Fundamental limits of information transfer

Communication technologies transfer information between two points in space. A message is encoded onto a physical carrier, sent through a channel that degrades it and recovered at the far end. In an earlier post, I gave each domain of information technology a dominant bound, and communications got the Holevo limit. Here I want to look at where that bound comes from and what it buys. It turns out that the answer depends almost entirely on the mean photon number per mode, which is small in the regime that parallel architectures are pushing links toward.

The size of the gap

Two figures set the gap, both normalized to 1 W and 1 km. An ultimate optical link in a single spatial mode is bounded at ~10¹⁶ bits per second (Giovannetti et al., 2004), while a fiber-optic link built in 2025 carries ~10¹⁰ bits per second (Optical Internetworking Forum, 2025). Latency is different, since a deployed link runs ~10⁻⁵ s of total latency (Optical Internetworking Forum, 2025), a factor of ~3 from the vacuum bound and ~2 from the fiber’s own group delay. Therefore, the throughput gap is the one worth attacking, and it is ~6 orders of magnitude wide. Most of those orders are ordinary engineering, such as unoccupied bandwidth, launch power below what is available and electronic noise in the receiver. The quantum step, from the Shannon limit to the Holevo limit, is a small part of the total in a conventional link. However, its size is set by the mean photon number per mode, and it grows without bound as that number falls.

Two ways to measure the same light

We send one symbol per mode, where a mode is one independent degree of freedom of the field, such as one time slot at one frequency, and we write its mean photon number as ⟨N⟩ = ⟨â^†â⟩. The capacity depends on what the receiver is allowed to do. A mode has two quadratures, the amplitudes of its cosine and sine components, and they do not commute. If we encode on both quadratures and read them out with heterodyne detection, which measures both at once, the link is an additive white Gaussian noise channel with a per-quadrature signal-to-noise ratio of ⟨N⟩, and its capacity is the two-quadrature Shannon limit (Shannon, 1948; Banaszek et al., 2020),

C_S2 = log₂(1 + ⟨N⟩) bits per mode.

If instead we encode in a single quadrature and read it out with coherent (homodyne) detection, we give up half the degrees of freedom but gain four times the signal-to-noise ratio in the one we keep (Banaszek et al., 2020),

C_S1 = ½ log₂(1 + 4⟨N⟩) bits per mode.

If we allow arbitrary quantum measurements, including a single joint measurement across many uses of the channel, the bound rises to the Holevo limit (Holevo, 1973; Schumacher & Westmoreland, 1997; Giovannetti et al., 2004),

C_Hol = g(⟨N⟩) = (⟨N⟩+1) log₂(⟨N⟩+1) − ⟨N⟩ log₂⟨N⟩ bits per mode,

where g(·) is the bosonic entropy function. It exceeds both Shannon limits for every ⟨N⟩ > 0 (Banaszek et al., 2020).

Origin of the gap

Part of the gap between the Shannon and Holevo limits can be recovered one symbol at a time by photon counting, and the rest is superadditive coding gain (Guha, 2011; Banaszek et al., 2020). Coherent states, the states of light a laser emits, are not orthogonal, so no measurement on one symbol at a time can fully resolve neighboring constellation points. A collective measurement acts on all n modes of a codeword at once, in a joint space where codewords can be made nearly orthogonal, so the information extracted from the block exceeds the sum of n single-symbol measurements. The difference is entirely in the measurement at the receiver, since coherent states with Gaussian modulation already saturate the Holevo limit for the lossy bosonic channel when read out jointly (Giovannetti et al., 2004). If the transmitter and receiver share entanglement ahead of time, the limit rises further to the entanglement-assisted classical capacity, which is twice the Holevo capacity on a noiseless channel and can be an arbitrarily large multiple of it on some noisy ones (Bennett et al., 1999; Bennett et al., 2002; Bennett & Wiesner, 1992), so the hierarchy is C_EA ≥ C_Hol ≥ C_S2.

The quantum step versus photon number

At large ⟨N⟩ the field behaves classically and the Holevo and Shannon limits converge. At ⟨N⟩ = 100 photons per mode, the Holevo limit is only 1.22 times the two-quadrature Shannon limit, so it has no bearing on a conventional high-power link. At small ⟨N⟩, however, the ratio of the two capacities grows like 1 − ln⟨N⟩ without bound (Banaszek et al., 2020). At ⟨N⟩ = 1 the Holevo limit is twice the Shannon limit, and at ⟨N⟩ = 0.01 it is 5.6 times.

The same behavior shows up in energy, where the energy per bit is E_b = ħω⟨N⟩/C. The two-quadrature Shannon-limited energy per bit has a minimum, E_Sh ≈ ħω ln2 (Verdu, 1990; Giovannetti et al., 2004), which at 1550 nm is about 89 zJ per bit, or 0.69 photons per bit, so no coding takes a two-quadrature coherent receiver past log₂e ≈ 1.44 bits per photon. The Holevo-limited energy per bit, E_Hol ≈ ħω/log₂(e/⟨N⟩), has no minimum, and at ⟨N⟩ = 0.001 joint detection extracts 11.4 bits per photon, about 11 zJ per bit. The ħω ln2 minimum is therefore a property of the measurement, and the Holevo limit holds for all measurements.

