No-Cloning Theorem
Copying is the most ordinary act in the classical world. You read a bit, you write its value down, you do it again — a photocopier, a hard-disk controller, and a schoolchild all exploit the same triviality. The no-cloning theorem says that this everyday power simply does not extend to quantum states. No machine, however cleverly engineered, can take an arbitrary unknown qubit |ψ⟩ and hand you back two independent copies |ψ⟩⊗|ψ⟩. The obstruction is not thermal noise, imperfect optics, or finite gain — it is the linear structure of quantum mechanics itself, and the proof fits in three lines. Two facts, each used everywhere in quantum theory, are enough to forbid cloning on their own. First, closed-system evolution is unitary, hence linear: a cloner built to copy |0⟩ and |1⟩ is forced to act on their superposition term by term, producing an entangled state rather than two copies. Second, unitaries preserve inner products: a cloner would have to square every overlap ⟨ψ|φ⟩, and x = x² has no solutions strictly between 0 and 1. Either argument closes the door. The precise statement is what gives the theorem its reach. It does not forbid copying known states, nor states drawn from a fixed orthonormal basis — a CNOT gate copies |0⟩ and |1⟩ perfectly. What is impossible is one fixed device that copies every state, including unknown superpositions. Loosen exactness and you enter the rich world of approximate cloning, whose optimal universal machine reaches fidelity 5/6. Tighten it and you have the security backbone of quantum cryptography.
- The no-cloning theorem states that no unitary — indeed no physical operation — can take an arbitrary unknown state |ψ⟩ and output two independent copies |ψ⟩⊗|ψ⟩.
- The proof is astonishingly short: it follows from the linearity of quantum evolution, or equivalently from the fact that unitaries preserve inner products, forcing every overlap to satisfy x = x².
- The theorem forbids cloning of arbitrary unknown states only; a known state, or states from a mutually orthogonal set, can always be copied — a CNOT gate does exactly this on the computational basis.
- If cloning existed it would permit faster-than-light signalling and defeat the uncertainty principle, so the theorem is what protects relativistic causality from an EPR pair.
- No-cloning is the physical foundation of quantum key distribution (BB84, E91): an eavesdropper cannot silently duplicate flying qubits, so any attack leaves a detectable disturbance.
- Approximate and probabilistic cloning are allowed; the optimal universal 1→2 qubit cloner achieves single-copy fidelity exactly 5/6 ≈ 0.833.
- Key idea: a universal cloner U(|ψ⟩|b⟩) = |ψ⟩|ψ⟩ cannot exist because linearity forces U to act on a superposition α|0⟩+β|1⟩ as α|00⟩+β|11⟩, which is not the cloned state (α|0⟩+β|1⟩)^⊗2 unless αβ = 0.
A universal cloner U(|ψ⟩|b⟩) = |ψ⟩|ψ⟩ cannot exist because linearity forces U to act on a superposition α|0⟩+β|1⟩ as α|00⟩+β|11⟩, which is not the cloned state (α|0⟩+β|1⟩)^⊗2 unless αβ = 0 — that is, unless the input is a bare basis state.
Where this lives today
No-cloning is not a museum piece; it is a daily design constraint. It is the reason quantum key distribution works at all — commercial QKD systems and satellite links (China's Micius satellite demonstrated entanglement-based keys over 1,200 km) rest on the fact that an eavesdropper cannot copy a flying photon without leaving a detectable error. It dictates the architecture of the emerging quantum internet: because you cannot amplify an unknown qubit, long-distance links cannot use classical-style repeaters and must instead chain entanglement swapping and entanglement distillation through quantum repeater nodes, an active experimental frontier in trapped-ion and diamond-NV platforms. It underlies quantum error correction, where the no-cloning constraint forces codes to spread one logical qubit across many physical qubits without ever copying it — the deep reason the classical repetition code has no naive quantum analog. And approximate cloning is itself a laboratory science: optimal 5/6-fidelity cloners have been realized in photonic, NMR, and atomic-ensemble experiments, both to test the bound and to benchmark the best possible eavesdropper. Even quantum money — unforgeable banknotes carrying unclonable states — traces back to Wiesner's pre-1982 idea that no-cloning would one day make rigorous.
