Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 10D (Quantum Information and Computation)

Apply the hypothetical quantum cloning device repeatedly to obtain $N$ copies of the unknown state. If $c=|⟨\varphi |\psi ⟩|<1$, the two possibilities arewhose overlap has modulus $c^N$. For equal prior probabilities, the Helstrom-Holevo bound gives optimal success probabilityThus, for any $\epsilon >0$, sufficiently many clones permit success probability above $1-\epsilon$. Orthogonal states already have $c=0$ and can be distinguished perfectly. This is clone-assisted asymptotic state discrimination.

Here distinct states means distinct rays. Vectors differing only by a global phase represent the same quantum state and cannot be distinguished.

Run the perfect discriminator on the supplied state and record whether the result is $i=0$ or $i=1$. For each $i$, choose a unitary $V_i$ satisfyingwhich is possible by extending $|{\varphi }_i⟩$ to an orthonormal basis. Prepare two fresh systems in $|0⟩|0⟩$ and, controlled by the classical outcome $i$, apply $V_i\otimes V_i$. The output isThe discriminator may destroy its input; the two freshly prepared outputs still implement cloning on this known pair. This is the construction that perfect discrimination implies cloning for a known state family.

Suppose first that the stated unitary and environment states exist. Preservation of the inner product givesThe environment states are normalized, so the Cauchy-Schwarz inequality gives $|⟨e_0|e_1⟩|\le 1$. Therefore

Conversely, writeand assume $|a|\le |b|$. If $b\ne 0$, put $c=a/b$, so $|c|\le 1$, and chooseThen $⟨e_0|e_1⟩=c$ and the desired output vectors have inner product $bc=a$, exactly matching the input vectors. If $b=0$, the inequality forces $a=0$, and one may take $|e_0⟩=|e_1⟩=|0⟩$.

In either case the input pair and output pair have the same Gram matrix. The isometry taking one pair to the other extends to a unitary matrix, so the required $U$ exists. Hencewhich is the environment-assisted two-state pure-state transformation criterion.