Fundamental limits of information storage

Storage technologies transfer information between two points in time. A message is encoded onto a physical carrier, the carrier sits in an environment that degrades it, and the message is recovered later. That is the same three-part structure as a communication link (Peterson & Davie, 2011), with elapsed time in place of distance. In an earlier post I grouped memory capacity under computing and gave it the Bekenstein limit, and in the previous post I worked through what the Holevo limit buys a communication link. Here I want to run the same exercise for storage. It turns out that the Bekenstein limit is a correct bound but not the one a memory reaches first, and that the cost of storage is set by the energy barrier that keeps a bit from thermalizing.

The size of the gap

Storage has three figures of merit, and they are not equally far from their bounds. Density is the first. A 44 TB heat-assisted magnetic recording drive in a 3.5-inch enclosure works out to ~9 × 10¹⁷ bits per cubic meter (Seagate Technology, 2026), and a 2 TB solid-state drive in an M.2 package is within a factor of five of that, so both deployed media sit near 10¹⁸ bits/m³ at the device level, or ~10¹⁵ bits per liter. Normalizing to 1 kg and 1 L, as in the earlier post, an ultimate memory holds ~2 × 10³¹ bits (Lloyd, 2000). That is a gap of ~16 orders of magnitude, and I will argue below that only the first 11 of them are reachable with ordinary matter.

Retention is the second, and unlike density it is already at its bound. A magnetic drive is specified for a decade, nanostructured glass claims room-temperature lifetimes past the age of the universe (Zhang et al., 2014), and DNA projects centuries under realistic degradation kinetics (Banal, 2026). There is no gap worth attacking here, and the reason is that the retention time enters the barrier requirement only logarithmically, which I will get to.

Energy is the third. A DRAM access costs ~10 pJ per bit (Horowitz, 2014), and the thermodynamic floor for a bit that has to survive a decade is ~2 × 10⁻¹⁹ J, so the gap is ~7 orders of magnitude. Write speed is a fourth figure and is far from limiting, since the Margolus-Levitin bound puts the minimum time to flip a state of energy 60 k_BT at ~0.7 fs (Margolus & Levitin, 1998), against actual write times of nanoseconds for magnetic media and hundreds of microseconds for flash. Therefore density and energy are the gaps worth attacking, and retention is the constraint that makes both of them expensive.

A channel through time

The mapping to a communication link can be made quantitative. A channel is a system that accepts a symbol at one point and returns a degraded version of it at another, and nothing in that definition requires the two points to differ in space rather than in time. Standard treatments of information theory list the disk drive as a channel alongside the telephone line and the radio link, and the transmission of a genome from parent to daughter cell is another instance (Shannon, 1948; MacKay, 2003). Information is stored in physical degrees of freedom (Landauer, 1991), so a memory is a communication link whose transmitter and receiver are the same physical system at two different times, and the amount of information it holds over an interval is the capacity of that link.

The microscopic channel can be written down. A stored bit is a system in one of two wells, and the bath flips it between them at a rate 1/τ, which for a magnetic grain is the Néel-Brown rate discussed below. Solving the two-state master equation gives the probability that the bit read at time t disagrees with the bit written at time zero,

p(t) = (1 − e^−2t/τ)/2,

which makes a memory cell a binary symmetric channel whose crossover probability grows with the storage time. The capacity per cell is C(t) = 1 − h(p(t)), with h the binary entropy function (Shannon, 1948). For t ≪ τ the loss is a small coding overhead, and for t ≫ τ the capacity decays as e^−4t/τ/(2 ln 2). Optical fiber attenuates as e^−αL in distance and a memory cell loses capacity as e^−4t/τ in time, and in both cases the exponent is a material rate constant multiplied by the length of the channel.