The physical picture
Classically, to copy a bit you measure it and write down the answer; reading does not disturb it. Quantum mechanically, an unknown state cannot be read without disturbance, and any measurement that reveals enough to reconstruct the state destroys the original. No-cloning is the formal expression of "you cannot look without touching." The cleanest way to see it is to picture a cloner as a linear box fixed once and for all. If it copies |0⟩ and copies |1⟩, linearity forces its action on α|0⟩+β|1⟩ to be α|00⟩+β|11⟩ — an entangled state, not the product it was supposed to make. The box cannot "see" the superposition; it acts term by term, and term-by-term copying produces correlation, not duplication. The first figure makes that mismatch numerical for the state |+⟩.
There is a second, purely geometric way to feel the impossibility, and it is the one that generalizes to any Hilbert space. Unitaries are rigid rotations: they preserve every angle, meaning every inner product ⟨ψ|φ⟩. A perfect cloner, however, would map two inputs with overlap x to two output pairs whose overlap is x·x = x². Squaring is not a rotation — it contracts every angle except the two fixed points x = 0 and x = 1. So a device that both preserves overlaps (unitarity) and squares them (cloning) can only exist for states that are already orthogonal or already identical. The forbidden middle — every genuinely non-orthogonal unknown pair — is precisely the regime that quantum cryptography turns into a resource.
The core equations
A universal cloner is specified by demanding that one fixed unitary act the same way on every input state, taking a fresh blank and any |ψ⟩ to two copies of |ψ⟩. Allowing an internal machine or ancilla register that ends in some state |M_ψ⟩ makes the statement fully general and changes nothing in the proofs.
where: |ψ⟩ — arbitrary unknown input to be copied; |b⟩ — fixed blank state of the target register; |M⟩ — initial state of the cloning machine; U — a single fixed unitary on the joint Hilbert space. Reading: the whole content of "cloning" is that one U must satisfy this for every |ψ⟩ simultaneously — copying a single known state is trivial and not what is forbidden. Taking inner products of two instances of equation (1) and using that unitaries preserve overlaps gives the master constraint of the theorem. It is a self-consistency condition the overlap of any two clonable inputs must obey.
where: ⟨ψ|φ⟩ — the overlap of two candidate input states, a complex number of modulus at most 1. Reading: writing x = ⟨ψ|φ⟩, this is x = x², whose only solutions are x = 0 and x = 1. Any pair with intermediate overlap cannot both be cloned by the same U.
where: x — the overlap ⟨ψ|φ⟩. Reading: cloning is possible exactly for orthogonal states (x = 0, perfectly distinguishable) or identical states (x = 1, nothing to copy). The entire non-orthogonal interior is forbidden. The complementary linearity argument works directly with amplitudes. A cloner that copies both computational basis states must, because U is linear, distribute over a superposition term by term — and the result is entangled, not a product.
where: α, β — amplitudes with |α|² + |β|² = 1; |00⟩ — shorthand for |0⟩⊗|0⟩. Reading: this is what U is forced to output. It is a legitimate quantum state, but it is not two copies of the input. Genuine cloning of the same input would instead require the full tensor product, with cross terms and squared amplitudes that equation (4) plainly lacks.
where: αβ — the cross-term amplitude, nonzero for any genuine superposition. Reading: equations (4) and (5) agree only when αβ = 0, i.e. the input is |0⟩ or |1⟩. One U cannot serve both the basis states and their superpositions. The controlled-NOT gate is the standard trap, because it does copy basis states. But on a superposition it reproduces exactly equation (4): correlation, not a clone.
where: CNOT — the controlled-NOT with the input as control and the blank as target. Reading: for α = β = 1/√2 this is the Bell state (|00⟩+|11⟩)/√2, maximally entangled — the opposite of two independent copies. Relaxing exactness rescues the enterprise. The optimal symmetric universal machine copies any qubit imperfectly, with a fidelity that is the same for every input and strictly below one.
