QuantumComputinglib learn · inspect · formalize
Checked on this commit 4,524 public declarations commit 4e2c5b2c0be5 Build record

Quantum Information / Quantum Information Apps

Quantum Information Apps

Exercise PLAN-405

intermediate · boundary · test_public

primary source nl only. Current content-bound mathematical second pass accepted. No new Lean verification is claimed for this NL view.

Assumptions and conventions

      Question

      Suppose a fixed unitary U on A tensor B with reference state |0>_B is required to clone two particular states, U(|psi>_A |0>_B) = |psi>_A |psi>_B and U(|phi>_A |0>_B) = |phi>_A |phi>_B, where |psi>, |phi> are normalized. Determine exactly which pairs (|psi>, |phi>) can be simultaneously cloned by a single such unitary. Derive the constraint on the overlap <psi|phi> from unitarity, solve it, and interpret the two surviving possibilities. Then explain why this makes exact cloning of an unknown state drawn from a non-orthogonal set impossible, and give a concrete two-state qubit example that cannot be cloned.

      Hint: mathematical route

      A single cloning unitary can serve a pair only if the states are orthogonal or equal, since unitarity forces the overlap to satisfy z = z^2; e.g. |0> and |+> cannot be cloned.

      1. Unitarity gives <psi|phi> = <psi|phi>^2.
      2. The only solutions are orthogonal or identical states.
      3. A non-orthogonal distinct pair such as |0>,|+> is unclonable.
      Reference solution

      Unitarity preserves inner products: <psi|phi> = (<psi| tensor <0|) U^dagger U (|phi> tensor |0>) = (<psi| tensor <psi|)(|phi> tensor |phi>) = <psi|phi>^2. Writing z = <psi|phi>, the constraint is z = z^2, i.e. z(z - 1) = 0, giving z = 0 or z = 1. Hence a single cloning unitary can serve the pair only if the two states are either orthogonal (<psi|phi> = 0) or identical (z = 1 forces |phi> = |psi> up to a phase). There is no middle ground: any pair with 0 < |<psi|phi>| < 1 is forbidden. Consequently, if a device must clone every state in a set containing two distinct non-orthogonal states, it cannot exist, because the fixed-point equation z = z^2 is violated for that pair. Concrete qubit example: |psi> = |0> and |phi> = |+> = (|0> + |1>)/sqrt(2) have overlap <0|+> = 1/sqrt(2), satisfying neither z = 0 nor z = 1; therefore no unitary can clone both |0> and |+>. (If one only ever needed to clone states from an orthonormal set, e.g. |0> and |1>, cloning is possible -- exactly the classical copyable case.) The boundary is sharp: clonability of a pair holds precisely on the set {orthogonal} union {equal}.

      Assessment rubric
      1. Derives <psi|phi> = <psi|phi>^2 from unitarity of U (1 weight)
      2. Solves z = z^2 to get z = 0 or z = 1 and interprets them as orthogonal or equal states (1 weight)
      3. Concludes a non-orthogonal distinct pair cannot be cloned by any single unitary (1 weight)
      4. Gives a concrete qubit counterexample (|0> and |+>, overlap 1/sqrt(2)) that is unclonable (1 weight)

      Graph-grounded prerequisites

      These are support/source anchors, not an automatically extracted theorem-dependency proof.

      • Operator and semigroup theory (frozen curriculum anchor; see graph registry)
      • Matrix analysis (frozen curriculum anchor; see graph registry)

      Download the shared core registry

      Lean view and evidence boundary

      No aligned Lean view has been admitted for this native core in this textbook checkpoint. This is not a claim that the underlying mathematics lacks a formal proof. Contributions must preserve this exact target contract and split.

      Read the Lean track and existing diagnostic warmups

      Sources and provenance

      Contract: 0809c8dd8df20e65c90d686ac2b236abdd0e2bb697d197fd95eca4bbda333ae5

      Item digest: c77d7ed511391ab1864f7706a6fc6b3a476a578c5cbf169260939dff0ae77fef

      Graph snapshot: qcb-snapshot-ffa644f0da3031a1ce754b9a

      Authored support steps: 3. This is not measured graph distance or model difficulty.