In the long-time limit the crossover probability approaches 1/2, because the cell relaxes to thermal equilibrium and the equilibrium state is the same Gibbs state whatever was written, so the mutual information between the bit that went in and the bit that comes out approaches zero. A system at equilibrium retains no information about its preparation, and every finite system in contact with a bath approaches equilibrium. The memory time of a medium is therefore set by the spectral gap of its relaxation dynamics, and a medium that holds data for geological time is a system whose ergodicity is broken over the timescale of observation (Palmer, 1982). An energy barrier is the standard way to break ergodicity, because the Kramers escape rate closes the relaxation gap exponentially in ΔE/k_BT (Kramers, 1940). Storage is therefore a problem of delaying thermalization, and the sections below work out what the delay costs.

One feature of the mapping is inverted. In communication, latency is a cost to be minimized, and the previous post found a deployed fiber link sitting within a factor of three of the vacuum bound. In storage, the delay is the quantity specified rather than minimized, and the difficulty is that the channel stays lossy over the entire specified interval.

The capacity of a memory at short times is set by its entropy. The maximum number of bits a physical system can hold is S/(k_B ln 2), where S is the thermodynamic entropy of the set of configurations we can prepare and later distinguish. A bound on storage capacity is therefore a bound on entropy, which is why the Bekenstein limit shows up at all.

The hierarchy of bounds

Three different bounds are called the storage limit in different contexts, and they differ by tens of orders of magnitude.

The holographic bound caps the entropy of a region by its boundary area in Planck units, S ≤ A/(4ℓ_P²) (Bekenstein, 1981; ’t Hooft, 1993; Susskind, 1995; Bousso, 2002). For a spherical liter, R = 6.2 cm, that is 6.7 × 10⁶⁷ bits. It is saturated only when the region is itself a black hole, which at that radius requires ~4 × 10²⁵ kg, about seven Earth masses. It constrains geometry and says nothing about a memory of ordinary mass.

The Bekenstein bound adds the energy constraint, S ≤ 2πk_BER/(ℏc) (Bekenstein, 1981). At 1 kg and R = 6.2 cm it gives 1.6 × 10⁴² bits, twenty-six orders below the holographic figure because a kilogram is a very small amount of energy on this scale.

The realizable bound is lower again. Lloyd computes the maximum entropy an actual kilogram of matter can carry in a liter by maximizing over matter and radiation states, which peaks at ~5.9 × 10⁸ K and gives 2.1 × 10³¹ bits (Lloyd, 2000). Eleven orders separate this from the Bekenstein bound, because the density of states of real matter does not come close to saturating it. Running the same maximization over the observable universe instead of a kilogram gives ~10⁹⁰ bits registered in matter, against ~10¹²⁰ if gravitational degrees of freedom are included (Lloyd, 2002).

One correction to my earlier post is worth making here. I described packing information into a volume until the system collapses into a black hole, which is the right picture at fixed volume with growing energy, but it is misleading at fixed mass. A 1 kg black hole has a Schwarzschild radius of 1.5 × 10⁻²⁷ m and holds 3.8 × 10¹⁶ bits, fifteen orders below the same kilogram spread through a liter. The Bekenstein bound scales with the product ER, and collapse shrinks R by twenty-five orders of magnitude while leaving E alone. Gravitational collapse therefore destroys storage capacity rather than maximizing it, and it destroys retention as well. The same 1 kg black hole has a Hawking temperature of 1.2 × 10²³ K and evaporates in ~10⁻¹⁶ s (Hawking, 1975), and at astrophysical masses, where the lifetime is long, the contents return as Hawking radiation, scrambled and no earlier than the Page time (Page, 1993; Hayden & Preskill, 2007). The object that saturates the entropy bound is itself a thermal state with a finite lifetime and a thermal read channel, so the thermodynamic constraints on storage apply at the bound as well.

The bound that matters in practice is lower than all three of these, and it is chemical. Condensed matter contains ~5 × 10²⁸ atoms per cubic meter, so a solid-state memory storing one bit per atom cannot exceed ~5 × 10²⁸ bits/m³. Two very different demonstrations come within about an order of it. DNA at its information-theoretic ceiling of 227.5 exabytes per gram of double-stranded DNA is ~2 × 10²⁷ bits/m³ (Banal, 2026), which follows directly from two bits per base pair in a volume of ~1 nm³. Single-atom magnetic memory, written and read on individual holmium atoms spaced 1 nm apart, is the same order (Natterer et al., 2017). Therefore the eleven orders between today’s media and one bit per atom are ordinary engineering, and the remaining five up to Lloyd’s figure require abandoning chemistry for nuclear degrees of freedom or for the relativistic plasma that figure assumes.

