No-Cloning Theorem
Hand a photocopier any document and it returns two; this is so ordinary that classical information is defined by its copyability — a bit is a thing you can duplicate, broadcast, and back up at will. Quantum information refuses. There is no machine, no matter how cleverly built or how richly resourced, that takes a single copy of an arbitrary unknown state |ψ⟩ and returns two. This is the no-cloning theorem, and it is not a limit of engineering but a hard corollary of the two most basic facts about quantum evolution: it is linear, and it is unitary. The result was crystallized in 1982 by William Wootters and Wojciech Zurek, and independently by Dennis Dieks, all three responding to a proposal that claimed to signal faster than light by copying half of an entangled pair. The proof they found is a single line. Assume a device copies |0⟩ and |1⟩; linearity then forces its action on the superposition |+⟩ to be the entangled Bell state (|00⟩ + |11⟩)/√2 — emphatically not the two copies |+⟩|+⟩. The machine duplicates the basis and mangles everything else. What no-cloning forbids is narrow and exact: a single fixed device that copies every state, including the unknown and the non-orthogonal. It does not forbid copying a known state, nor copying any set of mutually orthogonal — hence perfectly distinguishable — states; a CNOT does exactly that. The obstruction is the continuum of non-orthogonal states such as |0⟩ and |+⟩, which no measurement can perfectly tell apart and therefore no machine can perfectly duplicate. From this single "no" flows the security of quantum cryptography, the impossibility of superluminal signalling, the peculiar shape of quantum error correction, and an entire quantitative theory of how close to copying one is allowed to get.
- The no-cloning theorem: no unitary — more generally, no physical process — maps an arbitrary unknown |ψ⟩ and a blank ancilla to two perfect copies |ψ⟩|ψ⟩.
- It is a one-line consequence of linearity (a cloner defined on a basis acts wrongly on superpositions) or, equivalently, of unitarity preserving inner products.
- Cloning fails only for unknown or non-orthogonal states; any fixed known state, or any orthonormal set, can be copied perfectly — a CNOT copies |0⟩, |1⟩ but not |+⟩.
- Unitarity forces the overlap of any two cloneable states to satisfy x = x², whose only solutions are x = 0 (orthogonal) and x = 1 (identical).
- Approximate cloning is allowed: the optimal universal 1→2 qubit cloner reaches single-copy fidelity 5/6, strictly below the perfect value 1.
- No-cloning underwrites BB84 key distribution, forbids faster-than-light signalling via entanglement, and forces error-correcting codes to spread information instead of copying it.
- Key idea: linearity (equivalently unitarity) of quantum evolution is incompatible with a single device that maps an arbitrary |ψ⟩ to |ψ⟩|ψ⟩; perfect cloning is possible only for mutually orthogonal — i.e. distinguishable — states.
Where this lives today
No-cloning is the working assumption behind an entire industry of quantum technologies, invoked daily rather than admired from afar. It is the security foundation of quantum key distribution: in BB84 and its successors, deployed today over metropolitan fibre networks and satellite links such as Micius, an eavesdropper cannot copy the transmitted qubits to measure at leisure, because the encoding bases are non-orthogonal — any copying attempt disturbs the states and shows up as an elevated error rate. It shapes quantum error correction: because a logical qubit cannot simply be triplicated the way a classical repetition code copies a bit, codes such as Shor's, Steane's, and the surface code instead delocalise one logical qubit's information across many physical qubits using entanglement, never producing a copy of the unknown logical state — the design constraint that makes fault-tolerant computing hard and interesting. It is why quantum networks need entanglement-based repeaters rather than signal-amplifying ones, since an unknown qubit cannot be amplified without cloning. And it is an active experimental frontier: optimal universal cloners have been realised in photonic, NMR, and trapped-ion systems, reaching the 5/6 ceiling, both to test the bound and to cap the information an adversary can extract. The single "no" from 1982 is now one of the most quantitatively exploited facts in the field.
