Part 4: The Impossible Coin
Table of Contents
(Updated: July 2026)
This is Part 4 of Quantum Computing for Cybersecurity Professionals, an 11-part series that builds quantum computing from the ground up for security and IT professionals who know classical crypto but have no physics background. No lies-to-children, no hype, every claim checkable with arithmetic. Read the full series online or download the free ebook.
A courier delivers a sealed device to your lab. One input port, one output port, no manual. The job is the one your profession trains for: characterize the black box.
You feed it a definite 0 and measure the output: 0 about half the time, 1 the other half, and over ten thousand trials the tally is a textbook fair coin. You feed it a definite 1. Fair coin again. By every test Part 2 taught you, the verdict writes itself: the device is a randomizer, apparently the recordless kind, and Belief 3 has already told you everything it will ever do.
So you chain two of them in series and feed the pair a definite 0. Out comes 0. Again: 0. Ten thousand runs, ten thousand zeros. Feed the chain a 1 and it returns 1 with the same monotony. One copy of this device manufactures perfect randomness. Two copies in a row manufacture perfect certainty, and nowhere in the glass and silicon is any record of what passed through.
Part 2 proved that impossible. Not unlikely. Impossible, by arithmetic you checked yourself. This article is about the cleanest quantum interference experiment ever devised, and it does four things: replays the impossibility proof, shows the amplitude arithmetic that predicts the bench perfectly, closes the one loophole your training will reach for, and then collects on the receipt. To put it plainly: if you verify one thing in this whole series by hand, make it the box below.
Three doors out
The classical argument, compressed from Part 2’s arithmetic box: an operation using fresh, unremembered randomness acts on your probability pair by averaging. If it turns each definite input into a fair coin, its output carries no dependence on its input, so a second application receives a coin, ignores it, and emits a coin. Certainty in, coin out, coin forever. The bench says certainty comes back. Something in the setup has to give, and there are exactly three candidates.
Door one: fraud. The demonstration is rigged, the device is a stage prop. Except this is among the most repeated experiments in physics; it runs on optics benches in undergraduate labs, and in software it is the first instruction of nearly every quantum program ever written. Names, dates, and a Nobel Prize are waiting in a box below. Door one is closed.
Behind door two, the respectable candidate: a hidden record. Part 2 allowed exactly one escape from irreversibility, and the one-time pad taught it: un-mixing is possible where the randomness was remembered. Maybe the device keeps a pad. This door deserves a real test rather than a wave of the hand, and it gets one, two sections down.
And door three: the model. Perhaps a probability pair was never the right description of what travels between the two stages. That door leads somewhere.
Quantum interference, by the numbers
Part 3 ended on a cliffhanger built for this moment: two states, $$(1/\sqrt{2}, 1/\sqrt{2})$$ and $$(1/\sqrt{2}, -1/\sqrt{2})$$, identical under measurement, different by one invisible sign. Here is the resolution. Those two states are this device’s two outputs. In amplitude language, the device maps a definite 0, the state $$(1, 0)$$, to $$(1/\sqrt{2}, 1/\sqrt{2})$$, and a definite 1, the state $$(0, 1)$$, to $$(1/\sqrt{2}, -1/\sqrt{2})$$. Measure either output once and the Born rule squares away the sign: fair coin, both times, exactly as your first session found. The device manufactures the invisible sign.
The second stage is where the sign stops being invisible. The device acts on amplitudes by the same linearity Part 2 established for probabilities: feed it a general state and it processes each component, then adds. Except these components carry signs, and signed contributions meeting at the same output can do the one thing probabilities never could. Cancel. Run the second stage on $$(1/\sqrt{2}, 1/\sqrt{2})$$ and the two routes into the “1” output arrive carrying $$+1/2$$ and $$-1/2$$: they annihilate, and the output-1 amplitude is exactly zero. The two routes into “0” arrive carrying $$+1/2$$ and $$+1/2$$: they reinforce, to exactly one. The state after the second stage is $$(1, 0)$$. Certainty, reconstituted.
This collision has a name, quantum interference, and you have just watched the only mechanism by which a quantum computer ever beats a classical one. Wrong answers cancel; right answers reinforce. That was Part 1‘s thirty-second preview of the entire field, and it just happened in front of you in four lines.
THE ARITHMETIC: The four lines that matter most.
