QuantumComputingBench · graph-grounded exercise book
Study state and channel representations, symmetry and tensor structure through primary-source-grounded problems. Planned curriculum anchors are not local Lean closure for the complete answers.
For every positive integer d, work on C^d tensor C^d and let Phi=|Phi_d><Phi_d|, where |Phi_d>=(1/sqrt(d))*sum_{i=1}^d |i,i>. Thus Phi is a rank-one trace-one projector. Put rho_d(x)=(1-x)*Phi+(x/d^2)*I for any real x, w…
primary source nl only
Fix d>=3. A d-dimensional Werner state has parameter alpha in [-1,1] and is k-extendible exactly when alpha >= (1-d)/k. An isotropic state has valid parameter x>=0 and is k-extendible exactly when x >= [d/(d+1)](1-1/k), …
primary source nl only
It is a theorem that if rho_AB is k-extendible and N : B -> B' is any quantum channel, then (id_A tensor N)(rho_AB) is again k-extendible; applying a channel to one side can only preserve or improve extendibility. One mi…
primary source nl only
The pure-symmetric de Finetti theorem bounds the trace distance between the k-body marginal of an n-party symmetric pure state and a convex mixture of i.i.d. pure states by sqrt(dk/(n - k)). The permutation-invariant mix…
primary source nl only
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.…
primary source nl only
The optimal N -> M approximate cloner achieves worst-case fidelity F*(T) = d_N/d_M where d_L = dim Sym^L(H) = binom(d + L - 1, L) with d = dim H. It is stated that F*(T) >= 1 - Kd/N with K = M - N. (a) Prove the lower bo…
primary source nl only
Fix a density operator rho on C^d with ordered spectrum r=(r_1>=...>=r_d>=0),sum_i r_i=1, and let n>=1. Weak Schur sampling measures rho^tensor(n) with the orthogonal Schur-Weyl projectors{P_lambda}, where lambda=(lambda…
primary source nl only
Call a bipartite state rho_AB k-extendible if there is a state rho_{A B_1 ... B_k} on A and k copies of B whose marginal on A B_i equals rho_AB for every i. Prove two structural facts and relate them. (a) Every separable…
primary source nl only
Let |psi>_AB be an arbitrary pure bipartite state. Prove that any state rho_{A B B'} with tr_{B'} rho_{A B B'} = psi_AB (the projector |psi><psi|_AB) must have the product form rho_{A B B'} = psi_AB tensor omega_{B'} for…
primary source nl only
Let rho_AB be k-extendible and let N : B -> B' be a quantum channel (a completely positive, trace-preserving linear map). Prove that sigma_{A B'} := (id_A tensor N)(rho_AB) is again k-extendible, by constructing an expli…
primary source nl only
Let |Phi+> = (1/sqrt(d)) sum_{i=1}^d |i>_A |i>_B be the standard maximally entangled state on C^d tensor C^d. (a) Prove the transpose trick: for every operator X in L(C^d), (X_A tensor I_B)|Phi+> = (I_A tensor X^T_B)|Phi…
primary source nl only
Let d_sym = dim Sym^n(C^d) = binom(n + d - 1, n). Prove the integral formula Pi_sym = d_sym integral dphi |phi><phi|^{tensor n}, where Pi_sym is the orthogonal projector onto the symmetric subspace Sym^n(C^d) and dphi is…
primary source nl only
Consider the candidate N -> M cloning map T(X) = (d_N/d_M) Pi_M (X tensor I^{tensor (M-N)}) Pi_M, where Pi_M projects onto Sym^M(H), I is the identity on H, and d_L = dim Sym^L(H). Prove that this map achieves the worst-…
primary source nl only
Given a quantum channel T : L(H^{tensor N}) -> L(H^{tensor M}), define its twirled version by averaging over the Haar measure on U_d: Tbar(X) := integral_{U_d} dU (U^dagger)^{tensor M} T( U^{tensor N} X (U^dagger)^{tenso…
primary source nl only
One measures rho^{tensor n}, where rho is an unknown density operator on C^d, to estimate its sorted eigenvalue vector r(rho). Assess the unqualified assertion that restricting all measurement effects to commute with BOT…
primary source nl only
Let rho be a state on A^{tensor n}, and let rhobar = (1/n!) sum_{pi in S_n} Q_pi rho Q_pi^dagger be its symmetrization (Q_pi the permutation operators). Let Pi_n be the projector onto the symmetric subspace Sym^n(H_A). P…
primary source nl only
Let H=C^d with d>=1, and integers M>=N>=1. Let Pi_L project onto the symmetric subspace of H tensor L and d_L its dimension. Define T(X)=(d_N/d_M)Pi_M(X tensor I^{tensor(M-N)})Pi_M. Prove T is completely positive and der…
primary source nl only
In the de Finetti proof one meets the average squared fidelity integral dphi p(phi) F(tilde-phi, phi^{tensor k})^2 = R, where R = [ (m+d-1)!/m! ] / [ (k+m+d-1)!/(k+m)! ] simplifies to the ratio of two products of (d-1) c…
primary source nl only
Write Sym^n(C^d) for the vectors fixed by every tensor-factor permutation, and q_d(n) for its dimension. (a) For d>=1 and integer n>=0, construct and normalize an occupation-number basis and prove q_d(n)=binom(n+d-1,n) b…
primary source nl only