No-Deleting and No-Broadcasting Theorems

Classical information is promiscuous. A bit can be copied at will, overwritten at will, and shredded without trace. Quantum information obeys a far stricter accounting, and this article maps the exact ledger. The no-cloning theorem (Wootters–Zurek, Dieks, 1982) forbids duplicating an arbitrary unknown pure state. Its logical mirror image, the no-deleting theorem (Pati–Braunstein, 2000), is obtained by reversing the arrow of time: just as you cannot fabricate a copy of an unknown state from a blank, you cannot reversibly annihilate a known copy against a reservoir of identical copies and return that reservoir to a blank. The information is never expunged — at most it is relocated into an ancilla, the quantum analogue of moving a file rather than deleting it. The no-broadcasting theorem (Barnum, Caves, Fuchs, Jozsa, Schumacher, 1996) lifts these statements to mixed states and weakens the demand. Cloning insists on a perfect product copy; broadcasting asks only that each marginal of a joint output equal the input, allowing the two copies to be correlated or entangled. Even this softer request fails: a family {ρᵢ} can be broadcast by one fixed channel precisely when all the ρᵢ commute — when they are simultaneously diagonalizable and hence effectively classical. One engine drives every version: the linear, trace-preserving, completely positive (and for reversible dynamics, unitary) structure of quantum evolution. Cloning breaks linearity on superpositions; deleting breaks unitarity by the adjoint argument; broadcasting breaks the data-processing inequality for non-commuting states. These are constraints on deterministic, input-independent boxes — not on approximate, probabilistic, or commuting-family machines. We state each theorem precisely, derive it in full, and draw the exact boundary of what remains permitted.

Key takeaways

The no-cloning theorem forbids copying an unknown pure state; its time-reversed partner, the no-deleting theorem, forbids deterministically erasing one of two known copies against a blank reservoir.
The no-broadcasting theorem generalizes no-cloning to mixed states: a family of states is broadcastable by a single channel if and only if the states mutually commute, [ρᵢ, ρⱼ] = 0.
All three follow from linearity and unitarity of quantum evolution — equivalently, from the distinguishability constraints on completely positive trace-preserving maps.
Nothing is truly destroyed: the deleted copy reappears in the ancilla, so quantum information is conserved, not annihilated, by reversible dynamics.
The optimal universal 1→2 cloner reaches single-copy fidelity F = 5/6 ≈ 0.833; no-go theorems bound machines, not the approximate frontier.
Key idea: linearity plus unitarity (or trace preservation) forbids any fixed, state-independent process from copying an unknown pure state (no-cloning), reversibly erasing a known copy into a blank (no-deleting), or replicating the marginals of non-commuting states (no-broadcasting) — commuting, classical states escape all three.

The core insight

Linearity plus unitarity (or trace preservation) of quantum evolution forbids any fixed, state-independent process from copying an unknown pure state (no-cloning), reversibly erasing a known copy into a blank (no-deleting), or replicating the marginals of non-commuting states (no-broadcasting); commuting, classical states escape all three.

Where this lives today

The no-go family is not a museum of impossibility results — it is the working substrate of quantum technology. In quantum key distribution (BB84 and its descendants) no-cloning is the security proof itself: an eavesdropper intercepting qubits in unknown bases cannot copy them to keep one and forward another undisturbed, so any information gain leaves a measurable error footprint, and the 5/6 cloning ceiling directly bounds the adversary's knowledge of the key. No-deleting and its cousin no-hiding underwrite quantum error correction: a code protects information not by making it un-erasable but by spreading it, so environmental coupling merely moves errors into a correctable syndrome — recovery works because the information was never destroyed. No-broadcasting forbids any passive universal tap on a quantum channel and is why a circuit wire cannot fan out; engineers use SWAP to relocate states and controlled gates to build correlations, never a copy primitive. The approximate-cloning frontier is itself an active experimental field: optimal 1→2 and phase-covariant cloners have been realized in photonic, NMR, and trapped-ion platforms, both to benchmark eavesdropper power and as primitives in metrology. The open questions are quantitative — how these bounds trade against noise, finite resources, and network geometry in real devices.