A link at 1550 nm with 100 GHz of bandwidth and 10 mW of optical power has ⟨N⟩ ≈ 7.8 × 10⁵ photons per mode. On a lossless channel the Shannon, Holevo and entanglement-assisted limits evaluate to 1.96, 2.10 and 4.20 Tbps at 5.11, 4.76 and 2.38 fJ per bit, so the Holevo improvement is 7%. If the same 10 mW is spread across 10⁷ multiplexed spectral and spatial channels instead, ⟨N⟩ falls to ≈ 0.078 per mode and the Holevo energy per bit drops to ≈ 25 zJ, five orders of magnitude below the Shannon-limited 5.11 fJ at the same power. Almost all of that drop comes from how the photons are distributed across modes, since the Shannon limit at ⟨N⟩ ≈ 0.078 is already 92 zJ. The remaining factor of 3.7 comes from how they are measured.

The quantum step is therefore primarily an energy-efficiency result. The rate per mode falls with ⟨N⟩, so aggregate throughput has to come from many multiplexed low-power channels, which is the operating point of massively parallel architectures (Gurses et al., 2026). At small ⟨N⟩, rates above the capacity of the best symbol-by-symbol receiver are unreachable with that receiver at any blocklength, so the rest of the improvement has to come from a joint measurement.

What we can build today

Collective decoding at link scale is out of reach today. However, two things we can build move a real link closer to these bounds, and one of them can pass a Shannon limit:

1. We can change how we measure, by making the receiver quantum-limited, so that the quantum fluctuations of the light set the noise floor and the electronics contribute negligibly. A quantum-limited receiver puts a link on the ideal Shannon curve without passing it, and we need it to detect squeezed light at all (Gurses et al., 2026).
2. We can change what we send, by encoding information in nonclassical states of light. A displaced squeezed state suppresses noise along the decision axis at the expense of the conjugate one, so squeezing enters the signal-to-noise ratio inside the logarithm: C_sq = ½ log₂(1 + 4⟨N⟩e^2r) at unit detection efficiency, for squeezing parameter r, where ⟨N⟩ counts only the photons in the displacement (Gurses et al., 2026; Banaszek et al., 2020). The hierarchy is C_S1 ≤ C_sq ≤ g(⟨N⟩ + sinh²r), with the Holevo limit taken at the total photon number including the sinh²r photons of the squeezing, so squeezed light with coherent detection exceeds the one-quadrature Shannon limit without the collective measurement needed to saturate the Holevo limit.

We worked on the first one. We demonstrated an integrated photonic-electronic quantum-limited coherent receiver with 14.0 dB of shot noise clearance and 520 μW knee power, and a 32-channel array of the same design gives a median 26.6 dB of clearance and a 12.6 μW knee power per channel (Gurses et al., 2026). Running this receiver against a fiber-optic transmitter, we measured 0.15 ± 0.01 dB of squeezing below the shot noise limit, limited by off-chip losses (Gurses et al., 2026). The receiver came together over several iterations (Gurses & Hajimiri, 2022; Gurses et al., 2023; Gurses et al., 2024). Pablo Backer-Peral led the work that transmitted and received squeezed light over a fiber-optic link (Backer-Peral et al., 2025), and the corresponding free-space system is an on-chip phased array that images squeezed light across 32 pixels (Gurses et al., 2025).

We also propose a squeezed light communication scheme that can surpass the one-quadrature Shannon limit (Gurses et al., 2026). We have not run it on a link yet.

Remaining obstacles

Loss is the first obstacle, and it is the reason that squeezing figure is small. The measured 0.15 dB corresponds to an end-to-end detection efficiency of 0.046, or 13.3 dB of system loss, most of it off-chip in the source and tabletop components. Paired with a lower-loss source, the same chip supports about 3 dB of observable squeezing (Gurses et al., 2026). Loss also mixes vacuum noise into a squeezed state (Gurses et al., 2026), so squeezing is the property that degrades first.

The second obstacle is the cost of generating squeezing. For a nonlinear waveguide with parametric gain coefficient μ, the pump power required for a given r is (r/μ)² (Gurses et al., 2026), so squeezing improves energy per bit only when the factor e^2r outweighs that overhead. A μ ≈ 224 W^−1/2, equivalent to the second-harmonic generation efficiency demonstrated in periodically poled lithium niobate microrings (Gurses et al., 2026), puts the pump power for high squeezing levels within reach.

The third obstacle is the joint-detection receiver itself. Superadditive gains have been seen in small experiments, though not at the blocklengths or rates a deployed link needs, since getting there requires reconfigurable multimode interferometers and non-Gaussian operations at the receiver (Gurses et al., 2026). We are working through this for interconnects in high-performance computing (Gurses & others, 2026), where per-link energy budgets keep falling. At ⟨N⟩ ~ 1 photon per mode the Holevo limit is already twice the Shannon limit, and at that operating point the choice between Shannon-limited and Holevo-limited transceiver design stops being academic (Gurses, 2026).

The fourth obstacle is temporal coherence, and it underlies the other three. Saturating the Holevo capacity at low ⟨N⟩ requires the signaling modes to stay phase-coherent for the duration of a codeword, and joint detection across distributed receivers requires clock synchronization among them (Gurses, 2026), so temporal coherence is a primary resource alongside energy, bandwidth and photon number.

Summary

The Holevo limit is a curve in mean photon number, and how much it is worth depends on where a link sits on that curve. At a thousandth of a photon per mode, it is worth almost an order of magnitude in bits per photon over heterodyne detection, and collecting all of it requires a joint-detection receiver that has not been built at scale. The next post runs the same exercise for information storage.