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. The answer depends almost entirely on the mean photon number per mode, which is small in exactly the regime that parallel architectures are pushing us toward.

The size of the gap

The gap is set by two figures, both normalized to 1 W and 1 km. An ultimate communication link 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 a different story, because a deployed link already sits close to its bound. Standard single-mode fiber has a group index of about 1.47 at 1550 nm, giving a propagation delay of ~4.9 μs/km against the vacuum figure of ~3.3 μs/km, and a deployed link runs ~10⁻⁵ s of total latency (Optical Internetworking Forum, 2025). That leaves a factor of ~3 against the vacuum bound and only a factor of ~2 against the group index of the fiber itself. Therefore, the throughput gap is the one worth attacking, and it is ~6 orders of magnitude wide.

Most of those 6 orders are ordinary engineering: unoccupied bandwidth, launch power well below what is available, and electronic noise in the receiver that has nothing to do with the light. The quantum step, from the Shannon limit to the Holevo limit, turns out to be a small slice of the total in a conventional link. However, the width of that slice is set by the mean photon number per mode, and it grows without bound as that number falls. The quantum step has to be evaluated at the photon number where a given link actually operates.

Two ways to measure the same light

Take one mode per symbol and write its mean photon number as ⟨N⟩ = ⟨â^†â⟩. Throughput is bounded by the channel capacity, and the capacity depends on what we allow the receiver to do. If we use both quadratures and read them out with heterodyne detection, the link becomes a complex additive white Gaussian noise channel with per-quadrature signal-to-noise ratio ⟨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 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.

These two cross at ⟨N⟩ = 2. Above the crossing point heterodyne detection gives the higher capacity, and below it single-quadrature modulation does, approaching a factor of two as ⟨N⟩ falls. The squeezing result later in this post is a statement about C_S1.

The extra vacuum unit in the heterodyne noise floor is the price of measuring two non-commuting quadratures at once, and quantum mechanics does not require us to measure that way. 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),

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

where g(·) is the bosonic entropy function. It strictly exceeds both Shannon limits for every ⟨N⟩ > 0 (Banaszek et al., 2020). Against the two-quadrature limit, the difference is C_Hol − C_S2 = ⟨N⟩ log₂(1 + 1/⟨N⟩).

Origin of the gap

The gap between the Shannon and Holevo limits is superadditive coding gain (Guha, 2011). Coherent states are not orthogonal. Neighboring constellation points always retain some quantum-mechanical overlap, and no measurement performed on one symbol at a time can fully resolve them. Symbol-by-symbol detection measures each received mode independently, and the information extracted from a codeword is the sum over its symbols. A collective measurement instead acts on all n modes at once, as a single quantum system. The individual symbols still overlap poorly, but codewords of n symbols span a joint n-mode Hilbert space in which they can be made nearly orthogonal, and the information extracted from the block exceeds the sum of the n independent single-symbol measurements. Therefore, the distinction lies entirely in the measurement performed at the receiver. Coherent states with Gaussian modulation already saturate the Holevo limit for the lossy bosonic channel, provided we read them out jointly (Giovannetti et al., 2004).

There is a third bound above both of these. If the transmitter and receiver share entanglement ahead of time, the limit becomes the entanglement-assisted classical capacity (Bennett et al., 1999; Bennett et al., 2002), saturated by transmitting one mode of a two-mode squeezed vacuum and keeping the other as a receiver-side ancilla. It reduces to twice the Holevo capacity in the noiseless limit and recovers the two bits per channel use of superdense coding (Bennett & Wiesner, 1992). The hierarchy is C_EA ≥ C_Hol ≥ C_S2, and we buy the first gap with pre-shared entanglement and the second with joint measurement. The factor of two is bounded, but it holds at every photon number, which complements the unbounded low-photon-number gain of Holevo over Shannon. Collecting it requires an entanglement-distribution infrastructure, which is one reason classical and quantum links may end up sharing a single low-loss, phase-sensitive substrate.

The quantum step versus photon number

At large ⟨N⟩ the two bounds converge in relative terms as the field starts behaving classically, and the additive difference saturates at log₂e ≈ 1.44 bits per mode. At ⟨N⟩ = 100 photons per mode, the Holevo limit is only 1.22 times the two-quadrature Shannon limit. A fraction of a bit per mode is negligible against 6 orders of magnitude, and the Holevo limit has no bearing on the design of 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 exactly twice the two-quadrature Shannon limit, 2 bits per mode against 1, and at ⟨N⟩ = 0.01 it is 5.6 times. Joint detection becomes arbitrarily better than quadrature measurement as the photon number per mode falls.