The retention constraint

A bit is a system held on one side of an energy barrier, and thermal fluctuations knock it over the barrier at the Néel-Brown rate (Néel, 1949; Brown, 1963),

τ = τ₀ e^ΔE/k_BT,

which is the Kramers escape rate specialized to a magnetic grain (Kramers, 1940). Inverting for a target per-bit failure probability p over a retention time t gives the barrier requirement ΔE/k_BT = ln(t/τ₀p). With an attempt time τ₀ = 1 ns, a decade of retention at p = 10⁻¹⁵ needs 74.8, and the magnetic recording industry’s working figure is K_uV/k_BT > 60 (Weller & Moser, 1999).

Two things follow. The first is that additional retention time is cheap, because it enters the barrier requirement only logarithmically. Going from ten years to 13.8 billion years costs 21 k_BT of additional barrier, which is why media with geological lifetimes exist without exotic physics. They have deep barriers, which the logarithm keeps inexpensive.

The second is that density is expensive, because the barrier height scales with the volume of the cell. For a magnetic grain, ΔE = K_uV. FePt in the L1₀ phase, the highest-anisotropy material in practical use, has K_u ≈ 7 × 10⁶ J/m³, so a 60 k_BT barrier at room temperature needs V ≥ 35 nm³, a cube 3.3 nm on a side. One grain per bit at that pitch is 9.3 × 10¹⁶ bits/m², or 60 Tb/in². Real media use roughly ten grains per bit to get enough read signal-to-noise ratio, which puts the practical limit near 6 Tb/in². Shipping HAMR drives are at ~2.2 Tb/in² (Seagate Technology, 2026).

Magnetic recording is therefore within a factor of ~3 of its physical limit, which is an unusual position for an information technology. The previous post found communications six orders of magnitude from its bound; magnetic storage has less than one, and the remaining ten orders in density are not available in this medium at all.

The constraint is usually stated as a trilemma. Readability improves with many grains per bit, thermal stability requires large grains, and writability requires a coercive field the head can actually supply, which falls as K_u falls. Any two of the three can be met together. HAMR resolves the trilemma by heating the medium close to its Curie temperature during the write, so that writing happens at low coercivity while storage happens at high anisotropy (Kryder et al., 2008). Decoupling the write temperature from the storage temperature is the only known way to move all three at once, and it works by making the barrier height time-dependent.

Passive and active storage

There are two ways to keep a system out of equilibrium for a chosen time, and every storage technology uses one of them or a combination of the two.

In passive storage the barrier is raised once and the medium is then left unpowered. The write pays ΔE = k_BT ln(t/τ₀p) once, retention requires no further energy, and long retention times are inexpensive because t enters only logarithmically. Tape, optical media, nanostructured glass, and archival DNA are all passive.

In active storage the cell decays quickly and errors are corrected faster than the bath injects them, which is the function a repeater performs in a communication link. DRAM is the simplest example. A typical cell holds its charge for around a second, the specification assumes much less, and the controller rewrites every row on a 32 to 64 ms cadence, so a decade of storage in DRAM is a chain of ~5 × 10⁹ refresh spans. Error-corrected storage of any kind is the same strategy, building a reliable memory from unreliable components in von Neumann’s sense (von Neumann, 1956). Biological storage is also active. A genome is maintained by polymerase selectivity, proofreading, and mismatch repair, which together hold the replication error rate near 10⁻¹⁰ per base pair (Drake et al., 1998).

