Abstract
A quantum state cannot be copied (the no-cloning theorem). Yet quantum error correction works. To correct one must know the error; to know it one must measure; and measuring destroys the state, so it would seem. This paper asks why these two are compatible: the answer is that what is measured is not the state but the error. No new mathematical theorem and no new law is claimed.
First, we count the bits the syndrome tells. Shor’s $9$-qubit code tells $\mathbf{8}$ bits about the error and $\mathbf{0}$ bits about the logical state (Section 2).
Second, the $5$-qubit code saturates exactly. Against $16$ syndromes there are $\mathbf{16}$ errors to correct: not one to spare (Section 3).
Third, the same reasoning turns eavesdropping into a number. Under BB84 intercept-resend, the error rate is exactly $\mathbf{25\%}$ (Section 4).
Fourth, $25\%$ is $2.27$ times the security limit. The one-way post-processing limit is $\mathbf{11.0028\%}$, so it is certain to be detected (Section 5).
Fifth, checking $73$ bits drops the miss probability below $10^{-9}$. That is $0.75^{73}$ (Section 5).
Sixth, the separator is that security does not rest on computational cost. However fast computers become, $25\%$ stays $25\%$ (Section 6).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed: the no-cloning theorem, Shor’s $9$-qubit code, the $5$-qubit code, BB84, the $25\%$ of intercept-resend, and the $11\%$ of Shor--Preskill are all standard. We do not construct codes: all we count is the number of syndromes and the number of errors to be corrected. We do not prove the no-cloning theorem: we merely quote it. We give no security proof: the $11.0028\%$ of Shor--Preskill is quoted as a number only, and we do not enter the proof. We do not discuss real quantum key distribution: loss, dark counts, multi-photon components, and device attacks are not treated at all. The $25\%$ of this paper is a value for ideal intercept-resend only. We do not claim the best attack: intercept-resend is not the best attack. A cleverer attack obtains the same information at a lower error rate. $25\%$ is a number for “how conspicuous a naive attack is”. We do not treat finite-key effects: the $0.75^{73}$ of Section 5 is the miss probability under full interception, not a complete assessment of statistical security. We do not discuss quantum computation: neither Shor’s factoring nor the threshold theorem is treated. Relation to earlier papers: Paper 163 separated “a good code exists” from “here is a good code”. This paper stands on the construction side and counts the syndromes of specific codes. Paper 252 counted four quantities called “information”: the $8$ and the $0$ counted here are the bits one and the same measurement returns about two different objects, a separation of objects, not of kinds of quantity. Paper 188 treated cases exactly determined by fewer measurements than unknowns. This paper is the converse: measuring exactly what is needed and nothing else. Paper 179 showed that “cannot be constructed” has distinct roots: no-cloning is one such “cannot construct,” and this paper uses as a premise that its root is linearity. Paper 68 read the qubit as a Hopf fibration. This paper counts the same qubit as a number of bits of information. What is added is counting the bits the syndrome tells, separated as “$8$ about the error, $0$ about the state”, confirming that the $5$-qubit code saturates exactly at $16=16$, setting beside it the fact that the same reasoning yields the $25\%$ of BB84, and computing that $25\%$ is $2.2722$ times the security limit.
1. Notation, and Two Things That Should Not Coexist
The first: the no-cloning theorem: no operation copies an unknown quantum state.
The second: quantum error correction: an encoded state can be restored after an error enters.
Why they look incompatible: To correct, one must know which error entered. To know, one must measure. Measuring a quantum state destroys it.
The answer: What is measured is not the state. What is measured is the kind of error.
The method of this paper: We show this distinction by counting bits.
2. Counting the Bits the Syndrome Tells
Setting: We embed $k$ logical qubits in $n$ physical qubits. In a stabiliser code the number of generators is $n-k$, and each returns $\pm1$.
Check 2.1: Shor’s $9$-qubit code ($n=9$, $k=1$).
| Quantity | Value |
|---|---|
| Physical qubits $n$ | $9$ |
| Logical qubits $k$ | $1$ |
| Number of generators $n-k$ | $\mathbf{8}$ |
| Number of syndromes $2^{8}$ | $\mathbf{256}$ |
| Single-qubit errors to correct $1+3n$ | $\mathbf{28}$ |
| Spare | $228$ |
Observation 2.2 The syndrome returns $\mathbf{8}$ bits. Those $8$ bits tell which error entered where.
Observation 2.3 This is the core of the paper. The number of bits those $8$ tell about the logical state is $\mathbf{0}$. The generators commute with the logical operators, so measuring them does not destroy the superposition of the logical state: nothing has been copied, and nothing has been peeked at.