The same asymmetry shows up in energy, where the energy per bit is E_b = ħω⟨N⟩/C. The Shannon-limited energy efficiency bottoms out at a floor of E_Sh ≈ ħω/log₂e = ħω ln2 (Verdu, 1990; Giovannetti et al., 2004), which at 1550 nm is about 89 zJ per bit, or 0.69 photons per bit. Equivalently, no amount of coding gets a two-quadrature coherent receiver past a ceiling of log₂e ≈ 1.44 bits per photon. The Holevo-limited energy efficiency is E_Hol ≈ ħω/log₂(e/⟨N⟩), which has no floor underneath it, and at ⟨N⟩ = 0.001 joint detection extracts 11.4 bits per photon, about 11 zJ per bit. The ħω ln2 floor is therefore a property of the measurement, and it disappears as soon as the receiver measures jointly.

That ceiling binds the coherent receiver specifically. Pulse-position modulation read out by photon counting is a different measurement on a different channel, and deep-space links already operate at ~10 bits per photon (Banaszek et al., 2020). The Holevo limit is the bound that survives across all measurements.

A concrete link makes the photon-number dependence easier to see. Take 1550 nm, 100 GHz of bandwidth, and 10 mW of optical power, which puts ⟨N⟩ ≈ 7.8 × 10⁵ photons per mode. The Shannon, Holevo, and entanglement-assisted limits evaluate there to 1.96, 2.10, and 4.20 Tbps, a 7% Holevo improvement, at 5.11, 4.76, and 2.38 fJ per bit. Spread the same 10 mW across 10⁷ multiplexed spectral and spatial modes instead, and ⟨N⟩ falls to ≈ 0.078 per mode, where the Holevo energy per bit drops to ≈ 25 zJ. Same power and the same total photon budget, five orders of magnitude in energy efficiency, and the only difference is how the photons are distributed across modes and how they are measured.

The quantum step is primarily an energy-efficiency result. The rate per mode falls as ⟨N⟩ does, and aggregate throughput then has to come from many multiplexed low-power channels, which is what points at massively parallel architectures carrying a small photon budget in each channel (Gurses et al., 2026). Both capacities should also be evaluated at the photon number reaching the detector, and channel loss and detector inefficiency push the improvement factor back down.

Coding and blocklength

Capacity on its own says nothing about error rate, and Shannon’s theorem says only that we can push the error probability arbitrarily low by coding over longer blocks at any rate below capacity (Shannon, 1948), at a cost in latency. For blocklength n over a bandwidth B, the decoding latency is bounded below by n/B, which is separate from the propagation delay above.

Coding over longer blocks costs latency, so how far below capacity do we have to back off at a finite blocklength? The maximum rate at blocklength n and block error probability ε is approximately C − √(V/n)·Q⁻¹(ε), where V is the channel dispersion, the variance of the information density, and Q⁻¹ is the inverse Gaussian tail function (Polyanskiy et al., 2010). Inverting that expression, the minimum blocklength for a target rate R is n* ≈ V[Q⁻¹(ε)]²/(C − R)², and the coding latency is n*/B. Approaching capacity therefore carries a quadratic latency penalty.

Each capacity carries its own dispersion. For the bosonic channel these are V_Sh = ⟨N⟩(⟨N⟩+2)(log₂e)²/[2(1+⟨N⟩)²] and V_Hol = ⟨N⟩(⟨N⟩+1)[log₂((⟨N⟩+1)/⟨N⟩)]², with V_EA = 2V_Hol (Polyanskiy et al., 2010). At large ⟨N⟩, V_Hol → 2V_Sh, so at a matched backoff from capacity a Holevo-limited link needs roughly twice the blocklength of a Shannon-limited one. At a matched operating rate, however, the larger Holevo capacity enlarges the backoff, which can more than compensate for the larger dispersion and leave the Holevo-limited link with the shorter blocklength. Which effect dominates depends on how far the operating rate sits below the two capacities.

The photon-starved case behaves differently again. At small ⟨N⟩ the ratio C_Hol/C_S2 diverges, so rates between the two are reachable with vanishing error probability under joint detection and unreachable under symbol-by-symbol two-quadrature reception at any blocklength. Therefore, at low signal power a longer code cannot close a rate shortfall, and the payoff has to come from changing the measurement.