Active storage carries a thermodynamic floor that passive storage does not. The refresh controller is a Maxwell demon in the strict sense (Szilard, 1929). It determines which cells the bath has flipped, restores them, and must clear its own record before the next cycle. Bennett located the unavoidable dissipation in clearing the record rather than in the measurement, where Brillouin had placed it (Brillouin, 1956; Bennett, 1982; Bennett, 2003), and clearing the record is erasure at k_BT ln 2 per corrected error (Landauer, 1961). The minimum maintenance power is therefore k_BT ln 2 multiplied by the rate at which the medium generates errors, and over a retention time t the energy per bit is at least (t/τ) k_BT ln 2 for a cell that decays in τ, against the one-time ln(t/τ₀p) of a barrier. For a cell that decays in a second, a decade of active retention costs at least 2 × 10⁸ k_BT per bit, six orders of magnitude above the ~75 k_BT that buys the same decade passively. The maintenance energy is linear in the retention time while the barrier energy is logarithmic in it, and the storage hierarchy from DRAM down to tape is arranged around that difference.

This accounting is also why the Maxwell demon literature applies to storage directly. A memory under refresh is a system held in a nonequilibrium steady state by measurement and feedback, which is the system Szilard analyzed and Bennett resolved (Szilard, 1929; Bennett, 1982).

The energy floor

Erasing a bit dissipates at least k_BT ln 2, which is 2.9 zJ at room temperature (Landauer, 1961). The bound has been verified directly, in a colloidal particle in a modulated double-well trap (Bérut et al., 2012) and in nanomagnetic bits (Hong et al., 2016). Reversible computation carries no such floor (Bennett, 1973), and write-once archival media never erase, so they never pay it.

Write-once media still pay the barrier. Writing a bit means driving the system over ΔE, and in any dissipative switch that energy is lost to the environment. At the 60 to 75 k_BT required for a decade of retention, that is 0.25 to 0.31 aJ, which is 87 to 108 times the Landauer figure. The energy floor of storage is therefore set by retention rather than by erasure, and the retention cost is roughly two orders of magnitude larger. This distinction matters because the two scale differently. The Landauer cost is fixed at k_BT ln 2 per erasure, while the retention cost grows logarithmically with the retention time and with the size of the array, since a larger array demands a smaller per-bit failure probability.

Against that floor, the ~10 pJ per bit of a DRAM access (Horowitz, 2014) is a factor of ~5 × 10⁷. Almost all of it is interconnect, sense amplifiers, and array overhead rather than the bit itself, which is the same conclusion the previous post reached about a fiber link. Most of the gap to the fundamental bound is ordinary engineering, and the interesting question is what remains after the engineering is done.

Coding and retention

The barrier requirement ΔE/k_BT = ln(t/τ₀p) is logarithmic in the tolerated raw error rate, so relaxing p and correcting the resulting errors with a code buys barrier height directly. The cost is code rate. For a binary symmetric channel at raw error rate p, the usable fraction of the raw capacity is 1 − h(p), with h the binary entropy function (Shannon, 1948). The right figure of merit is therefore the barrier energy per useful stored bit, ln(t/τ₀p)/[1 − h(p)].

Over a decade of retention, uncoded operation at p = 10⁻¹⁵ costs 74.8 k_BT; at p = 10⁻³ the barrier falls to 47.2 k_BT and the code rate to 0.989, for 47.7 k_BT per useful bit; at p = 10⁻², 44.9 k_BT at rate 0.919, for 48.8; and at p = 10⁻¹, 42.6 k_BT at rate 0.531, for 80.2. The minimum is at p ≈ 2.5 × 10⁻³, where a 46.3 k_BT barrier and a code rate of 0.975 give 47.5 k_BT per useful bit. Coding therefore cuts the retention energy floor by ~37%, and the shape of the curve shows why pushing further does not help, since between p = 10⁻³ and p = 10⁻¹ the barrier falls by only 4.6 k_BT while the code rate falls by nearly half.

Modern NAND with LDPC coding operates at raw error rates between 10⁻³ and 10⁻² and delivers 10⁻¹⁵ to the host (Cai et al., 2017). That is the same operating point, arrived at from device physics and controller economics rather than from thermodynamics. The convergence is not a coincidence, because both are minimizing the same quantity. It is also the storage analog of the finite-blocklength discussion in the previous post, where coding bought error probability at a cost in latency, while here it buys barrier height at a cost in capacity.