The physical picture
Quantum states live in a linear space, and every allowed evolution must respect that linearity — this is the whole story. Suppose you build the most natural "copier" imaginable, one that behaves impeccably on the standard basis: it sends |0⟩ to |0⟩|0⟩ and |1⟩ to |1⟩|1⟩. A CNOT gate, with the input as control and a blank |0⟩ as target, does precisely this. But now feed it a superposition. Linearity leaves no freedom: the output must be the linear combination of the two basis outputs, which is the entangled Bell state (|00⟩ + |11⟩)/√2 — a state in which neither qubit alone carries |+⟩, and which is nothing like the intended product |+⟩|+⟩. The copier does not copy; it entangles. The first figure shows this failure in one diagram.
There is a second way to see the obstruction, and it is quantitative rather than pictorial. Cloning would let you cheat at distinguishing states: given many perfect copies of an unknown |ψ⟩, you could do tomography and read off its amplitudes. But non-orthogonal states cannot be perfectly distinguished by any measurement, so they cannot be perfectly copied either. The geometry is rigid. A genuine unitary preserves the overlap ⟨ψ|φ⟩ between two inputs, whereas a cloner would deliver two copies each and hence square the overlap to ⟨ψ|φ⟩². The only overlaps that survive both demands are the fixed points of x = x², namely 0 and 1. The second figure makes this the punchline of the whole theorem.
The theorem is a statement about physics, not about ignorance or budgets. Even with infinite energy, time, and engineering, the unknown qubit handed to you cannot be duplicated. That is exactly why quantum information behaves as a resource — scarce, unforgeable, impossible to broadcast — in a way the freely copyable classical bit never can.
The core equations
A cloning machine, if it existed, would be a fixed unitary acting on the data qubit together with a blank ancilla, and it would have to work for every input |ψ⟩. Writing that demand down is the entire premise from which the contradiction flows.
where: U — the hypothetical cloning unitary on the joint system; |ψ⟩ — the arbitrary unknown state to be copied; |b⟩ — a fixed blank ancilla, chosen independently of |ψ⟩; ⊗ — tensor product of data and ancilla registers. Reading: the single U and single |b⟩ must not depend on the input — that universality is what the theorem kills. Feed two different states into that same machine and compare the overlaps of inputs and outputs. Unitarity preserves the input overlap, while the cloned outputs carry it squared, forcing one algebraic equation.
where: ⟨ψ|φ⟩ — the overlap (inner product) of two candidate input states. Reading: the same number must equal its own square, so it can only be 0 or 1 — this single line is the theorem in its most compact form. Solving x = x² leaves exactly two possibilities, which is the dichotomy that carves cloneable pairs from the rest.
where: {0, 1} — the only overlap magnitudes a common cloner tolerates; 0 means orthogonal (distinguishable), 1 means identical. Reading: a continuum of qubit states takes every overlap in [0, 1], so no single U can serve them all. None of this forbids copying the computational basis. A CNOT with a blank target duplicates any basis state perfectly — which is exactly how classical, copyable computation embeds inside the quantum circuit model.
where: |x⟩ — a computational-basis state, x = 0 or 1; |0⟩ — the target ancilla initialised to |0⟩. Reading: orthogonal states can be copied — the theorem targets superpositions and unknowns, not the basis. Hand that same CNOT a superposition, though, and linearity turns the copier into an entangler. This is the concrete face of the no-go.
where: |+⟩ = (|0⟩ + |1⟩)/√2, the Hadamard-produced superposition; (|00⟩ + |11⟩)/√2 — a Bell state, entangled rather than two copies. Reading: the natural copier outputs a correlated pair whose halves are each maximally mixed — no copy of |+⟩ exists anywhere in it. Perfect cloning is out, but approximate cloning is a genuine, optimizable problem. The best symmetric universal machine that turns one qubit into two has a single-copy fidelity fixed by quantum mechanics.
where: F₁→₂ — fidelity of each output clone with the ideal |ψ⟩; η — the isotropic Bloch-vector shrinking factor of the optimal cloner, η = 2/3. Reading: 5/6 is the exact quantum speed limit on copying — better than the classical measure-and-resend value 2/3, worse than the forbidden 1. Finally, none of the argument leaned on U being reversible. Replacing the unitary by the most general physical process, a completely positive trace-preserving (CPTP) map, changes nothing: perfect cloning of non-orthogonal states remains impossible.
where: ρ_A — input density matrix, possibly mixed or unknown; ρ_A^out, ρ_B^out — reduced states of the two output registers. Reading: even allowing measurement, noise, and ancillas, no map perfectly duplicates an arbitrary ρ_A — linearity alone does the work.
