The No-Cloning Theorem

No state-independent physical device can perfectly copy every arbitrary unknown quantum state. The obstruction is linearity: a map that copies two basis states sends their superposition to an entangled state, not to two copies of the superposition. Equivalently, inner-product preservation would have to turn an overlap into its square. Black-hole duplication arguments invoke this theorem only if the interior and exterior records are independent copies in one common quantum description.

The proof is short, but applying it requires unusually careful identification of the two output systems.

Required background. Quantum Information supplies isometries, inner products, quantum channels, and reference systems used in the proof.

Helpful background. Semiclassical Gravity supplies the evaporation setting; Quantum Fields in Curved Spacetime supplies interior and exterior mode descriptions; Causality and Global Structure supplies the observer comparison regions.

No-cloning theorem for nonorthogonal states

Statement. Let a fixed physical device receive an arbitrary normalized pure state $\lvert\psi\rangle_A$, a fixed blank $\lvert0\rangle_B$, and a fixed machine state. Under linear quantum mechanics, no input-independent isometry can implement

$$\lvert\psi\rangle_A\lvert0\rangle_B \longmapsto \lvert\psi\rangle_A\lvert\psi\rangle_B$$

for every $\lvert\psi\rangle$ in a set containing two distinct nonorthogonal states.

The hypotheses matter: the input is unknown within the set, the blank and physical map are fixed, the outputs are independent perfect copies, and the dilation preserves inner products. The theorem does not forbid copying a known orthogonal basis, transferring a state while erasing the original, or encoding one logical state redundantly in a larger code.

Inner-product proof

Suppose the isometry clones both $\lvert\psi\rangle$ and $\lvert\phi\rangle$. If the machine returns to the same final state, preservation of inner products gives

$$\langle\psi\vert\phi\rangle =\left(\langle\psi\vert\phi\rangle\right)^2.$$

Writing $s=\langle\psi\vert\phi\rangle$, the equation $s=s^2$ permits only $s=0$ or $s=1$. Distinct nonorthogonal states have $0\lt|s|\lt1$, a contradiction.

Allowing an input-dependent final machine state does not help. If

$$V\lvert\psi,0,M\rangle =\lvert\psi,\psi,M_\psi\rangle,$$

then

$$s=s^2\langle M_\psi\vert M_\phi\rangle.$$

Taking absolute values and using $|\langle M_\psi\vert M_\phi\rangle|\leq1$ yields $|s|\leq|s|^2$, impossible for a nonzero overlap smaller than one. This is the complete linearity proof given by Wootters and Zurek Wootters and Zurek 1982, pp. 802–803.

The same result follows by expanding a superposition. If a linear map copies $\lvert0\rangle$ and $\lvert1\rangle$, then it sends $\alpha\lvert0\rangle+\beta\lvert1\rangle$ to

$$\alpha\lvert00\rangle+\beta\lvert11\rangle,$$

whereas two desired copies would contain four terms, including $\alpha\beta\lvert01\rangle$ and $\alpha\beta\lvert10\rangle$. They agree only in special cases.

Orthogonal labels are copyable

For a known computational basis, controlled-NOT implements

$$\lvert0\rangle\lvert0\rangle\mapsto\lvert0\rangle\lvert0\rangle, \qquad \lvert1\rangle\lvert0\rangle\mapsto\lvert1\rangle\lvert1\rangle.$$

This does not contradict the theorem because the two inputs are orthogonal. Applied to $\lvert+\rangle$, it produces a Bell state rather than $\lvert+\rangle\lvert+\rangle$. Measuring in the computational basis and preparing two copies likewise copies a classical outcome while destroying coherence between the outcomes.

Approximate cloning can trade fidelity between outputs, and probabilistic exact cloning can succeed on suitably linearly independent sets with a heralded failure probability. Neither supplies a deterministic perfect universal cloner. Quantum error correction also creates multiple recoverable representations of logical information without factorizing the physical state into independent copies available to arbitrary simultaneous operations.

Application to black holes

Suppose an arbitrary diary state falls into a black hole. If unitary evaporation later places a faithful independent copy in radiation while the original diary remains as an independent interior factor, a common global Hilbert space would contain two copies. The theorem rules out that state-independent map. Black-hole complementarity responds by denying that both descriptions are independent simultaneously accessible factors, not by modifying the algebraic proof.

That response inherits obligations. It must say which observer’s algebra contains which operator, how overlapping descriptions agree, and why arbitrary superpositions retain linear evolution. A causal claim that no observer can compare the records may block an experiment, but it does not make a globally defined cloning isometry mathematically consistent. Harlow explains the connection between cloning, complementarity, and observer comparisons in Harlow 2016, §§ 5.5–5.7.

The theorem also does not identify where a semiclassical description fails. The radiation record might be a reconstruction of the same logical degree of freedom rather than an independent copy; the interior operator might have a restricted domain; or the horizon state might differ from the semiclassical vacuum. These are different proposals and require separate consistency tests.

Counterexamples to overstatement

The controlled-NOT basis copier is a counterexample to the false claim that “quantum information can never be copied.” A classical bit encoded in orthogonal states can be copied perfectly. Teleportation is a counterexample to the false claim that “an unknown state can never appear elsewhere”: it transfers the state while measurement and entanglement ensure that no second usable copy remains.

Conversely, two systems having identical reduced density matrices does not mean they contain cloned pure states. The two marginals of a Bell pair are both maximally mixed, but the complete state is entangled and neither marginal equals an arbitrary input. The theorem concerns a physical map on all allowed inputs, not visual similarity of two reduced states.

Generic cloning and broadcasting theory belongs to Quantum Information, horizon mode construction to Quantum Fields in Curved Spacetime, and explicit microscopic encoding to later gravitational or holographic treatments.

Exercise

A proposed device clones two normalized states whose overlap magnitude is $1/2$. Its dilation leaves machine states with overlap magnitude $q$. What value of $q$ would inner-product preservation require, and why is the device impossible?

Solution

Taking magnitudes in the cloning equation gives $1/2=(1/2)^2q$, so $q=2$. Inner products between normalized machine states have magnitude at most one, so no such dilation exists. The contradiction does not depend on how large the machine is. For orthogonal inputs the corresponding equation is zero on both sides and imposes no obstruction, which locates the theorem’s boundary.

References

• Harlow, Daniel. 2016. “Jerusalem Lectures on Black Holes and Quantum Information.” Reviews of Modern Physics 88: 015002. Open text.
• Wootters, William K., and Wojciech H. Zurek. 1982. “A Single Quantum Cannot Be Cloned.” Nature 299: 802–803. DOI.