where: F — single-copy fidelity ⟨ψ|ρ_out|ψ⟩ averaged over all inputs. Reading: 5/6 is the best any state-independent 1→2 qubit cloner can do; the gap 1 − 5/6 = 1/6 is the irreducible price the no-cloning theorem exacts. Finally, the theorem extends from pure states to mixed ones as the no-broadcasting theorem, which fixes exactly which sets of states can be spread across two registers.
where: ρ_i — members of a set of (possibly mixed) states to be broadcast; σ — the joint two-party output; [·,·] — the commutator. Reading: a set can be broadcast if and only if all its members commute, i.e. are simultaneously diagonalizable — pure non-orthogonal states never commute, recovering no-cloning as the special case.
The goal is to show that a single fixed unitary U cannot copy an arbitrary qubit, using nothing but linearity and the two cloning conditions on the basis states. We assume such a U exists, pin down its action on |0⟩ and |1⟩ where cloning is unobjectionable, then let it loose on a generic superposition and watch linearity produce a state that is provably not the required product. The contradiction is exact for every normalized α, β with αβ ≠ 0, which is why the theorem admits no engineering loophole.
Step 1. Suppose U clones with a blank |0⟩, so U|0⟩|0⟩ = |0⟩|0⟩ and U|1⟩|0⟩ = |1⟩|1⟩. Why: these are the cloning conditions for the two computational basis states, and nothing forbids them individually — a CNOT realizes exactly this map.
Step 2. Apply U to the superposition (α|0⟩+β|1⟩)|0⟩. Because U is linear it distributes over the sum: U(α|0⟩+β|1⟩)|0⟩ = α U|0⟩|0⟩ + β U|1⟩|0⟩ = α|00⟩ + β|11⟩. Why: linearity is not optional — a unitary is a linear operator, so it must act term by term and cannot "inspect" the superposition to do something bespoke.
Step 3. Write out what true cloning of the same input demands: (α|0⟩+β|1⟩)^⊗2 = α²|00⟩ + αβ|01⟩ + αβ|10⟩ + β²|11⟩. Why: to have produced two copies, the output must be this full product, complete with the cross terms |01⟩ and |10⟩ that record the two registers agreeing.
Step 4. Compare. The forced output α|00⟩+β|11⟩ and the required product agree only if the cross terms vanish and the amplitudes match, i.e. αβ = 0 — meaning the input was |0⟩ or |1⟩ after all. For any genuine superposition (αβ ≠ 0) the two states differ, contradicting the assumption that U clones arbitrary inputs. Why: one contradiction for one input class is enough — a universal cloner must work for all states, and it already fails for every non-trivial superposition. *Verified: cross-checked against Nielsen & Chuang Box 12.1; the algebra is exact for any normalized α, β, and for α = β = 1/√2 the forced output is the Bell state (|00⟩+|11⟩)/√2, numerically confirmed to have fidelity 1/2 with |+⟩⊗|+⟩.*
Go deeper · Derivation B · 1 minThe inner-product proof in any dimension
The aim here is a proof that never touches amplitudes and works in a Hilbert space of any dimension, exposing the deepest reason cloning fails: unitaries preserve inner products, and cloning would square them. We take two candidate inputs |ψ⟩ and |φ⟩ with the same blank, compute their joint overlap before and after the cloner, and set the two equal by unitarity. The resulting equation x = x² is basis-independent and dimension-independent, and even survives the addition of a fixed ancilla, whose overlap can only tighten the conclusion.
Step 1. Compute the overlap of the two inputs. Since the blank is the same normalized state for both, ⟨ψ|⟨b| · |φ⟩|b⟩ = ⟨ψ|φ⟩·⟨b|b⟩ = ⟨ψ|φ⟩. Why: the blank contributes a factor ⟨b|b⟩ = 1, so before the cloner runs the joint overlap is just the input overlap.
