The Universe Has No Delete Key

Here's an experiment you can run with a toy.

Take an Etch A Sketch: the gray screen, the two white knobs, the single line that can never lift off the glass. Spend ten minutes drawing a house. Give it a chimney. Give it a door on the left and a little window with a cross in it. Be proud of the window.

Now turn it upside down and shake it for two seconds.

The house is gone. The shake spread the gray powder back over the glass, so the screen is blank again, as if no house had ever been drawn on it.

So where did the house go?

I don't mean the lines. I mean the information: the fact that there was a house on that screen, with a door on the left and a chimney on the right. That fact existed three seconds ago. Where is it now?

Same question for the trash on your computer. Same question for the factory reset on your old phone. Something was there, you pressed a button, and now it isn't.

The short answer

Here's the whole answer in five steps. The rest of this post takes them one at a time and shows how we know each one.

1. Erasing is a merge. Many different patterns go in, and one blank comes out. Eight different patterns of three coins, all reset to the same one: that's what erasing is.
2. Nothing is lost. It moves. Reality keeps a permanent record and never merges two different histories into one. Two different pasts never lead to the same present, so the difference between the starting patterns can't vanish. It leaves the memory (the coins, the screen, the drive) and lands in the room around it as heat. There is no silent erasure.
3. The price is fixed. Moving one bit (one yes-or-no fact, like one coin) into the room costs at least kT ln 2 of heat. Here T is the temperature and k is Boltzmann's constant. At room temperature that's 2.87 × 10⁻²¹ joules per bit. A lab has measured it.
4. The books close exactly. For a small memory and a small room, the heat that lands in the room is exactly three things added up. Count it in bits: one bit's worth of heat is the price from step 3. The three things are the bits the memory lost, the correlation the wipe builds between memory and room (how much looking at one tells you about the other), and the imbalance (how far the wipe pushes the room away from the odds it settles into). Every bit that leaves the memory turns up on the other side.
5. A copy helps. Pushing harder doesn't. Keep a copy and the wipe costs less. No amount of force gives the bits a place to go, and every wipe fills part of the room.

Why this matters: Part 1 of this series said reality has no eraser. A delete is what an eraser does, so a delete is where that claim gets tested. The test comes with a number that a lab has measured. Every bit your computer has ever truly erased warmed the room by at least a known amount. That amount is the least heat any computer must give off when it erases a bit. It's called Landauer's limit.

Previously, in Part 1

This post stands on one idea from Part 1, so here it is in a minute.¹

Reality is a permanent record. Every event is written down, and nothing is ever erased. Reality never merges two different histories into one. We call the record the ledger, like a bookkeeper's: entries go in one after another, and they stay. Part 1 said the ledger has no eraser and no delete key.

Here's the reason, and it's known physics. The basic laws of physics never destroy information. In quantum mechanics this is unitarity: the present fixes the past exactly, and running the laws backward recovers every earlier state. That's why physicists treat the black hole information problem as a real problem: losing information would break the laws. Hawking's 1976 calculation said a black hole destroys the information that falls into it. That clashed with unitarity, which is why it became a famous problem. Most physicists now expect the information to come back out in the radiation, and the open question is how.²

For this post, it comes down to one rule: two different pasts never lead to the same present. We use exactly that rule below.

Step 1: Erasing is a merge

To erase a memory is to take something that can hold many different things and make it hold one thing. Many different starting patterns go in. One ending pattern comes out. That's a merge, and every erase is one.

I'll be exact about the word. Here, erasing (I'll also say wiping) means setting a memory back to blank, whatever was in it, with no copy of the old contents left anywhere else.

Take three coins on a table. Three coins make eight possible patterns (H is heads, T is tails): HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. The pattern showing is what the coins hold, so the three coins are a memory that can hold eight different things.

Now wipe the memory. Set all three coins to heads, whatever they said before.

Eight patterns go in. One comes out. HTT and THH, which were different, are now the same. The wipe merged them.

Now count what the wipe did. A bit is one yes-or-no fact, like one coin. You don't know which of the eight patterns the coins show, and each pattern turns up equally often. How many yes-or-no questions would you need to ask to find out? "Is the first coin heads?" "Is the second?" "Is the third?" Three. So before the wipe, the coins hold 3 bits of uncertainty: three yes-or-no facts you don't know.

After the wipe, you need no questions. You know it says HHH. Three bits of uncertainty before, none after. (Physicists call this uncertainty entropy. I'll stick with uncertainty.)