The physical picture

Think of a quantum state as a direction on the Bloch sphere known only to whoever prepared it. A copier would have to point a second arrow the same way without ever measuring the first — but measurement is the only way to learn a direction, and any measurement of an unknown qubit collapses its superposition. The information you would need in order to copy is exactly the information the measurement postulate forbids you to read out. The obstruction is not a limit of engineering; it is baked into the linearity of the state space: a gate that copies the two basis states must, by linearity, entangle any superposition of them rather than duplicate it — and the first figure makes that failure concrete.

Figure

A CNOT copies the basis states 0 and 1 perfectly, but by linearity it turns the superposition +0 into the entangled Bell state, not into the product ++. The missing cross terms 01 and 10 are the no-cloning obstruction itself

No-deleting is this same movie run backwards. Feed a hypothetical cloning machine in reverse — two copies plus a blank in, one copy plus a blank out — and you have a deleting machine; reversing a deleter clones. Since cloning is impossible, so is reversible deletion, and the deleted information cannot vanish: it must be pushed into the ancilla. Broadcasting sits one rung softer, demanding only that each marginal reproduce the input. The second figure shows why even that fails except classically, and quantifies how close an approximate machine can come: the optimal universal cloner has a hard fidelity ceiling of 5/6 for two copies, sliding toward 2/3 as it spreads one input across ever more outputs.

Figure

The optimal universal symmetric cloner fidelity F = (2M+1)/(3M) for 1 input to M copies: a hard ceiling of 5/6 ≈ 0.833 at M = 2, falling monotonically toward the classical measure-and-prepare floor of 2/3 ≈ 0.667 as M grows. Perfect cloning F = 1 is unreachable for any M ≥ 2

The core equations

The cleanest statement of no-cloning starts from the machine one would build. A universal cloner is a fixed unitary U that copies every state onto a blank ancilla |b⟩. The demand that a single U work for all |ψ⟩ is exactly what collapses.

U\,(|\psi\rangle\otimes|b\rangle) = |\psi\rangle\otimes|\psi\rangle \quad \forall\,|\psi\rangle \qquad (1)

where: $|\psi\rangle$ — arbitrary unknown pure state to be copied; $|b\rangle$ — fixed blank ancilla; $U$ — a single unitary cloning machine, independent of $|\psi\rangle$ . The " $\forall\,|\psi\rangle$ " is the whole content: cloning one known state is trivial; the theorem is about one box copying every state. Apply the same U to two candidate inputs and take the inner product: unitarity preserves overlaps while the right-hand side squares them, so the overlap must satisfy an equation with only two solutions.

\langle\psi|\phi\rangle = \big(\langle\psi|\phi\rangle\big)^2 \qquad (2)

where: $\langle\psi|\phi\rangle$ — overlap (inner product) of two candidate input states. Reading: with $x = \langle\psi|\phi\rangle$ , the equation $x = x^2$ forces $x \in \{0,1\}$ — orthogonal ( $x=0$ ) or identical ( $x=1$ ); no arbitrary pair clones. The deleting machine is the time-reverse. Give it two identical copies plus an ancilla and ask it to overwrite the second with a fixed blank |Σ⟩ for every unknown |ψ⟩, letting the ancilla change.

U\,(|\psi\rangle\otimes|\psi\rangle\otimes|A\rangle) = |\psi\rangle\otimes|\Sigma\rangle\otimes|A_\psi\rangle \qquad (3)

where: $|\psi\rangle\otimes|\psi\rangle$ — two identical copies of an unknown state; $|\Sigma\rangle$ — fixed standard blank replacing the erased copy; $|A\rangle, |A_\psi\rangle$ — ancilla before and after, the latter possibly $\psi$ -dependent. Reading: linearity forces $|A_\psi\rangle = \alpha|A_0\rangle + \beta|A_1\rangle$ — the ancilla carries the very amplitudes that defined $|\psi\rangle$ , so deletion only relocates the state. Broadcasting weakens "perfect copy" to "correct marginals." A fixed channel 𝓔 broadcasts ρ if the joint output ρ_AB reproduces ρ on each subsystem after tracing out the other — the copies may be correlated, a strictly easier task than cloning.

\mathcal{E}(\rho)=\rho_{AB}, \qquad \mathrm{Tr}_B\,\rho_{AB}=\rho, \qquad \mathrm{Tr}_A\,\rho_{AB}=\rho \qquad (4)

where: $\mathcal{E}$ — fixed broadcasting channel, input-independent; $\rho_{AB}$ — joint two-system output; $\mathrm{Tr}_{A/B}$ — partial trace giving each marginal. Reading: cloning demands the product $\rho_{AB}=\rho\otimes\rho$ ; broadcasting demands only correct marginals — the weaker task, so forbidding it is the stronger theorem. The no-broadcasting theorem gives the sharp condition: a family is broadcastable by a single channel exactly when the states mutually commute — when they share an eigenbasis and are therefore, in that basis, ordinary classical probability distributions.