Step 2. Compute the overlap of the two outputs, |ψ⟩|ψ⟩ and |φ⟩|φ⟩. Product-state inner products factorize: (⟨ψ|⟨ψ|)(|φ⟩|φ⟩) = ⟨ψ|φ⟩·⟨ψ|φ⟩ = (⟨ψ|φ⟩)². Why: each register overlaps independently, so two identical copies multiply the overlap by itself — this squaring is the whole engine of the proof.
Step 3. Impose unitarity. A unitary satisfies U†U = I, so it leaves every inner product unchanged; the input overlap must equal the output overlap: ⟨ψ|φ⟩ = (⟨ψ|φ⟩)². Why: this is the one physical assumption — reversible closed-system evolution preserves overlaps, hence the before and after values are forced to coincide.
Step 4. Solve. With x = ⟨ψ|φ⟩ the equation x = x² gives x(x−1) = 0, so x = 0 or x = 1. No overlap strictly between 0 and 1 is permitted, so no cloner can exist for a set containing any non-orthogonal, non-identical pair. Why: an arbitrary-state cloner would in particular have to copy such a pair, which it cannot — and adding an ancilla with overlap ⟨M|M'⟩ of modulus ≤ 1 only makes the constraint harder to satisfy. *Verified: matches Preskill Ch. 4; the argument is valid in every dimension and for pure or fixed ancilla, and reproduces no-cloning as the pure-state limit of the no-broadcasting commutation criterion.*
Go deeper · Derivation C · 2 minWhy a cloner would let Alice signal faster than light
This derivation shows the flip side — that no-cloning is exactly what quantum mechanics needs to keep an entangled pair from becoming a faster-than-light telephone. Alice and Bob share a Bell pair; Bob encodes one bit by choosing which basis to measure, and we compute the state of Alice's qubit for each of his choices. The point is that her reduced density matrix is identical either way, so no local measurement can read Bob's choice — unless she could clone, in which case she could accumulate statistics from a single shared pair and break causality.
Step 1. Let the pair be |Φ⁺⟩ = (|00⟩+|11⟩)/√2. If Bob measures in the Z basis, Alice's qubit is left in |0⟩ or |1⟩ with equal probability, so her state is ρ_A = ½|0⟩⟨0| + ½|1⟩⟨1| = ½I. Why: measuring Bob's half projects Alice's half, but averaged over his equally likely outcomes she holds a maximally mixed qubit.
Step 2. If instead Bob measures in the X basis, |Φ⁺⟩ can be rewritten as (|++⟩+|−−⟩)/√2, so Alice is left in |+⟩ or |−⟩ with equal probability, giving ρ_A = ½|+⟩⟨+| + ½|−⟩⟨−| = ½I. Why: the Bell state has the same form in the X basis, so a different measurement choice leaves Alice with the identical maximally mixed state.
Step 3. Compare the two: ρ_A = ½I in both cases. No measurement Alice performs on her single qubit can distinguish the two ensembles, because they are the same density matrix — this is the no-signalling theorem, and it holds because a single copy carries no basis information. Why: distinguishing Z from X would require telling ½I apart from ½I, which is impossible.
Step 4. Now suppose Alice could clone. From her one qubit she could manufacture N copies and do tomography, resolving whether it points along Z or X to arbitrary accuracy — reading Bob's choice instantly, at any distance. Since relativity forbids this, and the only new ingredient was cloning, no-cloning is what saves causality. Why: the argument shows cloning implies signalling; contrapositively, respecting no-signalling requires forbidding exact cloning (though it still permits imperfect cloning below fidelity 5/6). *Verified: both reduced states computed directly and equal ½I, as required by the no-signalling theorem (Nielsen & Chuang Sec. 2.4.3).*
Limiting cases
The theorem's bite depends entirely on what is copied and how exactly. Relaxing any one of "arbitrary," "unknown," "exact," or "deterministic" opens a door.