That's what every erase looks like. A memory that was free to say many different things is forced to say one. That goes for the three coins, the powder in an Etch A Sketch and the bits on a hard drive. Being forced to say one thing is what makes it an erase.

Zoom out. Step 1 is done. Erasing is a merge: many starting patterns, one blank. The three coins start with 3 bits of uncertainty and end with none. Next, and why: a merge is a problem for a record that never erases. If the ledger can't merge two different histories into one, those 3 bits have to be somewhere. Where?

Step 2: Nothing is lost. It moves.

The ledger never merges two different histories into one. So the difference between the eight patterns can't just vanish. It leaves the coins and lands in the room around them, as heat.

Here's how we know. Look at the whole world at once: the coins, the table, the air, the floor. Everything that isn't the coins, I'll call the room.

Run two versions of the world. In version one, the coins start as HTT. In version two, they start as THH. Wipe the coins in both. Afterward the coins read HHH in both versions.

But the two worlds can't have become the same world. The ledger has no eraser, and two different histories don't turn into one. Look at the coins: identical. So the difference has to be somewhere else. Only one place is left: the room.

What does a difference in the room look like? The air is a crowd of molecules, always moving. In version one the crowd ends up moving in one pattern. In version two it ends up moving in a different pattern.

That crowd is where heat lives: energy spread across a huge number of molecules at random. A crowd that size can hold any difference you hand it. Averaged over all eight patterns the coins can start with, the room ends up warmer.

Here's the same mechanism in bookkeeping terms: every erased bit gets posted to the room's side of the ledger. A bookkeeper's ledger lives by one rule: whatever leaves one entry shows up in another. When a bit leaves the memory, it shows up in the room. An erase that cost the room nothing would be an event with no entry on the room's side: a merge nobody wrote down. The ledger never merges two histories, so that can't happen. There is no silent erasure.

Here's the delete button, drawn as what it really is.

So where did the house go? Into the room. It's in the motion of the powder and the air around the toy. That motion is scrambled so finely that in practice nobody can read a house back out of it. But scrambled isn't erased. Every difference between the old screen and the new one is in there, because the same rule covers the toy: a world where a house was drawn and shaken away and a world where none was drawn are two different pasts, so they can't lead to the same present.

(The shake also heats the toy through plain friction, far more than the heat the bits themselves cost. We'll keep the two apart.)

What about the trash? Once you empty it, isn't the file gone?

Your computer doesn't delete either, at least not at first. Emptying the trash usually just marks the file's space as free to reuse. The bits stay on the drive until the drive writes over that space or clears it. When it does, that's a true erase: a merge, and the room gets warmer.

Zoom out. Step 2 is done. The difference between the starting patterns can't vanish, so it moves: out of the memory and into the room, as heat. Nothing is erased silently. Next, and why: a rule that says "the room gets heat" isn't much use without an amount. The amount has a fixed minimum.

Step 3: The price of one bit

Moving one bit of uncertainty into a room at temperature T costs at least kT ln 2 of heat, on average. That's the floor: the least heat any erase can pay. At room temperature, about 300 kelvin (27 °C), the floor is 2.87 × 10⁻²¹ joules per bit. That's a decimal point, twenty zeros, then 287.

Three symbols, and you have all of it. T is the temperature in kelvin, a scale that starts at the coldest temperature there is. k is Boltzmann's constant, 1.38 × 10⁻²³ joules per kelvin: it converts temperature to energy. And ln 2 = 0.693 is one bit, the uncertainty in one fair coin, counted with the natural logarithm that these formulas use. Multiply: 1.38 × 10⁻²³ × 300 × 0.693 = 2.87 × 10⁻²¹ joules. (A joule is about the energy it takes to lift an apple one meter.)

Here's how we know. Three ways.

It follows from the books. Step 4 writes out the whole heat bill (all the heat the room receives), line by line. The first line is the uncertainty the memory lost, and the other two lines can never be negative. So the heat can't be less than that first line: kT ln 2 for every bit.

It was worked out in 1961. Rolf Landauer, a physicist at IBM, found the price. It's called Landauer's limit.³

It's been measured. In 2012 a team in Lyon held a single glass bead in a laser trap with two dips. Bead in the left dip meant 0. Bead in the right dip meant 1. To wipe the bit, they pushed the bead into the same dip, whichever dip it started in. They worked out the heat from the bead's motion. Wipe in a hurry, and the average heat lands above the limit. Wipe slowly, and it settles onto kT ln 2.⁴

One piece is still being worked out: getting the exchange rate between the ledger's entries and joules from the ledger's own rules instead of from measurement. That exchange rate is how many joules one entry in the ledger is worth.