The cost of cooling

Every quantity above scales with k_BT, so running the memory cold looks like it should cut the energy per bit in proportion. It does not, once the refrigerator is included in the accounting. A bit stored at temperature T_c needs a barrier of ~60 k_BT_c and dissipates that much into the cold space when written. Removing that heat to an environment at T_h costs at least the Carnot work W = Q(T_h − T_c)/T_c = 60 k_B(T_h − T_c). The total drawn from the room-temperature supply is 60 k_BT_c + 60 k_B(T_h − T_c) = 60 k_BT_h, independent of T_c. At the Carnot limit the cost of a thermally stable bit is set by the ambient temperature and not by the operating temperature, and the same cancellation applies to the Landauer term. Real cryogenic plants run one to three orders above Carnot, so cooling is a net loss on the energy budget.

Cooling buys density rather than energy. The barrier requirement is ΔE ≥ 60 k_BT_c, so the minimum grain volume falls linearly with temperature. At 77 K the FePt grain volume drops by a factor of 3.9 and the areal density limit rises by a factor of ~2.5. Whether the density is worth the cryogenic plant is an application question rather than a thermodynamic one.

What quantum mechanics buys

The central result of the previous post was that allowing arbitrary quantum measurements raises the channel capacity above the Shannon limit, without bound, as the photon number per mode falls. The corresponding question here is whether a quantum memory holds more classical bits than a classical memory of the same size. It does not, and the theorem that forbids it is the same Holevo bound.

Holevo’s bound caps the accessible classical information in a quantum system by the von Neumann entropy of the ensemble (Holevo, 1973), which for a d-dimensional system is at most log₂ d. Therefore n qubits hold at most n classical bits, however they are prepared and whatever measurement is used to read them. Superdense coding does not evade this, since the two classical bits per transmitted qubit require a second qubit already delivered as pre-shared entanglement (Bennett & Wiesner, 1992), and the two-qubit accounting is unchanged.

Two things are still bought. The first is that a quantum memory stores quantum states, which classical media cannot do at all, and which the previous post mentioned at the end but did not develop. The entanglement-assisted capacity that sits a factor of two above the Holevo limit requires the receiver to hold an ancilla until the signal arrives (Bennett et al., 1999; Bennett et al., 2002), so the coherence time of that memory is a hard constraint on the link (Gurses, 2026). Retention in a storage context and temporal coherence in a communication context are the same resource measured in the same units. The solid-state state of the art is six hours of optically addressable nuclear-spin coherence in Eu³⁺:Y₂SiO₅ (Zhong et al., 2015), extended past 13 hours at 125 mK (Wang et al., 2025). Six hours is 6.5 × 10¹² m at the speed of light, about 43 astronomical units, so coherence time does not limit the range of an entanglement-assisted link; the limits come from the storage efficiency and the number of modes a memory can hold.

The second is that error correction converts redundancy into retention, and the channel description above quantifies what the correction is working against. A quantum memory is a quantum channel from the present to the future, with a capacity for quantum states given by the regularized coherent information (Lloyd, 1997; Devetak, 2005). The classical capacity of the two-state cell above is positive at every finite time, but a quantum capacity can reach zero at a finite time. A memory whose carriers are lost with probability p has quantum capacity 1 − 2p, which reaches zero once half the carriers are gone (Bennett et al., 1997). The mechanism is the no-cloning theorem. At p = 1/2 the environment holds as much of the state as the memory does, so any decoder that recovered the state from the memory output would recover it equally well from the environment’s share, and the two decoders together would clone the state. Therefore, past one half-life of the carrier, no decoding applied at readout recovers a stored qubit, regardless of the encoding.

The correction therefore has to run before that point, and the no-cloning theorem also rules out the classical refresh strategy, since reading and rewriting each cell is copying. Quantum error correction meets both constraints by measuring the error syndrome without measuring the data, at intervals short against the carrier lifetime. It is the same exchange of code rate for effective barrier height that governs NAND, applied to a system whose raw barrier is too shallow to be useful on its own. Below the surface-code threshold, the logical error rate per cycle falls by a constant factor for every increase of two in code distance, measured at Λ = 2.14, with a distance-7 logical qubit outliving its best physical qubit by a factor of 2.4 (Google Quantum AI and Collaborators, 2025). That is a retention result, and it means every quantum memory demonstrated to date is active, maintained like a DRAM cell by correction that runs faster than the decay.