\{\rho_i\}\ \text{broadcastable} \iff [\rho_i,\rho_j]=0 \ \ \forall\, i,j \qquad (5)

where: $\{\rho_i\}$ — family of states one fixed channel must broadcast; $[\rho_i,\rho_j]=\rho_i\rho_j-\rho_j\rho_i$ — commutator, zero iff simultaneously diagonalizable. Reading: commuting = classical = broadcastable; any non-commuting pair (a $Z$ -basis and an $X$ -basis state) cannot be broadcast by one machine. Finally, the quantitative escape hatch: perfection is forbidden but approximation is not. The optimal universal symmetric cloner maps one copy to M copies with a fidelity exactly known and always below one.

F_{1\to M}=\frac{2M+1}{3M},\qquad F_{1\to2}=\frac{5}{6},\qquad F_{N\to M}=\frac{MN+M+N}{M(N+2)} \qquad (6)

where: $F_{1\to M}$ — single-copy fidelity of the optimal universal symmetric $1\to M$ qubit cloner; $F_{N\to M}$ — the Gisin–Massar generalization from $N$ input copies to $M$ ; $F$ — overlap of each output marginal with the ideal $|\psi\rangle$ . Reading: $F_{1\to2}=5/6\approx0.833$ ; as $M\to\infty$ , $F\to 2/3$ — the fidelity of simply measuring one qubit and re-preparing (the classical limit). Perfect cloning ( $F=1$ ) is unreachable for any $M\ge2$ .

Derivation ANo-cloning from linearity alone

The goal is to show that a universal cloner fails at the most elementary level — the bare linearity of quantum evolution, before unitarity is even invoked — and to locate the failure in a specific pair of missing amplitudes. We assume a fixed linear map that correctly copies the two computational basis states (a CNOT does exactly this), apply it to an equal superposition, and compare the output against what a genuine clone would require. The gap that appears is the no-cloning obstruction in its rawest form, and it is why a quantum wire cannot fan out.

Step 1. Assume a fixed linear map $L$ that clones the basis: $L(|0\rangle|b\rangle)=|00\rangle$ and $L(|1\rangle|b\rangle)=|11\rangle$ . Why: this is not a straw man — a CNOT with the blank as target achieves precisely this, so basis cloning is genuinely available and the obstruction must live elsewhere.

Step 2. Feed in the superposition $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ . Linearity distributes $L$ over the sum: $L\big((\alpha|0\rangle+\beta|1\rangle)|b\rangle\big)=\alpha|00\rangle+\beta|11\rangle$ . Why: a linear map cannot do anything except act term by term, so the output is rigidly fixed once the basis action is fixed — there is no freedom left to shape a superposition.

Step 3. Write down what a true clone demands: $(\alpha|0\rangle+\beta|1\rangle)^{\otimes2}=\alpha^2|00\rangle+\alpha\beta|01\rangle+\alpha\beta|10\rangle+\beta^2|11\rangle$ . Why: a genuine copy is a product state, and expanding the product exposes cross terms $|01\rangle$ and $|10\rangle$ weighted by $\alpha\beta$ that the linear output simply does not contain.

Step 4. Compare. The two expressions agree only when $\alpha\beta=0$ , i.e. for a basis state. For any genuine superposition the linear output $\alpha|00\rangle+\beta|11\rangle$ is an entangled state, not the product copy. Why: this pins the failure to the cross terms — a single fixed linear box cannot both copy the basis and copy their superpositions, which is the no-cloning theorem. *At $\alpha=\beta=1/\sqrt2$ the output is the Bell state $(|00\rangle+|11\rangle)/\sqrt2$ , whose reduced state has cloning fidelity $\langle+|\tfrac{I}{2}|+\rangle=1/2$ , far below one.*

Go deeper · Derivation B · 1 minNo-deleting and the relocation of information

Derivation BNo-deleting and the relocation of information

The aim is to show that reversible deletion is impossible and, more informatively, to see exactly where the "deleted" quantum information goes. We posit a unitary deleting machine that, fed two identical copies of an unknown qubit, overwrites the second with a fixed blank while leaving the first intact. Fixing its action on the basis and demanding consistency on superpositions forces the ancilla to encode the state that was supposedly erased — so annihilation becomes mere transport.