Here's what the floor looks like at 300 kelvin.

(A terabyte is a trillion bytes, and a byte is eight bits. In random data every bit is a fair coin, so a wipe removes the uncertainty in all 8 trillion bits.)

The floor is tiny. One joule pays it for wiping more than 43 million of those drives. But it's never zero, and it's the same per bit for every type of memory, from a glass bead to a chip. A colder room is cheaper. At 3 kelvin, a hundredth of room temperature, the floor per bit falls a hundredfold.

Doesn't a laptop get far hotter than that when it deletes things?

It does. The floor is the part of the heat that every erase must pay, however perfectly the machine is built. A real chip also wastes energy as ordinary heat in its wires and switches, and an Etch A Sketch shake wastes plenty as plain friction. All of that sits far above the floor, and all of it is separate from the bill for the bits.

Zoom out. Step 3 is done. Each bit of uncertainty moved into a room at temperature T costs at least kT ln 2: 2.87 × 10⁻²¹ joules at 300 kelvin. It's measured, it's tiny, and it's never zero. Next, and why: "at least" leaves a gap. What's in the heat above the floor? For a small memory and a small room, we can write the whole bill, line by line, and add it up.

Step 4: The books close exactly

For a small memory and a small room, the books close exactly. (Small means finite here: each has a limited number of states. Finite matters for the floor: with a finite room it is never quite reached, as the end of this step shows.) Count the heat that lands in the room in units of kT ln 2, the price of one bit, so the total comes out in bits. Then:

heat into the room = bits the memory lost + bits of correlation + bits of imbalance

Every bit that leaves the memory turns up on the other side.

It's an exact equation, proved for any small quantum memory and any small room, however the wipe is done.⁵ An ordinary coin is a special case. The equation holds under two conditions. First, the room starts out settled at its temperature (its odds are the ones it settles into at that temperature) and unlinked from the memory (looking at one tells you nothing about the other). Second, counting the memory and the room together, the wipe keeps every distinction: no two different histories merge. That's the ledger's rule anyway.

The right side has three lines, and each has a plain meaning:

• Bits the memory lost. How much uncertainty left the memory, in bits. The three-coin wipe: 3.
• Bits of correlation. After the wipe, are the memory and the room linked, so that looking at one tells you something about the other? A link costs heat on top of the wipe itself.
• Bits of imbalance. A room at a given temperature has settled odds: its atoms are calm or excited with fixed odds set by the temperature. A wipe can push the room away from those odds. The further it pushes, the more it costs. Relative entropy measures the push: how far, in bits, one set of odds is from another. It's zero only when they match.

Here's a case small enough to do on paper. The memory is one coin, heads or tails with equal odds: 1 bit of uncertainty. The room is one atom, a tiny room to keep the arithmetic on one line. It starts out settled, with odds at this temperature of 3 in 4 calm and 1 in 4 excited, and unlinked from the coin. By Boltzmann's odds rule in thermal physics, at temperature T every extra kT ln 3 of energy makes a state 3 times less likely. So odds of 3 to 1, calm to excited, mean the excited state holds kT ln 3 more energy than the calm one.

The wipe is a swap: heads and calm trade places, and so do tails and excited. So the coin takes on the atom's odds, and the atom takes on the coin's. This wipe is partial: the coin ends up 3 in 4 heads, not certain. The equation covers partial wipes and full ones alike. (Something has to drive the swap, a hand or a battery, and that something supplies the energy. We're counting where the heat lands.)

The two sides of the equation match: the heat that lands in the room equals the three lines on the right added up. Here are the numbers.

• Heat into the room: 0.3962 bits. The atom is now excited half the time instead of a quarter: an extra quarter of the time. An excited atom holds kT ln 3 more energy than a calm one, so the extra energy sitting in the room, the heat, averages ¼ × kT ln 3 = 0.2747 kT. Divide by kT ln 2 to count it in bits.
• Memory lost: 0.1887 bits. The coin comes up heads 3 times in 4, so you can guess it better than before. Its uncertainty fell from 1 bit to 0.811 bits.
• Correlation: 0 bits. The coin and the atom traded places, but looking at one still tells you nothing about the other.
• Imbalance: 0.2075 bits. The atom's new odds are (½, ½), and its settled odds are (¾, ¼). Their relative entropy is 0.2075 bits.