A quantum analog of the passive barrier has not been demonstrated. Passive classical storage works because anisotropy separates the two states of a bit by a macroscopic energy barrier. Whether any Hamiltonian protects a qubit at finite temperature without a controller is the self-correcting quantum memory problem, and its status depends on the spatial dimension. In two dimensions no stabilizer Hamiltonian can do it (Bravyi & Terhal, 2009). In four dimensions the toric code does, with an energy barrier that grows with the system size (Dennis et al., 2002). In three dimensions the best known construction retains its advantage only up to an optimal system size, and the general question is open (Haah, 2011; Brown et al., 2016). Whether passive quantum storage is possible in three dimensions is an open problem in Hamiltonian physics rather than an engineering question.

Remaining obstacles

Reading is a communication problem, and everything from the previous post applies to it unchanged. A medium at 10²⁷ bits/m³ has to be read out through some channel, and at that density the photon or electron budget per bit is small, which is the low-⟨N⟩ regime where the gap between Shannon-limited and Holevo-limited reception is widest (Banaszek et al., 2020). A quantum-limited receiver is therefore as relevant to a read head as to a fiber (Gurses et al., 2026), and the readout of a dense archival medium may be the most immediately practical place to spend joint-detection capability, because the channel is short, the loss is controllable, and the latency budget is generous.

The second obstacle is structural. Capacity scales with volume and bandwidth scales with surface, so the bits sit inside a boundary they all have to cross, and the ratio of delivered bits per second to bits stored falls as the linear dimension grows. This is the memory wall in its most basic form, and it is why the densest media occupy the cold end of the storage hierarchy. There is a latency floor underneath it as well. A one-liter device is 12 cm across, which is 0.4 ns of one-way light travel before any addressing or decoding.

The third is that raw density is not usable density. DNA reaches its numbers only with heavy coding against synthesis and sequencing errors, and the best current codecs recover 155.8 and 25.9 exabytes per gram under high- and low-fidelity conditions against the 227.5 exabytes per gram ceiling (Banal, 2026). Storage systems deliberately operate at raw error rates near 10⁻², and the distance between a medium’s physics and a system’s usable capacity is coding overhead, addressing, and wear management. That distance is not waste, since the coding section above shows it is where a third of the retention energy is recovered, but it does mean a medium’s raw density is never the number to design against.

The fourth is more basic than the others. Eleven orders of magnitude separate today’s media from one bit per atom, and every one of them is engineering, but the one-bit-per-atom bound is a discontinuity rather than another engineering step. Nothing in the hierarchy between one bit per atom and Lloyd’s 10³¹ bits per liter is a memory in any recognizable sense; the intermediate configurations are hot dense plasmas whose retention time is set by their own thermalization (Lloyd, 2000). An honest roadmap for storage density therefore stops at 5 × 10²⁸ bits/m³ and treats everything above it as a statement about physics rather than a target.

How you can help

The pattern that comes out of this is the reverse of the one in the previous post. Communications has six orders of magnitude of headroom against a bound that is a curve in photon number, and the payoff is concentrated at the low-photon-number end where nobody currently operates. Storage has a bound that is chemical, most media are far from it, and one medium is within a factor of ~3 of it. The interesting work in storage is in choosing which set of atomic degrees of freedom to encode into and in paying the retention barrier as efficiently as possible once that choice is made, rather than in closing a gap to a bound that no memory of ordinary matter will approach. The energy cost of storage is set by retention rather than by erasure, and coding is the cheapest way to reduce it. The channel picture in the second section organizes all of this, since every constraint in the post, from the refresh cadence of DRAM to the evaporation of a black hole, is a statement about the capacity of a channel run through time.

I recommend this tool if you want to move along these curves yourself and compare the ultimate and current versions of computing, communication, and sensing systems. If you think I have put a bound in the wrong place, or you know of work on atomic-scale or molecular media 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 processing.