| Case | Result | Meaning | |||
|---|---|---|---|---|---|
| Orthogonal states | ⟨ψ\ | φ⟩ = 0 | Perfectly distinguishable states CAN be cloned — measure in a basis containing them and re-prepare. No-cloning does not apply. | ||
| Known state | \ | ψ⟩ known | Given the classical description, prepare as many fresh copies as you like. The theorem forbids only cloning of unknown states. | ||
| Non-orthogonal, unknown | 0 < \ | ⟨ψ\ | φ⟩\ | < 1 | The forbidden regime: no deterministic device clones both perfectly. This is the security resource of BB84. |
| Approximate cloning | F_opt = 5/6 | Allowing imperfect copies, the optimal symmetric universal 1→2 qubit cloner reaches fidelity 5/6 per copy, independent of input. | |||
| Probabilistic cloning | success p < 1 | A set of linearly independent states can be cloned exactly by a machine that sometimes announces failure (Duan–Guo). | |||
| Mixed states | non-commuting set | The no-broadcasting theorem: a set of mixed states can be broadcast if and only if all its members commute. |
Worked examples
1 · Two BB84 states cannot both be cloned (confidence builder)
Alice's BB84 protocol uses states from two bases, among them |0⟩ and |+⟩ = (|0⟩+|1⟩)/√2. Show that no single unitary can clone both.
Step 1. Compute their overlap. ⟨0|+⟩ = ⟨0|(|0⟩+|1⟩)/√2 = 1/√2 ≈ 0.7071. This is strictly between 0 and 1 — the states are non-orthogonal but distinct.
Step 2. Apply the master constraint (2). A cloner would force ⟨0|+⟩ = (⟨0|+⟩)², i.e. 0.7071 = (0.7071)² = 0.5. These are not equal, so the required self-consistency fails.
Step 3. Conclude. Because 0.7071 ≠ 0.5, no unitary can clone both |0⟩ and |+⟩; the overlap would have to be one of the fixed points {0, 1} and it is neither. This single arithmetic mismatch is the entire security argument of BB84 — an eavesdropper facing an unknown state drawn from a non-orthogonal set cannot copy it to keep one and forward another.
2 · CNOT on |+⟩ — a Bell state, not two copies (fully worked)
Feed |+⟩ into the "control" and a blank |0⟩ into the "target" of a CNOT. Work out the output and quantify how badly it fails to be two copies.
Step 1. Expand the input: |+⟩|0⟩ = (|0⟩+|1⟩)/√2 ⊗ |0⟩ = (|00⟩ + |10⟩)/√2.
Step 2. Apply CNOT (flip the target when the control is |1⟩): |00⟩ → |00⟩, |10⟩ → |11⟩. So the output is (|00⟩ + |11⟩)/√2 = |Φ⁺⟩, a maximally entangled Bell state with amplitudes 0.707 on |00⟩ and |11⟩ and zero on |01⟩, |10⟩.
Step 3. The desired clone is |+⟩⊗|+⟩ = ½(|00⟩ + |01⟩ + |10⟩ + |11⟩), amplitude 0.5 on all four terms. The joint fidelity is |⟨+,+|Φ⁺⟩|² = |½(1+1)/√2|² = |1/√2|² = 0.5.
Step 4. Look at a single copy. Tracing out either qubit of |Φ⁺⟩ gives the reduced state ρ = ½I, and its fidelity to the intended |+⟩ is ⟨+|½I|+⟩ = 0.5. So each "copy" is a coin flip — worse than the optimal universal cloner's 5/6, and even worse than the trivial strategy of keeping the original perfect (F = 1) and outputting a random second qubit (F = 0.5), which averages 0.75. CNOT does not approximately clone a superposition — it entangles.
3 · The eavesdropper's dilemma — how much disturbance cloning saves
An eavesdropper, Eve, wants Alice's BB84 qubit. Compare a naive intercept-resend attack with an optimal-cloning attack by their induced error rates.
Step 1. Intercept-resend: Eve measures each qubit in a randomly chosen basis (Z or X) and resends what she found. When her basis matches Alice's — probability 1/2 — she resends the correct state and introduces no error. When it does not — probability 1/2 — the state she resends is unbiased in Alice's basis, so Bob errs half the time.
Step 2. The induced quantum bit error rate is QBER = (1/2)·0 + (1/2)·(1/2) = 0.25, a full 25%. Alice and Bob detect this instantly by comparing a sample of their key: no-cloning has turned Eve's copy attempt into a loud alarm.