Add the three lines: 0.1887 + 0 + 0.2075 = 0.3962. They match the heat to every digit shown. The books close.

If you want to check the arithmetic: a coin with odds p and 1 - p has uncertainty p · log₂(1/p) + (1 - p) · log₂(1/(1 - p)) bits. That's 1 for p = ½ and 0.811 for p = ¾. (The log₂ of a number asks how many times you'd double 1 to reach it: log₂ 8 = 3.) The imbalance is ½ · log₂(½ ÷ ¾) + ½ · log₂(½ ÷ ¼) = 0.2075.

Here's the same equation drawn as a scale: the heat on the left, the three lines on the right.

Now read the equation as a rule. The last two lines are never negative. So the heat can never fall below the first line: the uncertainty the memory lost, times kT ln 2. That's the floor from Step 3. The other two lines are the price of mess. A wipe that links the memory to the room, or knocks the room out of balance, pays more than the floor. In our example the floor was 0.1887 bits and the real bill was 0.3962, about twice as much.

Here's why small matters. With a finite room, the floor can't be touched. If the memory really loses uncertainty, the last two lines can't both be zero. The floor is a limit you approach and never reach.

Zoom out. Step 4 is done. The books close exactly: heat into the room equals the uncertainty the memory lost, plus the correlation built, plus the imbalance. Every bit that leaves the memory turns up on the other side. Next, and why: with the books open, we can ask what moves the bill. One trick works, one doesn't, and one rule binds every wipe.

Step 5: What helps, what doesn't

A copy helps. Pushing harder doesn't. And every wipe fills part of the room.

A copy helps. Wipe a notebook when no copy exists anywhere, and the room is billed for every bit. Wipe it while a perfect copy survives, and the floor drops to zero. With the copy in hand, you can always undo the wipe, so nothing merged. The bits didn't go to the room. They stayed in the copy. The bill comes due when the last copy goes. It's the no-silent-erasure rule from the other side: information that still exists somewhere hasn't been erased.

Pushing harder doesn't. Say you throw everything you have at the wipe: a giant lever, a huge battery. It doesn't help, because force isn't what's missing. A place for the difference to go is. Seal the three coins in a box with nothing but a lever, and all the lever can do is shuffle them. HTT becomes THT, and every pattern is still different from every other. To merge eight patterns into one, the difference needs a room.

The equation from Step 4 agrees. The lever and the battery are a work source, the part that supplies the work. If it starts out as mixed up as it can be (every one of its states equally likely), it already holds all the uncertainty it can, so it can't absorb the bits. All it adds to the bill is lines that are zero or more. None of them ever takes anything off. The bits need a receiver.

Every wipe fills part of the room. A wipe that keeps the total energy has to put each level somewhere distinct in the room, so every wipe uses up fresh room. A slot is a place that holds one level, and every slot a wipe fills is no longer empty. (In the drawing below, each slot is a column, filled to its level.) It's the same rule as before, counted in places instead of bits: two different levels can't share a slot, because that would merge them.

Here's the count, for a wipe that keeps the total energy. Say the memory has four slots, each filled to its own level. The wipe pours the four levels into four empty slots in the room, intact. It needs four, not three: four different levels can't be squeezed into three slots and told apart afterward. Fill the memory again and wipe it again, and you need four more. K wipes of an N-slot memory need at least K × N empty slots.

So deleting is moving, and moving takes space. Nothing gets deleted, which means everything you've ever wiped is still in the room. That's Part 1's rule, that nothing is ever erased, with a price tag on it. Part 2 follows that rule to its conclusion about time: yesterday still exists.⁶ The past still exists.

The whole trip, in one breath

Erasing is a merge: many starting patterns, one blank. The ledger never merges two different histories, so the difference leaves the memory and lands in the room as heat. There is no silent erasure. The price is at least kT ln 2 per bit, 2.87 × 10⁻²¹ joules at room temperature, and a lab in Lyon has measured it. For a small memory and a small room, the books close exactly. The heat into the room equals the bits the memory lost, plus the correlation built, plus how far the room was knocked out of balance. A copy makes the wipe cheaper. Pushing harder doesn't, because a wipe needs the room to move into, and every wipe fills more of it. The universe has no delete key. It has a move key.

So, where did the house go? Into the room. The screen forgot it. The universe didn't.