The device’s rule on definite inputs: $$(1,0) \rightarrow (1/\sqrt{2},\ 1/\sqrt{2})$$ and $$(0,1) \rightarrow (1/\sqrt{2},\ -1/\sqrt{2})$$. On a general state it acts by linearity, exactly as in Part 2’s box: process each component, add the results. So the new amplitudes are:$a_0′ = \frac{a_0 + a_1}{\sqrt{2}} \qquad a_1′ = \frac{a_0 – a_1}{\sqrt{2}}$
Start from a definite 0, state $$(1, 0)$$. First pass: $$a_0′ = 1/\sqrt{2}$$, $$a_1′ = 1/\sqrt{2}$$. Second pass: $$a_0” = (1/\sqrt{2} + 1/\sqrt{2})/\sqrt{2} = 1$$ and $$a_1” = (1/\sqrt{2} – 1/\sqrt{2})/\sqrt{2} = 0$$. Final state $$(1, 0)$$: the output is 0 with certainty. Start from $$(0, 1)$$ instead and the signs flow through to give $$(0, 1)$$ back. Check it; it is six additions.
Now the peek. Measure between the stages and the Born rule fires: you get 0 or 1, evens, and the state becomes the matching definite pair. The second stage then does what it does to every definite input: produces a fair coin. Peeked chain: 50/50. Unpeeked chain: certainty. Your own arithmetic predicts that looking in the middle changes the answer at the end.
The record loophole, tested
Door two remains, and it is the respectable one. Part 2’s one-time-pad box established that apparent un-mixing is possible when the randomness is remembered somewhere, so the professional hypothesis is that the device pads: it secretly records what it did on pass one and undoes it on pass two. The hypothesis is testable, because a record is a physical thing, and you can add one of your own.
Install a detector between the two stages, any mechanism at all that registers which value went past. The bench result is unambiguous, and it has been since the interference experiments of the 1980s: the certainty vanishes. The chain’s output becomes a fair coin, precisely the peeked-chain arithmetic from the box. And the detail that should stop you mid-coffee is that nobody needs to read the detector. Its record can sit unexamined forever; the mere existence of the information, anywhere, in anything, produces the 50/50 outcome. That is [Part 3](note: linked above)’s definition of measurement doing live ammunition work: an interaction that leaves a record is a measurement, minds optional.
So run the logic. If the device relied on an internal record to un-mix, an extra record should not hurt; a pad plus a bystander’s photocopy of the pad still decrypts. Instead, any record kills the effect. The un-mixing exists only in the total absence of records, which is the exact configuration Part 2 proved cannot un-mix. Classically, the record was your only road back. Here, a record is what closes the road. The reversal runs on silence.
MYTH AUTOPSY: “It’s really 0 or 1, we just don’t know which.”
The most respectable myth in this series was granted a stay of execution in Part 3, because single measurements could not refute it. The stay expires here. Give the story every courtesy and let it speak precisely, about the moment between the two stages, in a chain now certified record-free.
Premise: between the stages, the qubit is secretly definite, 0 or 1, even odds, and the pair $$(1/\sqrt{2}, 1/\sqrt{2})$$ merely describes your ignorance. Established behavior: this device turns each definite input into a fair coin; that was your first characterization session, ten thousand trials a side. Prediction: the second stage receives a definite value and therefore outputs a fair coin. Observation: the second stage outputs the original input, every time.
The premise did the predicting, and the premise is dead. Between the stages there is no secret value; there is the state $$(1/\sqrt{2}, 1/\sqrt{2})$$ itself, one definite mathematical object whose halves met signed halves and cancelled. Belief 1’s picture, a value plus ignorance, cannot produce a minus sign, and the minus sign is measurably there: it is the difference between the certainty you observe and the coin the story predicts.
Say instead: a superposition has no underlying value, and the proof is operational: treating it as a hidden value predicts the wrong statistics for an experiment you can run.
FOR THE RECORD: This is a real bench, and also line one of every program.
On an optics table the device is a half-silvered mirror, and two of them bracket a Mach-Zehnder interferometer; the “coin” is which path the photon takes. Grangier, Roger, and Aspect ran the single-photon version in 1986, one photon at a time through the apparatus, and the interference held with near-perfect visibility. Aspect later shared the 2022 Nobel Prize for the experimental program, with entangled photons, that closed out the hidden-variable question at large; that larger story, Bell’s theorem included, arrives with entanglement in Part 6.