What we can build today

Collective decoding is out of reach with the receivers we can build today. However, there are two things we can build that move a real communication link closer to these bounds, and in one case past 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 while the electronics behind the photodiodes contribute negligibly. A quantum-limited receiver puts a link on the ideal Shannon curve without passing it, and it is what lets us 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, with the local oscillator phase locked to the squeezed quadrature, 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 (Gurses et al., 2026; Banaszek et al., 2020). The hierarchy is C_S1 ≤ C_sq ≤ C_Hol. Squeezed light with coherent detection occupies an intermediate regime, exceeding the one-quadrature Shannon limit without requiring 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, 520 μW knee power, 2.57 GHz 3-dB bandwidth, and 90.2 dB of common-mode rejection ratio (Gurses et al., 2026). Scaling the design to a 32-channel array gives a median 26.6 dB of clearance and a 12.6 μW knee power per channel, and automatic common-mode rejection correction yields a median 76.8 dB of rejection at minimum (Gurses et al., 2026). Low knee power is the figure that matters for the parallel architectures above, because it is what lets each channel reach the shot-noise-limited regime on a microwatt of local oscillator. 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). This 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 over free space 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, and the receiver above therefore operates as a measurement device.

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, almost all of it in off-chip coupling (Gurses et al., 2026). The receiver’s own optical loss is 2.7 dB, and paired with a lower-loss source the same chip supports about 3 dB of observable squeezing (Gurses et al., 2026). That improvement is a packaging problem. However, loss acts differently on coherent and squeezed light. With detection efficiency η, the measured quadrature variance of a squeezed state is ¼(ηe^−2r + 1 − η), where the first term is the attenuated squeezed variance and the second is vacuum noise mixed in by the loss (Gurses et al., 2026). A coherent state comes out of a lossy channel as a weaker coherent state, while a squeezed state comes out with less squeezing and in a mixed state, which makes squeezing the property that degrades first. Loss hurts on the classical side too. The exact classical capacity of the lossy bosonic channel with transmissivity η is g(η⟨N⟩) − g((1−η)⟨N⟩) (Giovannetti et al., 2004), which falls as the transmissivity falls.

The second obstacle is the cost of generating squeezing. The squeezing parameter scales as r = μ√P_pump for a nonlinear waveguide with parametric gain coefficient μ, which puts the pump power required for a given r at (r/μ)² (Gurses et al., 2026). Squeezing improves energy per bit only when the exponential gain e^2r outweighs that overhead, which makes μ and the detection efficiency η the two parameters that decide whether the trade is favorable. μ is the dominant one, and conversion efficiencies corresponding to μ ≈ 224 W^−1/2 have been demonstrated in periodically poled lithium niobate microrings (Gurses et al., 2026), which puts high squeezing levels within reach. There is also an optimum local oscillator power, which enters the power budget directly and also sets η through the shot noise clearance.

The third obstacle is the joint-detection receiver itself. Structured designs exist, such as Hadamard-coded coherent-state sequences decoded through a mesh of order n log n elements followed by photon counting (Guha, 2011). Superadditive gains have been seen in small experiments, though not at the blocklengths or rates a deployed link needs. Getting there requires reconfigurable multimode interferometers and non-Gaussian operations at the receiver (Gurses et al., 2026), which is a harder requirement than either low-loss packaging or high parametric gain. We are working through this for interconnects in high-performance computing, where the distances are short and the power budgets are tight (Gurses & others, 2026). Intra-data-center and chip-to-chip links are the most immediate case, because their per-link energy budgets are approaching ⟨N⟩ ~ 1 photon per information bit, and at that operating point the choice between Shannon-limited and Holevo-limited transceiver design stops being academic (Gurses, 2026). That work is in preparation.

A fourth constraint runs underneath the other three, and the energy and bandwidth accounting above does not surface it. Approaching the Holevo capacity at low ⟨N⟩ requires the signaling modes to stay phase-coherent for at least the duration of a codeword, and joint detection across distributed receivers additionally requires clock synchronization among them (Gurses, 2026). Laser linewidth and frequency stability bound the first, and for entanglement-assisted operation the coherence time of whatever memory holds the ancilla bounds the second. Temporal coherence is therefore a primary resource alongside energy, bandwidth, and photon number, and an honest accounting of the distance to the Holevo ceiling has to carry it on equal footing.

How you can help

The Holevo limit is a curve in mean photon number, and how much it is worth depends entirely on where a link sits on that curve. At the photon numbers a conventional fiber link uses, it is worth a fraction of a bit per mode, which no link designer needs to account for. Down at a thousandth of a photon per mode, it is worth almost an order of magnitude in bits per photon, and collecting it requires a joint-detection receiver that has not been built at scale. The concrete version of the whole program is a link whose operating point comes out of a Holevo capacity budget, where today it would come out of eye-diagram margins.

I recommend this tool if you want to move along that curve yourself and compare the ultimate and current versions of communication, computing, and sensing systems. If you think I have put a bound in the wrong place, or you know of work on joint-detection receivers I should be reading, I would like to hear it, and comments are open below. Next I want to run the same exercise for information storage.