Step 1. Define the machine on basis inputs: $U|0\,0\,A\rangle=|0\,\Sigma\,A_0\rangle$ and $U|1\,1\,A\rangle=|1\,\Sigma\,A_1\rangle$ . The first slot keeps the surviving copy, the second is reset to the blank $|\Sigma\rangle$ , and the ancilla lands in some $|A_0\rangle$ or $|A_1\rangle$ . Why: any deleting machine must at least handle the classical cases, and these fix its behaviour on a spanning set.

Step 3. Demand the machine output the fixed blank in the second slot for every $\psi$ : $U(|\psi\rangle|\psi\rangle|A\rangle)=|\psi\rangle|\Sigma\rangle|A_\psi\rangle$ . Matching against the linear image forces $|A_\psi\rangle=\alpha|A_0\rangle+\beta|A_1\rangle$ . Why: the only way the second register can become a $\psi$ -independent blank is for the amplitudes $(\alpha,\beta)$ to be carried off into the ancilla — the information is moved, not destroyed.

Step 4. Impose unitarity. Norm preservation of $\alpha|A_0\rangle+\beta|A_1\rangle$ for every $(\alpha,\beta)$ requires $\langle A_0|A_1\rangle=0$ with $|A_0\rangle,|A_1\rangle$ each normalized — an orthogonal ancilla encoding that is just the state copied elsewhere. Any attempt to make $|A_\psi\rangle$ a single fixed blank collapses $\alpha,\beta$ and breaks linearity. Why: reversible deletion would demand the ancilla both encode the state and not encode it; the contradiction is the no-deleting theorem, and its resolution is that quantum information is conserved. *The ancilla necessarily carries the deleted state — a conclusion echoed by the no-hiding theorem (Braunstein–Pati, 2007), which shows that lost information always reappears in the environment.*

Go deeper · Derivation C · 1 minNo-broadcasting forces commutativity

Derivation CNo-broadcasting forces commutativity

The goal is to prove the sharp criterion that a single channel can broadcast a family of states only if they mutually commute, using the one tool that makes the argument airtight: monotonicity of quantum relative entropy under any CPTP map, including the partial trace. We take two states, assume one channel broadcasts both, and squeeze their relative entropy between the input value and itself from two sides. That squeeze saturates the data-processing inequality on both marginals at once, which is possible only for commuting states.

Step 1. Let a fixed channel 𝓔 broadcast both $\rho_0$ and $\rho_1$ , so $\mathrm{Tr}_B\,\mathcal{E}(\rho_i)=\rho_i=\mathrm{Tr}_A\,\mathcal{E}(\rho_i)$ . Write the joint outputs as $\sigma_i=\mathcal{E}(\rho_i)$ . Why: this is the full broadcast hypothesis — both marginals of each joint output must return the original input, on both A and B.

Step 2. Apply monotonicity of relative entropy to the map 𝓔: $S(\rho_0\Vert\rho_1)\ge S(\sigma_0\Vert\sigma_1)$ . Distinguishability cannot grow under a physical channel. Why: the relative entropy $S(\rho\Vert\sigma)=\mathrm{Tr}\,\rho(\log\rho-\log\sigma)$ is the information-theoretic measure of how tellable-apart two states are, and no CPTP box can make two states more distinguishable than their inputs.

Step 3. Apply monotonicity again to the partial trace $\mathrm{Tr}_B$ : $S(\sigma_0\Vert\sigma_1)\ge S(\mathrm{Tr}_B\sigma_0\Vert\mathrm{Tr}_B\sigma_1)=S(\rho_0\Vert\rho_1)$ , using the broadcast condition. Chaining Steps 2–3 gives $S(\rho_0\Vert\rho_1)\ge S(\sigma_0\Vert\sigma_1)\ge S(\rho_0\Vert\rho_1)$ , so every inequality is an equality. Why: the two bounds pin the joint relative entropy exactly to the input value, saturating data processing — a rigid condition, since generically discarding system B loses distinguishing information.