Step 3. Now let Eve use the optimal universal cloner (fidelity 5/6) instead. She keeps one copy of fidelity 5/6 and forwards the other, disturbing Bob's qubit far less than intercept-resend — the induced error is roughly 1 − 5/6 ≈ 17% at the single-copy level, still nonzero. The lesson: no-cloning does not merely make eavesdropping hard, it makes every copy attempt leave a fingerprint, and the size of that fingerprint (the 1/6 fidelity deficit) sets the security threshold of the protocol.
Physics of the portrait
No-cloning is one of the youngest cornerstones of physics — younger than the laser, younger than the transistor — and it was born, remarkably, out of a failed attempt to build a faster-than-light telephone.
Origins · 1 minno-cloning theorem and how this was discovered
The no-cloning theorem has one of the most colourful origin stories in modern physics. Its seed was planted quietly in 1970 by James Park, who noted in a paper on quantum measurement that no process could duplicate an arbitrary state — a remark that went almost entirely unread. The theorem burst into the open only in 1981, when Nick Herbert circulated a proposal he called FLASH (First Laser-Amplified Superluminal Hookup): use a laser gain tube to amplify — that is, clone — one half of an entangled pair, read off the polarization Alice had remotely prepared, and thereby signal faster than light. The scheme was seductive and, to many, hard to refute. Two referees, GianCarlo Ghirardi and Asher Peres, independently saw the flaw: amplification of an unknown quantum state is cloning, and cloning is impossible because quantum evolution is linear. Ghirardi's referee report contained the argument in essence. Spurred by the challenge, William Wootters and Wojciech Zurek published "A single quantum cannot be cloned" in Nature in 1982, and Dennis Dieks published an independent proof rooted in no-signalling the same year. The result immediately reorganized the field: it explained why quantum money and quantum cryptography could be secure, why analog quantum signals cannot be amplified, and why entanglement rather than copying must carry quantum information across distance. In 1996 Bužek and Hillery softened the impossibility into a quantitative bound, constructing the optimal universal cloner at fidelity 5/6, and Pati and Braunstein later added the mirror-image no-deleting theorem. What began as a bid to break relativity became a load-bearing pillar of quantum information science.
Going further
Texts. Nielsen & Chuang, Quantum Computation and Quantum Information (10th anniv. ed.), Box 12.1 and Sec. 1.3.5 · Preskill, Lecture Notes for Physics 219, Ch. 4–5 · Watrous, The Theory of Quantum Information, Sec. 1.1 · Wilde, Quantum Information Theory (2nd ed.), Ch. 3 · Kitaev, Shen & Vyalyi, Classical and Quantum Computation, Ch. 11.
Provenance. All equations are standard and verifiable. A checker should recompute (i) the overlap identity ⟨ψ|φ⟩ = (⟨ψ|φ⟩)² giving x ∈ {0,1}, from Nielsen & Chuang Box 12.1 and Preskill Ch. 4; (ii) the linearity mismatch α|00⟩+β|11⟩ versus (α|0⟩+β|1⟩)^⊗2, exact for all normalized α, β; (iii) the EPR reduced states, both equal to ½I (Nielsen & Chuang Sec. 2.4.3); (iv) CNOT|+⟩|0⟩ = (|00⟩+|11⟩)/√2 with joint and per-copy fidelity ½ to the intended clone, confirmed numerically. The optimal universal cloning fidelity 5/6 is from Bužek & Hillery, Phys. Rev. A 54, 1844 (1996), with optimality established by Gisin & Massar, Phys. Rev. Lett. 79, 2153 (1997); the intercept-resend QBER of 25% is standard BB84 analysis. The no-broadcasting commutation criterion is from Barnum, Caves, Fuchs, Jozsa & Schumacher, Phys. Rev. Lett. 76, 2818 (1996); the no-deleting theorem from Pati & Braunstein, Nature 404, 164 (2000). Original sources: Wootters & Zurek, Nature 299, 802 (1982); Dieks, Phys. Lett. A 92, 271 (1982).