The goal here is to show, with nothing but the linearity of quantum evolution, that a device which correctly copies the two basis states |0⟩ and |1⟩ cannot possibly copy their superposition, so that no single machine copies every input. We do not assume anything exotic — not even that the copier is unitary at this stage; we use only that quantum evolution is a linear map, which every allowed process is. The strategy is to pin the copier down on the basis, then let linearity dictate its action on |+⟩ and read off the contradiction between the entangled output it is forced to produce and the product state cloning would require.
Step 1. Suppose a single operation U clones each basis state with a blank ancilla |0⟩, so that U|0⟩|0⟩ = |0⟩|0⟩ and U|1⟩|0⟩ = |1⟩|1⟩. These are exactly the two conditions a CNOT satisfies, so such a U demonstrably exists on the basis. Why: we grant the copier everything it can legitimately have — perfect action on a distinguishable set — and show even that cannot extend to superpositions.
Step 2. Apply U to the input |+⟩|0⟩ = (|0⟩ + |1⟩)|0⟩/√2. Because U is linear, it distributes across the sum: U((|0⟩ + |1⟩)|0⟩/√2) = (U|0⟩|0⟩ + U|1⟩|0⟩)/√2. Why: linearity is not optional — it is the defining property of quantum time evolution, so the action on a superposition is fixed the moment the action on the basis is.
Step 3. Substitute the Step 1 values to get (|00⟩ + |11⟩)/√2, the Bell state |Φ⁺⟩. This is the machine's actual output, and it is entangled: neither qubit, taken alone, is in any pure state. Why: the copier had no freedom left after Step 1, so this entangled pair is what it must produce — the linearity has already decided.
Step 4. Compare with the goal. Genuine cloning of |+⟩ would require |+⟩|+⟩ = (|00⟩ + |01⟩ + |10⟩ + |11⟩)/2, a product state with four equal terms. The output (|00⟩ + |11⟩)/√2 has only two terms and is entangled, so the two states are different; U does not clone |+⟩. Contradiction. Why: one state is a product and the other is maximally entangled — they cannot be equal, so the assumption that one device copies both basis and superposition fails. *Verified: CNOT|+⟩|0⟩ = (0.7071, 0, 0, 0.7071) = (|00⟩+|11⟩)/√2 numerically, and |+⟩|+⟩ = (0.5, 0.5, 0.5, 0.5); the two differ. Reproduces Nielsen & Chuang Box 12.1 and Preskill Ch. 4.*
Go deeper · Derivation B · 2 minThe unitarity proof and the overlap dichotomy
The aim of this second, sharper argument is to show that the impossibility is not tied to any particular basis or gate but is a rigidity of inner products under unitary evolution: any two states a common machine can clone must be either orthogonal or identical, with no overlap allowed in between. This is stronger than the linearity proof because it quantifies the obstruction — it produces the equation x = x² whose solution set {0, 1} is the exact boundary of what is cloneable. We assume a unitary cloner acts on two arbitrary states with the same blank ancilla, take the inner product of the two output equations, and let unitarity collide with the tensor structure of the outputs.
Step 1. Assume U clones both |ψ⟩ and |φ⟩ with the same blank |b⟩: U|ψ⟩|b⟩ = |ψ⟩|ψ⟩ and U|φ⟩|b⟩ = |φ⟩|φ⟩. The ancilla and the machine are fixed — the whole point of a universal copier. Why: reusing one U and one |b⟩ for two different inputs is precisely the universality claim under test.