Step 4. Saturation of relative-entropy monotonicity under $\mathrm{Tr}_B$ requires the recovery (Petz) condition, and demanding it simultaneously for the trace over A and over B forces the two states to share a common eigenbasis: $[\rho_0,\rho_1]=0$ . Commuting states, being jointly diagonal, are classical distributions a measure-and-copy channel broadcasts trivially. Why: only mutually diagonal states carry no complementary information to be lost on discarding a subsystem, so commutativity is necessary and sufficient — the Barnum–Caves–Fuchs–Jozsa–Schumacher theorem. *"Commuting iff broadcastable" is the central result, and it contains no-cloning as the pure-state special case.*

Limiting cases

Every one of these theorems binds only under a precise set of conditions. Relax any single condition — exactness, universality, determinism, non-commutativity — and a matching loophole opens. The table is the map of those escapes.

Case	Result	Meaning
Orthogonal states	⟨ψ	φ⟩ = 0 ⇒ cloning allowed	Distinguishable states can be measured and re-prepared; a CNOT copies the computational basis exactly.
Commuting states	[ρᵢ, ρⱼ] = 0 ⇒ broadcasting allowed	Simultaneously diagonalizable states are classical; measure in the common eigenbasis and copy the outcome.
Approximate cloning	F₁→₂ = 5/6 ≈ 0.833	The optimal universal symmetric 1→2 qubit cloner (Bužek–Hillery) copies each output with fidelity 5/6, below 1.
Probabilistic cloning	linearly independent {	ψᵢ⟩} ⇒ exact clone with p < 1	Duan–Guo: a finite set of linearly independent states can be cloned exactly, but only sometimes — flagged by a heralding outcome.
Information relocation		ψ⟩ₐ	0⟩_B →	0⟩ₐ	ψ⟩_B	A SWAP-type "deletion" is fine: the state is moved to the ancilla, never annihilated — fully consistent with no-deleting.

Common trapAssuming the no-go theorems forbid all copying, all deleting, and all tapping of quantum signals

They forbid far less than that. Each theorem rules out only a deterministic, state-independent, exact operation on unknown or non-commuting states. Orthogonal (distinguishable) states can be cloned freely — that is just measure-and-prepare. Commuting states can be broadcast freely — they are classical. Approximate cloning up to fidelity 5/6 is allowed and routinely built; probabilistic cloning of a linearly independent set is allowed with heralded success. And "no-deleting" is not violated by measuring a qubit and resetting it to |0⟩: that is irreversible and leaks the erased information to the apparatus — the relocation the theorem predicts, not the reversible annihilation it forbids.

Worked examples

1 · How badly CNOT fails to clone, and how well the optimal cloner does

Send |+⟩ = (|0⟩+|1⟩)/√2 through a CNOT (blank target |0⟩). Compute the output, the single-copy cloning fidelity of the naive scheme, and compare with the optimal universal cloner.

Step 1. Apply CNOT term by term: |+⟩|0⟩ = (|0⟩|0⟩ + |1⟩|0⟩)/√2 → (|00⟩ + |11⟩)/√2. The output is the Bell state Φ⁺, maximally entangled, not the product |+⟩|+⟩.

Step 2. Ask how good a "copy" register B is on its own. Trace out A: ρ_B = Tr_A |Φ⁺⟩⟨Φ⁺| = (|0⟩⟨0| + |1⟩⟨1|)/2 = I/2, the maximally mixed state. All memory of |+⟩ is gone from the marginal.

Step 3. Compute the fidelity of that marginal with the target: F = ⟨+| ρ_B |+⟩ = ⟨+| (I/2) |+⟩ = 1/2. The naive CNOT "cloner" scores only 0.5 on a superposition input — no better than a coin flip about the phase.

Step 4. Compare the best possible machine. The optimal universal symmetric 1→2 cloner reaches F₁→₂ = (2·2+1)/(3·2) = 5/6 ≈ 0.833 on every input, including |+⟩ — beating the naive scheme but still short of 1, the no-cloning ceiling. Naive CNOT fidelity = 1/2; optimal universal cloner = 5/6; perfect cloning = impossible.

2 · The overlap identity rules out a would-be cloner

Could a single unitary clone both |0⟩ and |+⟩? Test the necessary condition ⟨ψ|φ⟩ = ⟨ψ|φ⟩² directly.

Step 1. Overlap of the two candidate inputs: ⟨0|+⟩ = ⟨0|(|0⟩+|1⟩)/√2 = 1/√2 ≈ 0.707.