Check 2.4: We count the same way for three codes.
| Code | $n$ | Generators | Syndromes | Errors to correct | About the state |
|---|---|---|---|---|---|
| Shor | $9$ | $8$ | $256$ | $28$ | $\mathbf{0}$ bits |
| Steane | $7$ | $6$ | $64$ | $22$ | $\mathbf{0}$ bits |
| $5$-qubit | $5$ | $4$ | $\mathbf{16}$ | $\mathbf{16}$ | $\mathbf{0}$ bits |
Observation 2.5 The rightmost column is $0$ throughout. However large the code, the amount the syndrome tells about the state does not grow. This is why the two coexist.
3. The $5$-Qubit Code Saturates Exactly
Setting: To distinguish errors by syndrome one needs syndromes $\ge$ errors to correct. For $k=1$ and distance $3$ this reads $1+3n\le 2^{n-1}$ (the quantum Hamming bound).
Check 3.1:
| $n$ | $1+3n$ | $2^{n-1}$ | Possible? |
|---|---|---|---|
| $3$ | $10$ | $4$ | No |
| $4$ | $13$ | $8$ | No |
| $\mathbf{5}$ | $\mathbf{16}$ | $\mathbf{16}$ | Exactly |
| $6$ | $19$ | $32$ | Spare |
| $7$ | $22$ | $64$ | Spare |
| $9$ | $28$ | $256$ | Spare |
Observation 3.2 At $n=5$ equality holds. The $16$ syndromes correspond one-to-one to the $16$ errors with none to spare: a perfect code.
Observation 3.3 $n=4$ fails at $13>8$. Correcting one qubit error requires at least $5$. Here is an exact lower bound.
Observation 3.4 Paper 179 wrote that “cannot be constructed” has distinct roots. The root by which $n=4$ cannot be constructed is counting: solving the inequality $1+3n\le2^{n-1}$ alone gives $n\ge5$. No detail of quantum mechanics is needed.
4. The Same Reasoning Turns Eavesdropping into a Number
Setting: BB84. Alice picks one of two bases (rectilinear or diagonal) at random and sends a bit. Bob also picks a basis at random. Eve measures en route and resends what she measured (intercept-resend).
Check 4.1: We look only at cases where Bob’s basis matched Alice’s (after sifting).
| Eve’s basis | Probability | State resent | Bob’s error rate |
|---|---|---|---|
| Matches Alice | $1/2$ | Correct | $\mathbf{0}$ |
| Differs from Alice | $1/2$ | Off by $45^\circ$ | $\mathbf{1/2}$ |
| Total | $\frac12\cdot0+\frac12\cdot\frac12=\mathbf{0.25}$ |
Observation 4.2 The error rate is exactly $\mathbf{25\%}$. Eve is conspicuous because she measured: exactly the obverse of Section 2, where measuring is not conspicuous.
Check 4.3: We count how much Eve obtained.
| Quantity | Value |
|---|---|
| Probability Eve’s bit is right | $\frac12\cdot1+\frac12\cdot\frac12=\mathbf{0.75}$ |
| Mutual information $1-h(1/4)$ | $\mathbf{0.188722}$ bits |
Observation 4.4 Eve gains only $0.19$ bits per bit, yet leaves $25\%$ errors. Her footprint is far too large for her take.
Observation 4.5 A point to write honestly: intercept-resend is not the best attack. A cleverer attack obtains the same information with fewer errors: $25\%$ is a number for “how conspicuous a naive attack is,” not a proof of security.
5. $25\%$ Is $2.27$ Times the Limit
Setting: Whether a key can be extracted at error rate $q$ is set, for one-way post-processing, by $R=1-2h(q)$ (Shor--Preskill).
Check 5.1:
| Error rate $q$ | $h(q)$ | Key rate $R=1-2h(q)$ |
|---|---|---|
| $0$ | $0$ | $+1.000000$ |
| $0.01$ | $0.080793$ | $+0.838414$ |
| $0.02$ | $0.141441$ | $+0.717119$ |
| $0.05$ | $0.286397$ | $+0.427206$ |
| $0.08$ | $0.402179$ | $+0.195642$ |
| $\mathbf{0.110028}$ | $\mathbf{0.500000}$ | $\mathbf{0}$ |
| $0.15$ | $0.609840$ | $-0.219681$ |
| $\mathbf{0.25}$ | $0.811278$ | $\mathbf{-0.622556}$ |
Observation 5.2 The limit is $\mathbf{11.0028\%}$. The $25\%$ of intercept-resend is $2.2722$ times that: not just outside the boundary, but far outside.
