures that the transmission itself remains undetectable. The entanglement dimension follows a square root law (SRL) in the covert setting, i.e., O(n)O(\sqrt{n})O(n​) Einstein-Podolsky-Rosen (EPR) pairs can be distributed covertly and reliably over nnn channel uses. We begin with covert communication of classical information under a secrecy constraint. We then leverage this result to construct a coding scheme for covert entanglement generation. Single-letter expressions are derived for the covert key-assisted and unassisted secrecy capacities, as well as for the covert entanglement-generation capacity.
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.