Step 2. A cloner preserves inner products but squares them on the copied side, so consistency demands ⟨0|+⟩ = ⟨0|+⟩², i.e. 0.707 =? (0.707)² = 0.5. The equality fails.

Step 3. Read off the general rule: x = x² has only the roots x = 0 and x = 1. The pair {|0⟩, |+⟩} has overlap strictly between these, so no unitary can clone both. Only orthogonal or identical states escape — which is why a QKD eavesdropper who does not know the basis cannot copy the intercepted qubit.

3 · Commuting states broadcast, non-commuting states do not

Compare two diagonal states with two states from complementary bases, and check the commutator criterion.

Step 1. Take ρ₀ = diag(0.8, 0.2) and ρ₁ = diag(0.3, 0.7) in the Z-basis. Both are diagonal, so [ρ₀, ρ₁] = 0: they commute. A measure-in-Z-then-copy channel outputs a classically correlated ρ_AB whose marginals are exactly ρ₀ or ρ₁ — they broadcast.

Step 2. Now take σ₀ = |0⟩⟨0| (a Z eigenstate) and σ₁ = |+⟩⟨+| (an X eigenstate). Their commutator [σ₀, σ₁] = (1/2)(|0⟩⟨1| − |1⟩⟨0|) ≠ 0: they do not commute.

Step 3. By the theorem, no single channel can broadcast {σ₀, σ₁}: doing so would let one party measure copy A in Z and another measure copy B in X, jointly extracting complementary information forbidden to a single qubit. Diagonal pair: broadcastable; Z-vs-X pair: not — exactly as the commutator predicts.

Physics of the portrait

The no-go theorems are usually filed under Wootters and Zurek, but the completion of the family — the deleting half, the conservation-of-information half — belongs to a later generation. The natural portrait for this node is the physicist who supplied the time-reversed partner of no-cloning and showed where lost quantum information hides.

Origins · 1 minArun K. Pati and how this was discovered

Arun K. Pati

b. 1966, Indian theoretical physicist · no-deleting theorem and the conservation of quantum information

Arun Kumar Pati works at the intersection of quantum information, foundations, and the geometry of quantum states. His most cited contribution, with Samuel Braunstein in Nature (2000), is the no-deleting theorem: the exact complement of no-cloning, proving that an unknown quantum state cannot be reversibly deleted against a copy — the information can only be relocated, never annihilated. Where Wootters, Zurek, and Dieks had shown in 1982 that quantum information cannot be freely created by copying, Pati and Braunstein closed the loop by showing it cannot be freely destroyed, a conservation law for quantum information under reversible dynamics. They extended the theme in 2007 with the no-hiding theorem, which proves that information apparently lost from a system (say, to decoherence) is not erased but resides entirely in the environment, in principle recoverable — later confirmed by NMR experiment. Together with the 1996 no-broadcasting theorem of Barnum, Caves, Fuchs, Jozsa, and Schumacher, this work turned a handful of "you cannot do that" slogans into a coherent account of what deterministic quantum maps can and cannot do, and made those limits the load-bearing walls of quantum cryptography and error correction.

History

1982

— Wootters, Zurek, and Dieks independently prove the no-cloning theorem

1996

— Barnum, Caves, Fuchs, Jozsa, and Schumacher prove the no-broadcasting theorem

1996

— Bužek and Hillery construct the optimal universal approximate quantum cloner

1998

— Duan and Guo show linearly independent states admit probabilistic exact cloning

2000

— Pati and Braunstein prove the no-deleting theorem

2007

— Braunstein and Pati prove the no-hiding theorem, locating lost information in the environment

Going further

Texts. Nielsen & Chuang, Quantum Computation and Quantum Information, §1.3.5 and Box 12.1 · Watrous, The Theory of Quantum Information, Ch. 4 · Wilde, Quantum Information Theory, Ch. 3 and 11 · Preskill, Lecture Notes on Quantum Computation, Ch. 4 · Kitaev, Shen & Vyalyi, Classical and Quantum Computation, §11. Key results to keep in mind: (1) the overlap identity ⟨ψ|φ⟩ = ⟨ψ|φ⟩² has roots {0,1}; (2) the linearity cross-term mismatch α|00⟩+β|11⟩ vs (α|0⟩+β|1⟩)^⊗2, and the Bell-state marginal fidelity ⟨+|(I/2)|+⟩ = 1/2; (3) the deleting-machine ancilla relation |A_ψ⟩ = α|A₀⟩ + β|A₁⟩; (4) the commuting-iff-broadcastable criterion via relative-entropy monotonicity; (5) the optimal cloner fidelities F₁→₂ = 5/6 and F₁→M = (2M+1)/(3M), and the Gisin–Massar N→M formula (MN+M+N)/(M(N+2)). Note: a labelling sometimes seen in the literature, "(2N+1)/(3N) for N→2N", mislabels the 1→M result; the correct formula is as given here.