Step 2. Form the overlap of the two output states. Because the outputs are product states, the inner product factorises: (⟨ψ|⟨ψ|)(|φ⟩|φ⟩) = ⟨ψ|φ⟩·⟨ψ|φ⟩ = ⟨ψ|φ⟩². Why: the tensor inner product multiplies the overlaps of the two registers, and both registers hold the same states, so the overlap gets squared.
Step 3. Compute the same overlap on the input side using unitarity. Since U†U = I and ⟨b|b⟩ = 1, we have (⟨ψ|⟨b|)U†U(|φ⟩|b⟩) = ⟨ψ|φ⟩⟨b|b⟩ = ⟨ψ|φ⟩. Why: a unitary preserves every inner product, so the overlap of the two inputs passes through U unchanged.
Step 4. Equate the two computations of the same overlap: ⟨ψ|φ⟩ = ⟨ψ|φ⟩². Writing x = ⟨ψ|φ⟩ gives x = x², i.e. x(x − 1) = 0, so x ∈ {0, 1}. Only orthogonal (x = 0) or identical (x = 1) states can be jointly cloned by one U. Why: a continuum of qubit states — the whole Bloch sphere — realises every overlap in [0, 1], and no single unitary can satisfy x = x² across that continuum, so a universal cloner cannot exist. *Verified: x = x² ⇒ x ∈ {0, 1}. For |0⟩ and |+⟩ the overlap is ⟨0|+⟩ = 1/√2 ≈ 0.7071, which fails 0.7071 = 0.5, so these two states admit no common cloner. Matches Wootters–Zurek (1982) and Wilde Ch. 3.*
Go deeper · Derivation C · 2 minBest possible approximate cloning and the fidelity 5/6
Since perfect cloning is forbidden, the natural question is how close a machine can come, and the goal of this derivation is to pin the answer for the most-studied case — one qubit copied symmetrically into two — at the single-copy fidelity 5/6. The route is to exploit universality: a cloner that must treat every input state alike can only shrink each output's Bloch vector isotropically toward the centre of the sphere, so its whole action is captured by one number, the shrinking factor η. We fix the form of the output from symmetry, quote the optimal η from the Gisin–Massar bound, and convert it into a fidelity.
Step 1. By universality, each output clone of a pure input |ψ⟩ with Bloch vector r must be the isotropically depolarised state ρ_out = ½(I + η r·σ), with a single shrinking factor 0 ≤ η ≤ 1 independent of which state |ψ⟩ was. Why: if the machine favoured any direction on the Bloch sphere it would not be universal, so the only allowed degradation is a uniform contraction of the Bloch vector.
Step 2. Compute the fidelity of one clone with the ideal |ψ⟩. Using |ψ⟩⟨ψ| = ½(I + r·σ) and (r·σ)² = I for a unit vector, F = ⟨ψ|ρ_out|ψ⟩ = ¼ Tr[(I + η r·σ)(I + r·σ)] = ¼(2 + 2η) = (1 + η)/2. Why: the fidelity depends on the output only through the shrinking factor, so the entire copying quality is now one algebraic function of η.
Step 3. Insert the optimal shrinking factor. The Gisin–Massar bound gives the best symmetric N→M universal fidelity F = (MN + M + N)/(M(N + 2)); for N = 1, M = 2 this is (2 + 2 + 1)/(2·3) = 5/6, corresponding to η = 2/3. Hence F₁→₂ = (1 + 2/3)/2 = 5/6 ≈ 0.833. Why: 5/6 is optimal, not merely achievable — no symmetric universal machine copies one qubit into two any better, and the Bužek–Hillery cloner attains it.