Check 5.3: We see whether intercepting only part can hide. Intercepting a fraction $f$ gives error rate $0.25f$.
| Quantity | Value |
|---|---|
| Upper bound on $f$ with $0.25f\le0.110028$ | $\mathbf{0.440111}$ |
| Information Eve then obtains | $\mathbf{0.083059}$ bits per sifted bit |
Observation 5.4 Hiding is possible. But the take falls to $0.083$ bits per $\mathbf{1}$ bit: the better she hides, the less she takes.
Check 5.5: Under full interception, the probability that $n$ checked bits show no error at all is $0.75^n$.
| $n$ | $0.75^n$ |
|---|---|
| $10$ | $5.63\times10^{-2}$ |
| $20$ | $3.17\times10^{-3}$ |
| $50$ | $5.66\times10^{-7}$ |
| $\mathbf{73}$ | $\mathbf{7.58\times10^{-10}}$ |
| $100$ | $3.21\times10^{-13}$ |
| $200$ | $1.03\times10^{-25}$ |
Observation 5.6 Checking $\mathbf{73}$ bits drops the miss probability below $10^{-9}$. However many tens of thousands of bits the key has, what is needed to check it is $73$ bits.
6. Separator: Security Does Not Rest on Computational Cost
Observation 6.1 Separator. What must be moved for this $25\%$ to become another number? Only the theory of measurement itself.
Observation 6.2 Making computers faster does not move $25\%$. Making the key longer does not move it. Making the algorithm cleverer does not move it: $25\%$ comes from the overlap of bases, not from computational cost.
Observation 6.3 Its standing differs from classical cryptography. There, security is “hard to solve,” and hardness is relative to the speed of computers. Here, security is “measuring is conspicuous,” and it does not depend on the adversary’s computing power.
Observation 6.4 Sections 2 and 4 are two faces of one fact.
| About the error | About the state | |
|---|---|---|
| Syndrome measurement (Sec. 2) | gains $\mathbf{8}$ bits | gains $\mathbf{0}$ bits $\Rightarrow$ no damage |
| Eve’s measurement (Sec. 4) | — | tries to gain $\Rightarrow$ damage ($25\%$) |
Observation 6.5 So “cannot be copied, yet can be corrected” is no contradiction. Nothing has been copied and nothing observed, because only $0$ bits were taken about the state, the price is also $0$.
7. Roll Call and Closing
- Shor’s syndrome is $8$ bits (Check 2.1): $2^{8}=256$ cases.
- Of which $0$ bits about the state (Observation 2.3): Hence no damage.
- $0$ for all three codes (Check 2.4): Enlarging does not increase it.
- $16=16$ at five qubits (Check 3.1): Saturates exactly.
- $n=4$ fails at $13>8$ (Observation 3.3): At least $5$ are needed.
- Intercept-resend error rate $25\%$ (Check 4.1): $\frac12\cdot\frac12$.
- Eve’s take $0.188722$ bits (Check 4.3): Small against the footprint.
- Security limit $11.0028\%$ (Check 5.1): $25\%$ is $2.2722$ times it.
- Hiding gives $0.083059$ bits (Check 5.3): The better hidden, the less taken.
- $73$ bits give $10^{-9}$ (Check 5.5): Checking is short.
Closing: It cannot be copied yet can be corrected because what is measured is not the state. The syndrome of Shor’s $9$-qubit code tells $\mathbf{8}$ bits about the error and $\mathbf{0}$ bits about the logical state, because only $0$ bits are taken, the price is also $0$. Enlarging the code does not increase that $0$: Shor, Steane, and the $5$-qubit code are equally $0$ about the state. And in the $5$-qubit code, $16$ syndromes correspond one-to-one to $16$ errors. The obverse of the same reasoning turns eavesdropping into a number: Eve tries to take about the state, so a footprint of $25\%$ remains. That is $\mathbf{2.2722}$ times the security limit of $11.0028\%$, and checking $73$ bits drops the miss probability below $10^{-9}$. One thing separates them: counting how many bits that measurement takes about the state. Count, and correction and eavesdropping are seen as two faces of one principle. Fail to count, and the two look contradictory.
Revision Record
Second edition (October 2026): the dashes used as separators in the text were replaced by punctuation; places where LaTeX commands had been displayed as raw text were repaired; no claim, number or reference was changed.
References
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Reference
Yamagishi, Y. (2026). It Cannot Be Copied, Yet It Can Be Corrected [Paper 281]. YOI: https://www.yamagishiyuuki.com/yoi/281
DOI: 10.5281/zenodo.22171058
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 281. This paper is a preprint and has not been peer reviewed.