No-Cloning and No-Broadcasting

No-cloning and no-broadcasting identify precise limits on copying quantum states. They do not prohibit amplification of a measurement result, preparation of many copies of a known state, or redundant records of a commuting observable. The no-broadcasting theorem supplies the mixed-state proof, and the no-cloning theorem treats pure-state copying.

Helpful background. Density Operators distinguishes states from preparation ensembles; Partial Trace defines output marginals.

Copying a state and copying its marginals

An exact cloner for a specified family would use one physical channel $\mathcal C$ satisfying

$$\mathcal C(\rho)=\rho\otimes\rho$$

for every input in the family. The same apparatus must work without being told which member was supplied. A broadcaster only requires

$$\operatorname{Tr}_B\mathcal B(\rho)=\rho, \qquad \operatorname{Tr}_A\mathcal B(\rho)=\rho.$$

Its outputs may be correlated. In finite dimensions a family of density operators is exactly broadcastable if and only if its members commute pairwise. This is weaker than cloning: even a commuting family can fail to admit independent product copies.

A classical family that broadcasts without cloning

For diagonal qubit states $\rho_p=p|0\rangle\langle0|+(1-p)|1\rangle\langle1|$, measure in the computational basis and write the obtained bit into two registers. The output is

$$\sigma_{AB}=p|00\rangle\langle00|+(1-p)|11\rangle\langle11|.$$

Both marginals equal $\rho_p$. Yet the registers always agree, whereas two independent samples from $\rho_p\otimes\rho_p$ disagree with probability $2p(1-p)$. A single classical sample can be copied, but it cannot reveal the unknown distribution well enough to manufacture independent samples from that distribution by a universal exact rule.

This example prevents a common misreading of no-broadcasting as a ban on ordinary classical communication. A detector can copy a stable pointer label to several memories. What it cannot do is preserve every incompatible quantum degree of freedom in all those memories simultaneously.

Why an unknown pure state cannot be read and rebuilt

If a procedure determined an arbitrary unknown pure state exactly from one input, it could feed the description to two preparation devices and clone it. No-cloning excludes this possibility under the quantum-channel assumptions. Tomography instead estimates a preparation from repeated trials and a statistical model. It is not a single-system readout of all amplitudes.

For intuition, an isometry that copied both $|\psi\rangle$ and $|\phi\rangle$ without a state-dependent environment would preserve their overlap while changing it to its square:

$$c=\langle\psi|\phi\rangle=c^2.$$

The full proof handles general ancillary systems; the conclusion is that distinct clonable pure states must be orthogonal. A device calibrated to clone computational-basis inputs therefore need not clone their coherent superpositions.

Foundational consequences and boundaries

The result helps explain why objective classical records are tied to a selected family of effectively compatible alternatives. It does not determine why one particular result happens or prove a specific account of classical emergence. Those questions remain in Macroscopic Records and Quantum Darwinism.

Approximate cloning, probabilistic success on restricted families, and state-dependent devices are different tasks. An experiment must compare its claimed performance to the appropriate task definition. The no-cloning slogan alone is also not a security proof: cryptographic security requires a protocol, noise assumptions and an adversary model.

Exercise: basis copying and entanglement

Apply a controlled-NOT with target $|0\rangle$ to a control $|+\rangle$. Compare the two-register output with $|+\rangle|+\rangle$ and compute a marginal.

Solution

The result is $(|00\rangle+|11\rangle)/\sqrt2$, not the four-term product $|++\rangle$. Each marginal is $I/2$, so neither retains the pure $|+\rangle$ state. The device copies basis labels while correlating superposed inputs; this is exactly the distinction between record formation and universal state cloning.

References

• Barnum, Howard, Carlton M. Caves, Christopher A. Fuchs, Richard Jozsa, and Benjamin Schumacher, “Noncommuting Mixed States Cannot Be Broadcast,” Physical Review Letters 76, 2818–2821 (1996), doi:10.1103/PhysRevLett.76.2818 — exact broadcasting.
• Wootters, William K., and Wojciech H. Zurek, “A single quantum cannot be cloned,” Nature 299, 802–803 (1982), doi:10.1038/299802a0 — pure-state copying obstruction.