Step 4. Place the number. The classical strategy — measure the qubit and re-prepare two copies of the guessed state — averages only F = 2/3 ≈ 0.667 over all inputs, while perfect cloning would give F = 1. The quantum-optimal 5/6 sits strictly between: quantum mechanics permits some copying, more than any classical scheme, but never all of it. Why: the gap 2/3 < 5/6 < 1 is the precise sense in which no-cloning is a soft ceiling on fidelity rather than an absolute ban on approximate duplication. *Verified: Gisin–Massar F(1→2) = 5/6 = 0.8333; η = 2/3 gives (1+η)/2 = 0.8333; the classical measure-and-resend bound is 2/3. Bužek & Hillery (1996), Gisin & Massar (1997).*
Limiting cases
The theorem is sharpest when read as a map from the input overlap to what is and is not possible — from perfect copying at the extremes to a graded impossibility in between.
| Case | Result | Meaning | |||
|---|---|---|---|---|---|
| Orthogonal states | ⟨ψ\ | φ⟩ = 0 | Perfect cloning IS possible: a CNOT or a measure-and-prepare device copies any orthonormal basis exactly. | ||
| Identical / known state | \ | φ⟩ = e^{iθ}\ | ψ⟩ | Trivially cloneable — once a state is known it can be re-prepared any number of times. | |
| Non-orthogonal, unknown | 0 < \ | ⟨ψ\ | φ⟩\ | < 1 | Perfect cloning IMPOSSIBLE. This generic case is the content of the theorem. |
| Optimal universal cloner | F₁→₂ = 5/6 | The best symmetric approximate 1→2 qubit cloner; perfection F = 1 is unreachable. | |||
| Classical measure-and-resend | F = 2/3 | Copying by measuring first tops out at 2/3 — the benchmark the quantum cloner beats. | |||
| Probabilistic exact cloning | p < 1 | Linearly independent states can be cloned exactly but only sometimes; for two states of overlap s the optimal success is p = 1/(1 + s) (Duan–Guo). |
Worked examples
1 · How badly does the naive CNOT copier fail? (fully worked)
The natural copier is a CNOT with a blank target. Push the superposition |+⟩ = (|0⟩ + |1⟩)/√2 through it and measure the quality of the "copies" quantitatively: compute the reduced state of each output register and its fidelity with the intended |+⟩. Compare with the optimal universal cloner's 5/6.
Step 1. Run the gate. |+⟩|0⟩ = (|00⟩ + |10⟩)/√2, and CNOT flips the target when the control is 1, giving CNOT|+⟩|0⟩ = (|00⟩ + |11⟩)/√2 = |Φ⁺⟩, a Bell state. Already the output is entangled, not a product — a warning that no clean "copy" lives inside it.
Step 2. Extract register A's state by tracing out register B: ρ_A = Tr_B|Φ⁺⟩⟨Φ⁺| = ½(|0⟩⟨0| + |1⟩⟨1|) = I/2, the maximally mixed state. By symmetry ρ_B = I/2 as well. Each "copy" has been degraded all the way to a coin flip — it retains none of the phase information that made |+⟩ different from |−⟩.
Step 3. Compute the fidelity of that output with the target: F = ⟨+|ρ_A|+⟩ = ⟨+|(I/2)|+⟩ = 1/2. The naive copier scores only F = 0.5 on |+⟩ — no better than blind guessing, and it does not even beat the classical measure-and-resend bound of 2/3.
Step 4. Weigh it against the optimum. The best universal cloner reaches F = 5/6 ≈ 0.833 on every input by deliberately entangling the two outputs with an ancilla in a balanced way, rather than dumping all the correlation into one Bell pair. The CNOT is a perfect basis copier (F = 1 on |0⟩, |1⟩) but a terrible universal cloner (F = 0.5 on |+⟩); optimality requires the extra machinery that caps out at 5/6.
2 · When may non-orthogonal states be cloned — and how often?
You are promised the incoming qubit is one of just two states, |0⟩ or |+⟩ — non-orthogonal, but only two possibilities. Perfect deterministic cloning is still impossible; quantify why, and then find the best probability with which they can be cloned exactly.
Step 1. Find the overlap. ⟨0|+⟩ = ⟨0|(|0⟩ + |1⟩)/√2 = 1/√2 ≈ 0.7071. This is strictly between 0 and 1, so the pair is non-orthogonal — the two states cannot be perfectly distinguished by any single measurement.
Step 2. Test the cloning dichotomy. A common deterministic cloner would need the overlap to satisfy x = x²: here 0.7071 ≠ (0.7071)² = 0.5. The condition fails, so no deterministic unitary clones both |0⟩ and |+⟩ — the general theorem applies even to this two-element set.
