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 screen is blank gray 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 things go in, and one blank comes out. Eight different patterns of three coins, all reset to the same pattern: that's what erasing is.
2. Nothing is lost. It moves. Reality keeps a permanent record and never merges two different histories into one, so the difference can't vanish. It leaves the memory and lands in the room around it as heat. There is no silent erasure.
3. The price is fixed. Moving one bit into the room costs at least kT ln 2 of heat, where 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, counted in bits, equals the bits the memory lost, plus the correlation the wipe builds between memory and room, plus how far the wipe pushes the room out of balance. 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 uses up room.

Why this matters: Part 1 said reality has no eraser. A delete is what an eraser does, so a delete is the test, and this 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, which is why it has a name: 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. We call that record the ledger, like a bookkeeper's: entries go in one after another, and they stay. If the ledger had a personality, it would be unflappable, extremely thorough, and armed with one answer to everything that has ever happened: Noted. The Ledger's face is a drawing. What it stands for isn't: the actual, permanent record of everything that has ever happened.

Part 1 listed the things the ledger doesn't have: an eraser, white-out, a "just kidding." And a delete key. That last one is today's subject.

Two regulars are back. Gary is a drummer who flips a coin before every beat to decide whether to play. (Gary is not getting the gig.) Today he's useful because he owns coins, and we need coins. Yeah-But is the voice in the back of your head that has never once said wow.

Here's today's route. The Ledger will keep the map, and I'll stop after each step to zoom out.

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 starting states go in. One ending state comes out. That's a merge, and every erase is one.

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

Take Gary. He keeps three coins on his drum. Heads means he plays that beat, and tails means he rests. Three coins make eight possible patterns: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Gary's setlist is whichever pattern is showing.

Now Gary wipes his setlist. He sets all three coins to heads, whatever they said before. (Heads means play, so a wiped Gary plays every beat. Honestly, it's an improvement.)

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. How many yes-or-no questions would you need to ask to find out which of the eight patterns Gary's coins show, assuming each pattern turns up equally often? "Is the first coin heads?" "Is the second?" "Is the third?" Three. So before the wipe, Gary's 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. Gary's coins, the powder in an Etch A Sketch, the bits on a hard drive: a memory that was free to say many different things is forced to say one. The forcing is what makes it an erase.

Zoom out. Step 1 is done. Erasing is a merge: many starting patterns, one blank. Gary's 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 Gary's eight patterns can't just vanish. It leaves the coins and lands in the room around them, as heat. Every erased bit gets posted to the room's side of the ledger. There is no silent erasure.

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

Run two versions of the world. In version one, Gary's 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 has room for any difference you hand it. Average over every pattern the coins can start with, and the room ends up warmer.

In the Ledger's language, the room's side gets an entry. 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 doesn't allow that. 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, 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. (The shake also heats the toy through plain friction, far more than the heat the bits themselves cost. We'll keep the two apart.)

Yeah-But clears his throat. "Yeah, but I emptied the trash. It's gone."

The Ledger, unmoved: Noted.

Here's the good news: your computer doesn't delete either. 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. And 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 turns out to be exact.

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 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: the exchange rate between temperature and energy. And ln 2 = 0.693 is the uncertainty in one fair coin, in the units physicists use for this. 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, 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.²

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, and 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 2.³

One piece is still being worked out: getting k, and the exchange rate between the ledger's entries and joules, from the ledger's own rules instead of from measurement.

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

(A terabyte is a trillion bytes, and a byte is eight bits.)

It's tiny. One joule pays the floor for wiping more than 43 million of those drives. But it's never zero, and it's the same 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.

Yeah-But again: "Yeah, but my laptop gets hot when it deletes things. Way hotter than that."

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 bill closes exactly. Count the heat that lands in the room in units of kT ln 2, the price of one bit, so the count 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 identity, proved for any small quantum memory and any small room, however the wipe is done.⁴ An ordinary coin is a special case. It holds when the room starts out settled at its temperature and unlinked from the memory, and the wipe keeps every distinction, which is the ledger's rule anyway. Physicists call a room like that a heat bath.

Three lines, three plain meanings:

• Bits the memory lost. How much uncertainty the memory shed, in bits. Gary's 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.

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. At this temperature its settled odds are 3 in 4 calm and 1 in 4 excited.

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 identity 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.)

Now the books. After the swap, start with the left side.

The heat. The atom is now excited half the time instead of a quarter. That's extra energy sitting in the room: heat. At this temperature an excited atom holds kT ln 3 more energy than a calm one, which is what makes the settled odds 3 to 1. So the heat is ¼ × kT ln 3 = 0.2747 kT. Divide by kT ln 2 to count it in bits: 0.3962 bits.

Now the right side, line by line.

• Memory lost. 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. 0.1887 bits.
• Correlation. None. The coin and the atom traded places, but looking at one still tells you nothing about the other. 0.
• Imbalance. The atom's new odds (½, ½) against its settled odds (¾, ¼). Physicists measure that gap with a formula called relative entropy. Here it comes to 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 identity drawn as a scale: the heat on the left, the three lines on the right.

Now read the identity 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.

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 uses up 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 Gary throws everything he has 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 his 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 exact books agree. A work source that starts out as mixed up as it can be can't hold any more uncertainty, so it can't absorb the bits. All it adds to the bill is lines that are zero or more, and none of them ever takes anything off. The bits need a receiver.

Every wipe uses up room. A wipe moves the contents into the room, and every slot it fills is a slot that's no longer empty. Here's the exact rule 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. 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.⁵

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, and 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: 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 a room to move into, and every wipe takes up 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.

(The Ledger would like you to know it noted every file you've ever deleted, and exactly where each one went. It isn't going to tell you. It just wanted you to know.)