The lossy bosonic channel (attenuator) whose environment modes are prepared in a correlated state — a multimode squeezed vacuum whose inter-mode correlations decay with channel use, giving non-Markovian memory — has an energy-constrained classical capacity that is known only under the restriction to Gaussian input states and Gaussian modulations. Whether non-Gaussian encodings can increase the capacity of this channel, i.e. whether the Gaussian-optimizer property proven for memoryless gauge-covariant bosonic channels extends to the correlated-environment setting, remains open. Partial resolutions exist: the memoryless limit is settled in favor of Gaussian optimality (Giovannetti et al., 2014–2015), and for a zero-temperature correlated environment matching all-input upper and Gaussian lower bounds give the exact capacity in a parameter range, including a proven perfect-memory asymptote that explains the observed convergence of capacity curves at large memory strength. The open core is Gaussian optimality (whose affirmative proof would yield the exact classical capacity) for thermal correlated environments and for zero-temperature environments outside the proven range.

Background

A lossy bosonic channel, or attenuator, models each use of an optical communication link as a beam splitter of transmissivity η\eta coupling the input field mode (a harmonic oscillator described by quadrature operators) to an environment mode; energy is lost to the environment, and environment noise degrades the signal. Using the channel nn times means sending nn modes. In the standard memoryless model the environment modes are independent vacua or thermal states; the channel with memory instead prepares the environment modes in a correlated joint state. Pilyavets, Zborovskii and Mancini (Phys. Rev. A 77, 052324 (2008)) introduced an analytically tractable version: the environment is a multimode squeezed vacuum (a pure Gaussian state produced by coupled squeezing of all environment modes) with covariance matrix 12[esΩ⊕e−sΩ]\frac{1}{2}[e^{s\Omega}\oplus e^{-s\Omega}], where Ω\Omega is the tridiagonal matrix coupling contiguous modes and s∈Rs\in\mathbb{R} is the memory strength (s=0s=0 recovers the memoryless channel). The induced noise correlations between channel uses decay with mode distance — the environment covariance-matrix elements along a fixed row are well fitted by a Gaussian profile of width 2∣s∣2\sqrt{|s|} in the mode separation — but the correlation structure is not of the Gauss–Markov form a∣i−j∣a^{|i-j|} with 0<a<10<a<1 (the covariance of a discrete Gaussian Markov process), and the channel is not forgetful (the correlations, though decaying, never vanish exactly, so the multi-use channel never factorizes into independent single-use channels); the memory is therefore called non-Markovian. The classical capacity of such a channel is the supremum of asymptotically achievable rates for transmitting classical information, computed for Gaussian channels via the Holevo quantity χ=S(∑kpk ωk)−∑kpkS(ωk)\chi = S(\sum_k p_k\,\omega_k) - \sum_k p_k S(\omega_k), where ωk\omega_k are the output states of the signal ensemble, SS is the von Neumann entropy, and the input states obey an energy constraint of NN photons per channel use on average. A Gaussian encoding modulates a (possibly nn-mode entangled) Gaussian seed state by Gaussian-distributed displacements; entangled inputs across uses can raise the achievable rate when memory is present. The central technical obstruction is that the celebrated Gaussian-optimizer theorem — solved for memoryless, gauge-covariant bosonic channels by Giovannetti, Holevo and García-Patrón (Commun. Math. Phys. 334, 1553 (2015)) by proving that the vacuum minimizes the output entropy, which makes Gaussian encodings optimal and the capacity additive — does not apply here: a channel whose environment correlations mix the modes is not gauge-covariant (it is not invariant under independent phase rotations of each mode), so both the minimum-output-entropy step and the additivity step lose their proofs. The 2008 paper therefore maximized χ\chi only over a restricted subclass of Gaussian inputs, explicitly as an assumption, and flagged the convergence of the resulting capacity-versus-memory curves at large ss as unexplained. Subsequent work completed the maximization over all Gaussian encodings (Pilyavets, Lupo, Mancini, IEEE Trans. Inf. Theory 58, 6126 (2012)) and proved exact capacities at zero environment temperature in a parameter range (Lupo, Pilyavets, Mancini, New J. Phys. 11, 063023 (2009)), but the Gaussian-optimality assumption for the memory channel has never been proven or refuted.

Problem Statement

Determine the energy-constrained classical capacity of the lossy bosonic channel with non-Markovian memory — the attenuator with transmissivity η\eta whose environment modes are prepared in a correlated multimode Gaussian state, such as the multimode squeezed vacuum with covariance matrix 12[esΩ⊕e−sΩ]\frac{1}{2}[e^{s\Omega}\oplus e^{-s\Omega}] with Ω\Omega the contiguous-mode coupling matrix and memory strength ss, or, at environment temperature T≥0T\ge 0, the multimode squeezed thermal environment with covariance matrix (T+12)[esΩ⊕e−sΩ](T+\tfrac{1}{2})[e^{s\Omega}\oplus e^{-s\Omega}] (reference 3), which recovers the squeezed vacuum at T=0T=0 — under an input energy constraint of NN photons per channel use. In particular, resolve the assumption under which all existing values of this capacity were derived: either prove that Gaussian encodings are optimal, i.e. that the true capacity equals the Holevo quantity maximized over Gaussian input states and Gaussian modulations (as computed by Pilyavets, Lupo and Mancini, IEEE Trans. Inf. Theory 58, 6126 (2012)), or exhibit channel parameters η\eta, ss, environment temperature TT and input energy NN together with an explicit non-Gaussian encoding whose achievable Holevo rate strictly exceeds the optimal Gaussian value at those parameters. The question is open for thermal correlated environments (T>0T>0) over the full range of memory strength, and for zero-temperature correlated environments outside the parameter range in which matching upper and lower bounds have been proven; the memoryless limit (s=0s=0) is settled in favor of Gaussian optimality and is not part of the open question.

References

  1. O. V. Pilyavets, V. G. Zborovskii, S. Mancini, 'A lossy bosonic quantum channel with non-Markovian memory', Phys. Rev. A 77, 052324 (2008). DOI: 10.1103/PhysRevA.77.052324; arXiv:0802.3397, https://arxiv.org/abs/0802.3397
  2. O. V. Pilyavets, V. G. Zborovskii, S. Mancini, A. I. Lvovsky, 'A Lossy Bosonic Quantum Channel with Non-Markovian Memory', AIP Conf. Proc. 1110, 123–126 (2009), QCMC 2008. DOI: 10.1063/1.3131287, https://doi.org/10.1063/1.3131287 (proceedings version)
  3. C. Lupo, O. V. Pilyavets, S. Mancini, 'Capacities of lossy bosonic channel with correlated noise', New J. Phys. 11, 063023 (2009). DOI: 10.1088/1367-2630/11/6/063023; arXiv:0901.4969, https://arxiv.org/abs/0901.4969
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