Step 3. Allow occasional failure. Because |0⟩ and |+⟩ are linearly independent, Duan–Guo probabilistic cloning permits exact copies on success, with optimal per-state probability p = 1/(1 + s) where s = ⟨0|+⟩. Substituting, p = 1/(1 + 1/√2) = 1/1.7071 = 0.586.
Step 4. Interpret. About 59 % of the time a heralded device outputs two perfect copies; the rest of the time it reports failure and produces nothing usable. As the states approach orthogonality (s → 0) the success rises to p → 1 (deterministic cloning of a distinguishable set), and as they approach identity (s → 1) it falls toward 1/2. Non-orthogonal states are cloneable only probabilistically, and the closer they are, the rarer the success.
Physics of the portrait
No-cloning was not sought for its own sake; it fell out of an attempt to debunk a faster-than-light signalling scheme, and the physicist who pinned it down had already made a habit of finding the sharp structural limits of quantum information.
Origins · 1 minWilliam Wootters and how this was discovered
William Wootters is an American theoretical physicist, long at Williams College, who took his PhD under John Archibald Wheeler at the University of Texas — an intellectual lineage steeped in the question of how information underlies physics. In 1982, with Wojciech Zurek, he wrote the one-page Nature paper "A single quantum cannot be cloned," and the origin story is telling: the physicist Nick Herbert had circulated a proposal called FLASH that claimed to signal faster than light by taking one half of an entangled pair and cloning it many times to read out, at a distance, which basis a partner had measured in. Refereeing and reacting to that scheme, Wootters and Zurek — and, independently, Dennis Dieks, and GianCarlo Ghirardi — realised the flaw was fundamental: linearity simply forbids the cloning step, and with it the superluminal telephone. A debunking turned into a founding theorem. Wootters went on to become an architect of quantitative entanglement theory, giving the closed-form "concurrence" and entanglement of formation for two qubits in 1998 and developing mutually unbiased bases. His signature is the same throughout: take an operational impossibility and turn it into an exact, computable constraint on what quantum information can and cannot do.
Going further
Texts. Nielsen & Chuang, Quantum Computation and Quantum Information (10th anniv. ed.), §1.3.5 and Box 12.1 · Preskill, Lecture Notes for Physics 219, Ch. 4 · Watrous, The Theory of Quantum Information, Ch. 3 · Wilde, Quantum Information Theory (2nd ed.), Ch. 3 · Kitaev, Shen & Vyalyi, Classical and Quantum Computation, Ch. 11.
Provenance. A checker should reproduce: (1) the linearity proof, confirming CNOT|+⟩|0⟩ = (|00⟩ + |11⟩)/√2 ≠ |+⟩|+⟩ numerically; (2) the inner-product identity ⟨ψ|φ⟩ = ⟨ψ|φ⟩² via U†U = I, giving overlaps {0, 1}; (3) the naive-copier fidelity F = ⟨+|(I/2)|+⟩ = 1/2 from the reduced Bell-state marginal; (4) the optimal universal cloner fidelity 5/6 from the Gisin–Massar formula (MN + M + N)/(M(N + 2)) at N = 1, M = 2, equivalently (1 + η)/2 with η = 2/3; (5) the Duan–Guo probability p = 1/(1 + s) = 0.586 for s = ⟨0|+⟩ = 1/√2. Primary sources: Wootters & Zurek, Nature 299, 802 (1982); Dieks, Phys. Lett. A 92, 271 (1982); Bužek & Hillery, Phys. Rev. A 54, 1844 (1996); Gisin & Massar, Phys. Rev. Lett. 79, 2153 (1997); Duan & Guo, Phys. Rev. Lett. 80, 4999 (1998).
Prerequisites. Quantum Circuit Model · CNOT Gate · Pauli Gates · Hadamard Gate. Leads to. Quantum Teleportation · BB84 Quantum Key Distribution · Quantum Error Correction · SWAP Gate.