Prerequisites. Quantum Circuit Model · CNOT Gate · SWAP Gate · Controlled-U Gates. Leads to. Quantum Key Distribution · Quantum Error Correction · Quantum Teleportation · Holevo Bound.

Ask the tutor: Ask me why the same cross-terms that block cloning are what let quantum error correction hide a logical qubit where no local measurement can reach it.

Isn't the no-deleting theorem just the no-cloning theorem stated backwards?

Almost, and that is exactly the point — but the two are genuinely independent statements, not one theorem written twice. No-cloning forbids the forward process (blank → copy); no-deleting forbids the reverse (copy → blank). Because unitary dynamics is reversible, the impossibility of one strongly suggests the impossibility of the other, and Pati and Braunstein made the argument rigorous. The extra content of no-deleting is where the information goes: it proves the erased amplitudes must reappear in the ancilla, giving a conservation law for quantum information that no-cloning alone does not state — complementary faces of the same reversibility.

How can no-broadcasting be stronger than no-cloning if it asks for less?

Because forbidding an easier task is a stronger statement. Cloning demands a perfect product copy, ρ_AB = ρ⊗ρ; broadcasting demands only that each marginal equal ρ, allowing the two outputs to be correlated or entangled. Broadcasting is strictly easier to achieve, so a theorem that rules it out rules out cloning automatically as the special case. The remarkable content is that even this weakened, correlation-tolerant copying is impossible for non-commuting states — so no-broadcasting contains no-cloning as the special case where the inputs are pure.

Doesn't measuring a qubit and resetting it to |0⟩ delete quantum information?

It deletes it for you, but not reversibly, so no-deleting is untouched. Measurement is irreversible and dumps the collapsed information into the apparatus and environment; resetting to |0⟩ then overwrites the register. The quantum information about the original superposition is not annihilated — it has leaked into the classical record and the environment, exactly the relocation the no-deleting and no-hiding theorems describe. What the theorem forbids is a unitary, reversible machine that takes two known copies and returns a clean blank with no trace anywhere. That would violate reversibility; measure-and-reset does not, because it is a lossy, entropy-increasing process from the start.

If perfect cloning is impossible, how do quantum repeaters and amplifiers work?

They never copy an unknown state. A quantum repeater extends entanglement by entanglement swapping and purification, consuming and re-establishing shared Bell pairs rather than duplicating a qubit — teleportation moves a state without cloning it. And there is no deterministic, noiseless quantum amplifier for an unknown state: no-cloning forbids it, and phase-insensitive amplifiers necessarily add noise. Where amplification is needed one either accepts that fundamental added noise or uses heralded, probabilistic schemes. Every working quantum-network primitive is built to relocate or entangle information, never to make an unauthorized second copy.

What exactly does "commuting" mean physically for two density operators?

Two density operators commute, [ρ₀, ρ₁] = 0, precisely when they can be diagonalized in the same basis. In that shared eigenbasis they are just two classical probability distributions over the same list of outcomes — there is no complementarity between them, no pair of incompatible measurements that trades sharpness on one for blurriness on the other. That is why they behave classically: you can measure once in the common basis, read out a definite outcome, and copy it freely. Non-commuting states, like a Z-eigenstate and an X-eigenstate, encode complementary information that no single measurement can capture, and complementarity is exactly what blocks broadcasting.

Does the no-deleting theorem conflict with thermodynamic erasure and Landauer's principle?

No — they describe different operations. Landauer's principle concerns irreversible erasure of a known classical bit, allowed at a cost of at least kT ln2 of dissipated heat per bit. No-deleting concerns reversible, unitary deletion of an unknown quantum copy, impossible at any cost. The two are compatible: Landauer erasure works precisely because it is irreversible and pushes information and entropy into the environment as heat — the relocation no-deleting permits. What no-deleting forbids is a cost-free, trace-free, reversible annihilation, which Landauer's setup never claims to provide.