Imagine a key hidden in one of eight drawers. You know that each drawer was equally likely to receive it. Someone who knows the location can guide you with three yes-or-no answers: left half or right half, which pair within that half, then which drawer within the pair.
Those three answers identify one possibility among eight. The same arrangement could hold a worthless brass washer instead of a key. The number of possible locations would still be eight, and the three-answer procedure would still work. Your reason for wanting the object would have changed. The information problem would not.
This invented example gives us a useful starting point for the phrase “information is physical.” Information can be described mathematically, while a record of it requires a physical arrangement. Those are connected ideas. They do not make every use of the word information interchangeable.
Count alternatives before judging a message
Claude Shannon’s 1948 theory of communication treated the reproduction of a selected message as an engineering problem. Its mathematics separated that problem from the message’s meaning. A system had to handle the possible selections, including ones its designer could not know beforehand. Shannon used logarithms to quantify the uncertainty associated with their probabilities. Original paper, introduction and section 6.
For eight equally likely drawers, the uncertainty is three bits because two multiplied by itself three times gives eight. A bit is the unit associated with a base-two logarithm. It is not a tiny parcel of wisdom. A warning about a real danger and a meaningless randomly selected label can each require three binary answers to distinguish among eight possibilities.
Probabilities matter as well as the number of alternatives. Suppose the key is always in drawer one. Learning that arrangement resolves no uncertainty for someone who already knows it. The entropy is zero. If drawer one receives the key half the time and the other drawers share the remaining probability, the entropy lies between zero and three bits.
A compact formula expresses this: multiply each outcome’s probability by the base-two logarithm of that probability, add the terms, and change the sign. An outcome with zero probability contributes zero. The calculation applies to the distribution we specified. It cannot assess whether our account of the drawer-selection process was truthful.
That last distinction matters in ordinary data work. A confidently wrong report can be easy to encode. A genuinely uncertain report can faithfully describe incomplete knowledge. Entropy, accuracy and practical usefulness ask different questions.
A longer record can contain the same choice
Write the drawer number as a three-digit binary code. Drawer one might be 000, drawer two 001, and so on through 111. Now copy that code a hundred times. The record has become longer, but it still distinguishes the same eight possible drawer locations.
The repeated copies might serve an engineering purpose. For example, you could compare them when noise corrupts part of the record. But they do not create a hundred new independent choices about where the key is. Treating record length as information content would miss the redundancy.
Here is another invented comparison. One file contains a million digits generated by a fair random process. Another contains a short, accurate instruction for finding a shutoff valve. The first may have much higher uncertainty before it is received. The second may be the one you urgently need. To explain its usefulness, we need the situation, the decision and the consequences of getting it wrong.
The business vocabulary of “reducing information entropy” often means removing duplication, contradictions and stale claims from a knowledge system. That can be useful editorial language. A rigorous entropy calculation would additionally require specified alternatives and probabilities. The metaphor alone does not supply them.
The drawer becomes a memory cell
The three answers can be recorded as states of physical memory. A switch can be up or down; an electrical arrangement can represent a zero or a one. The abstract distinction becomes available to a machine because some physical states reliably stand for it.
Consider just one bit, initially zero or one with equal probability. Now reset it to zero regardless of its previous state. Both inputs arrive at the same output. Looking at the final zero will not tell you which input was present. The operation has discarded a distinction.
This is logical irreversibility: the output does not uniquely determine the input. It differs from simply changing a zero into a one while preserving enough information to reverse the change. A reversible transformation can move information around without deleting the distinction in the same way.
Landauer’s principle connects complete erasure of an initially unbiased classical bit with a minimum average heat cost in the usual thermal-reservoir setting. The familiar bound is kT ln 2, where k is Boltzmann’s constant and T is absolute temperature. The conditions belong with the expression. Known inputs, incomplete erasure, correlations and different physical resources require more careful accounting. Landauer’s original analysis of logical irreversibility.
The heat need not appear as a spectacular event. At small scales, it takes a deliberately controlled experiment to resolve the relation.
A bead made the distinction visible
In a 2012 experiment, Antoine Bérut and colleagues used a silica bead held in a laser-created double-well potential as a one-bit memory. The bead’s well represented zero or one. Lowering the barrier and tilting the potential drove it toward the selected reset state; restoring the barrier completed the memory operation. The researchers measured trajectories and dissipated heat. For long erasure cycles, the mean heat approached the Landauer bound. Experimental paper, protocol and results.
Their result concerned a physical procedure with a defined starting distribution and success rate. Thermal fluctuations meant an individual trajectory could differ from the average. Incomplete erasure also changed the appropriate bound. Those qualifications help explain what the measurement established; they are part of the result itself.
A bead in either well can carry the same abstract bit as an electrical memory. The mechanisms are different. That is the point of giving the bit a physical realization: an information operation must be performed by something with dynamics, constraints and an environment.
The experiment did not measure how much electricity a modern data center needs to delete a file. It isolated one relation that a much larger system must accommodate along with many other costs.
Put the small number in its proper place
At 300 kelvin, using the SI value of Boltzmann’s constant, kT ln 2 is approximately 2.87 × 10⁻²¹ joules per bit. For one trillion independent, initially unbiased bits completely erased under those conditions, multiplying the ideal bound gives about 2.87 × 10⁻⁹ joules.
These are calculations from the formula, not measurements made for this article. Three hundred kelvin is about 27 degrees Celsius. The trillion-bit example is a deliberately specified hypothetical memory, rather than a claim about any file or product.
The tiny result is easy to misuse. It does not say a computer should consume only that much power. A computer also moves signals, maintains reliable states, performs operations, communicates with other components and manages heat. The bound is a floor for the specified erasure task. It is not a bill for a complete computing service.
Nor can you multiply the storage capacity printed on a drive by this number and infer the cost of deleting its contents. You would need to know what physical operation the deletion actually performs, what uncertainty is removed, and which other processes consume energy. Marking space as available and resetting every physical bit are different procedures.
Temperature appears in the bound, but lowering temperature does not make the rest of the apparatus free. A system-level comparison would have to include the resources used to maintain that temperature and the performance required of the device.
Ask which quantity the claim actually names
A statement about information may concern a probability distribution, a physical memory, a message’s accuracy, or a decision’s usefulness. Moving between them requires an explanation.
If someone says a database contains more information, ask whether they mean more bytes, more independent distinctions, better evidence, or more useful answers. If they claim to reduce entropy, ask what distribution or physical process is being measured. If they claim to create value, ask whose decision changes and what the change accomplishes.
These questions do not drain interest from the physical connection. They make its achievement clearer. An abstract distinction can be stored in a bead’s position, and discarding that distinction has consequences for the bead and its surroundings. That is already a striking relation between mathematics and matter.
The key in the drawer gives the relation another boundary. Three answers can locate it. Thermodynamics can constrain the cost of resetting their record. What the key opens—and what opening it is worth—requires a different account.
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