Ask the tutor: Want to walk through the Gisin–Massar bound that fixes the 5/6 fidelity, or see exactly how no-cloning turns an eavesdropper's best attack into BB84's security proof?
Does no-cloning mean I can never copy a qubit at all?
No — it is far narrower than that. If you know the state, you can copy it as many times as you like by simply preparing fresh qubits in the same state; knowledge is what a preparation procedure encodes. If your states are drawn from a mutually orthogonal set, a CNOT or any measure-and-prepare device copies them perfectly — this is exactly how classical bits are stored and processed inside a quantum computer. The one forbidden act is a single fixed machine that copies an arbitrary unknown state, including superpositions whose amplitudes you cannot see. So "quantum information cannot be copied" really means "an unknown quantum state cannot be copied," which is a precise and much weaker statement.
How is teleportation not a violation of no-cloning?
Because teleportation moves a state rather than duplicating it, and the bookkeeping is exact. In the protocol, Alice's Bell-basis measurement destroys the original qubit — after it, the input register retains no trace of the amplitudes — at the same moment the state is reconstructed on Bob's side. At every instant exactly one copy of |ψ⟩ exists: first at Alice's location, then at Bob's, never both simultaneously. If the original somehow survived, you would have two copies and the no-cloning theorem would indeed be broken, which is precisely why the destruction of the input is a requirement of the protocol and not an accident. Teleportation is a hand-off, not a photocopy.
Why can't I just measure the unknown qubit and prepare copies from the result?
Because a single measurement cannot reveal the state. A qubit |ψ⟩ = α|0⟩ + β|1⟩ hides two continuous complex amplitudes, but any measurement returns just one classical outcome and collapses the state, telling you almost nothing about α and β. If you measure |+⟩ in the computational basis you get "0" or "1" at random and lose the phase entirely. The best measure-and-prepare strategy, optimised over all measurements, reproduces the unknown qubit at only F = 2/3 average fidelity — strictly worse than the quantum-optimal cloner's 5/6, and nowhere near perfect. Measurement destroys exactly the information you would need to make a faithful copy.
Does the theorem still hold for noisy, non-unitary machines?
Yes, and this is what makes it robust. The unitary case is the cleanest to prove, but the argument really rests on linearity, which every physical process shares. A general open-system evolution is a completely positive trace-preserving (CPTP) map, and such maps are linear on density matrices; the same reasoning shows they cannot perfectly clone non-orthogonal states either. Adding ancillas, allowing measurements, tolerating noise — none of it opens a loophole, because none of it breaks linearity. That is why no-cloning is stated as a fact about quantum mechanics as such, rather than about idealised reversible devices, and why it constrains real, imperfect hardware just as firmly.
If cloning is impossible, why do people build "quantum cloning machines"?
Because approximate and probabilistic cloning are genuinely possible and genuinely useful. A universal quantum cloning machine does not produce perfect copies — it produces two imperfect ones, each with fidelity 5/6 for a qubit, which is the best any universal device can do. These machines matter both fundamentally, as a way to test the exact fidelity bounds quantum mechanics imposes, and practically, in quantum cryptography: the optimal cloner models the best possible eavesdropper, so bounding what a cloner can extract bounds how much an attacker can learn. Probabilistic cloners go further, producing perfect copies on a fraction of runs for linearly independent input sets. "Quantum cloning" is the quantitative theory of how close to the forbidden ideal one can legally get.
Why does no-cloning stop faster-than-light signalling?
Because cloning would let one party read a distant measurement instantly. Take an entangled pair shared by Alice and Bob. Whatever basis Alice measures in, Bob's local qubit alone is maximally mixed — he sees nothing. But if Bob could clone his qubit many times, he could do tomography on the copies and infer which basis Alice chose, learning her bit the instant she measured, regardless of distance. That would be a superluminal telephone, in conflict with relativity. No-cloning slams the loophole shut: Bob cannot make the copies, so his marginal stays uninformative until a classical, light-speed-limited message arrives. This is not a coincidence — the theorem was discovered precisely while debunking such a scheme.