For completeness: hidden-variable theories that survive this experiment do exist. De Broglie and Bohm built one by letting the amplitudes steer hidden values nonlocally, which means the amplitudes stay load-bearing either way, and nothing in this series changes under that reading. The casualty here is the specific, common-sense version your Part 2 receipt encoded.
In software, the device is the Hadamard gate, typically the first instruction of any quantum program you will ever read. When Part 8 breaks RSA, the opening move is this article, applied to every qubit in the register.
WHERE THIS BREAKS: The coin has no faces.
The impossible coin is this series’ keystone analogy, so its limits get stated plainly. Between measurements there is nothing two-sided in flight; the “sides” are amplitudes, and heads-or-tails imagery smuggles Belief 1 back in through the fire exit. And the clean numbers above are the noiseless limit. A real bench leaks, every leak is a partial record, and partial records buy partial coins: real devices return the input most of the time rather than all of it. The distance between “most” and “all,” multiplied by a few million qubits, is Part 10‘s entire subject.
Stamping the receipt
Part 2 closed with four beliefs on the record and a promise that exactly one would leave this room unchanged. The audit:
Belief 1, dead for qubits. There is no definite value under the pair; the hidden-value premise predicts a coin where the bench delivers certainty. (Your classical bits keep the belief. Scope matters.)
Belief 2, dead. Contributions can subtract. The update rule keeps its shape, linearity and all, except it now runs on amplitudes, and $$+1/2$$ met $$-1/2$$ in front of witnesses.
Belief 3, dead on its own terms. A recordless process un-mixed, which Part 2 proved no probability process can do, and the escape clause inverted: records no longer enable the reversal, they are the one thing that prevents it.
Belief 4, unchanged today. This experiment never touched it. Part 3 already bent its reading clause, and Part 5 takes the copying clause. It should not feel safe.
What stands in the wreckage is the description that predicted everything: the amplitude pair of Part 3, now promoted from bookkeeping to the thing the machine actually computes with. The state between the stages is $$(1/\sqrt{2}, 1/\sqrt{2})$$, definite, valueless, and cancellable, and every quantum algorithm you will ever hear about is a scheme for arranging such states so that the wrong answers cancel. The receipt is stamped.
ATTACKER’S-EYE VIEW: One trick, at scale.
Everything on Part 1’s casualty list is this bench trick, industrialized. Shor’s algorithm in Part 8 arranges a computation over an astronomical number of routes so that amplitudes flowing toward wrong answers meet with opposing signs and annihilate, exactly like the $$+1/2$$ and $$-1/2$$ you just watched, while routes agreeing on the hidden structure of the key reinforce. The general choreography, and why it works only for problems with structure, is Part 7‘s subject.
The record test cuts the other way too, and defenders of future systems should file it. An interference computation dies the moment any record of its intermediate values exists anywhere, so one planted probe, one leaky coupling, one over-curious diagnostic is a denial-of-service attack on a quantum computer; measuring the machine is breaking the machine. The environment mounts this attack for free, continuously, with no adversary required, and holding it off is the engineering mountain of Part 10.
One more consequence worth sixty seconds of paranoia: the sign channel from Part 3 is now a working covert channel. Information parked in relative signs is invisible to any inspector who only measures, and extracting it takes an interferometer aimed by someone who knows the encoding. Storage the auditor cannot audit, readable only by the party holding the layout: file that under both threats and tools.
What to remember
A single device turns each definite bit into a fair coin, yet two in series return the original input with certainty and no record anywhere, which Part 2’s arithmetic proves impossible for any classical random process. The amplitude description predicts it in four additions: the device manufactures Part 3’s invisible sign, and the second pass makes signed halves collide, cancelling the wrong output and reinforcing the right one. Any record of the intermediate value, read or unread, destroys the effect, so the reversal exists precisely where classical probability forbids it. The hidden-value story, granted every courtesy, predicts a coin where the bench delivers certainty; between the stages there is no secret value, only a superposition. And this cancellation is the sole engine of quantum computational advantage, the same move that eventually breaks RSA.
Next, in Part 5, “Vast State, Narrow Door”: we scale from one qubit to many, meet the 2^n amplitudes that make quantum states vast, and the n-bit measurement that keeps them almost unreadable. The “tries every answer at once” myth finally gets its autopsy, and Belief 4 loses its last clause: quantum information cannot be copied at all.