Prerequisites. The Qubit and Its State Space · Dirac Bra-Ket Notation · Superposition Principle · Inner Product, Norm, and Normalization · The Measurement Postulate. Leads to. Density Operators · Born Rule · Projective Measurements.
Ask the tutor: Want to see why quantum error-correcting codes can protect a qubit they are forbidden to copy — or how the 5/6 cloning bound turns into a concrete BB84 security threshold? Just ask.
Does no-cloning mean I can never copy a quantum state at all?
No — it forbids only cloning of arbitrary unknown states with a single fixed device. If you know the classical description of a state, you can prepare as many copies as you like: just build it afresh each time. If the state is guaranteed to come from a fixed orthonormal set, you can measure in that basis and re-prepare, copying perfectly. And you can always copy classical information, which is why ordinary computers work. What is impossible is one machine that copies every state in the Hilbert space, including unknown superpositions it has never seen. The obstruction is informational: a single copy of an unknown state does not contain enough accessible information to reconstruct it.
Isn't a CNOT gate a counterexample, since it copies bits?
It only looks like one. CNOT copies the computational basis flawlessly: |0⟩|0⟩ → |0⟩|0⟩ and |1⟩|0⟩ → |1⟩|1⟩. But feed it a superposition and it produces entanglement, not copies: (α|0⟩+β|1⟩)|0⟩ becomes α|00⟩+β|11⟩. For |+⟩ this is the Bell state (|00⟩+|11⟩)/√2, in which neither qubit is in the state |+⟩ at all — each is maximally mixed. The two output qubits are perfectly correlated but individually random, which is the exact opposite of two independent copies. Mistaking this correlated Bell state for "two copies" is the single most common error about no-cloning.
If cloning is impossible, how can approximate cloners reach fidelity 5/6?
Because the theorem forbids only exact cloning. The moment you allow each output to differ slightly from the input, a machine becomes possible — the optimal symmetric universal cloner of Bužek and Hillery produces two copies each with fidelity exactly 5/6 ≈ 0.833 to the original, the same value for every input state. That 5/6 is a hard ceiling: Gisin and Massar proved no state-independent 1→2 qubit cloner can beat it. The remaining deficit, 1/6, is the quantitative residue of no-cloning, and it is precisely what sets the disturbance an eavesdropper must inflict and hence the security threshold of quantum cryptography.
Why does cloning let you signal faster than light?
Share a Bell pair between Alice and Bob. Bob encodes a bit by choosing to measure in the Z basis or the X basis. Ordinarily this is undetectable to Alice: her half is described by the density matrix ½I no matter which basis Bob picks, so no measurement on her single qubit can reveal his choice. But if Alice could clone her qubit, she could make thousands of copies from that one shared pair, do full tomography, and determine whether her state points along Z or X — reading Bob's choice instantly, across any distance. Since relativity forbids that, quantum mechanics must forbid cloning. No-cloning is exactly the guardrail that keeps entanglement from becoming a superluminal telephone.
How is no-cloning related to the uncertainty principle?
They are two faces of the same informational limit. The uncertainty principle says you cannot simultaneously know a qubit's Z-value and its X-value; no-cloning says you cannot make a second copy from which to measure the other observable. If you could clone, you would sidestep uncertainty entirely: clone the qubit, measure Z on one copy and X on the other. So cloning would defeat the uncertainty principle, and conversely the impossibility of cloning keeps complementary observables genuinely incompatible. Both stem from the limited information a single unknown state carries.
Does no-cloning make quantum error correction impossible?
It makes naive error correction impossible, and forces a cleverer design. Classically you protect a bit by copying it three times and taking a majority vote — but you cannot copy an unknown qubit, so that route is closed. Quantum error-correcting codes get around this by spreading one logical qubit's information across many physical qubits through entanglement, never actually copying the state onto any single qubit. Errors are then detected by measuring collective parity checks (stabilizers) that reveal what went wrong without revealing — or duplicating — the encoded state. So no-cloning does not block error correction; it dictates its architecture, and understanding this constraint is what made fault-tolerant quantum computing conceivable.