We propose an efficiently measurable lower bound on quantum process fidelity of N-qubit controlled-Z gates. This bound is determined by average output state fidelities for N partially conjugate product bases. A distinct advantage of our approach is that only fidelities with product states need to be measured while keeping the total number of measurements much smaller than what is necessary for full quantum process tomography. As an application, we use this method to experimentally estimate quantum process fidelity F of a three-qubit linear optical quantum Toffoli gate and we find that F≥0.83. We also demonstrate the entangling capability of the gate by preparing Greenberger-Horne-Zeilinger-type three-qubit entangled states from input product states.

U J ⊗ I m (j) J , (S.6) which induces the Hilbert space decomposition H j ⊗H = J∈Irr(U * j ⊗U ) H J ⊗ H m (j) J . Let us denote by j JK the set of values of j such thatJ |. Given this the left hand side of Eq. (S.22) reads ψ|R s |ψ = J λ J |I I| + ν J I − 1 d |I I| , (S.7) U J ⊗ I m (j) J , (S.23) which induces the Hilbert space decompositionFirst, we notice that the multiplicity spaces H m (j) J and H m (j) K are one dimensional and therefore I m (j) J are rank one. From the Schur-Weyl duality, any irreducible representation U j of SU (d) is in correspondence with a young diagram Y j . The defining representation U is represented by a single box . One can verify that there cannot be two equivalent Young diagrams in the decomposition Y j × = K Y K . For a more detailed treatment we refer to [4]. Then we have thatj) J ) = 1 we stress that I m (j) J |I m (j ) J = δ j,j and H m JJ = span({|I m (j) J }, j ∈ j JJ ). From Eq. (S.25) the commutation relation of Eq. (S.21) becomes [R s , JK U J ⊗ V K ⊗ I m JK ] = 0, which, thanks to the Schur's lemma, gives R s = J,K I J ⊗ I K ⊗ s (JK) , (S.26) where s (JK) ∈ L(H m JK ), s (JK) ≥ 0. Due to |I I| being a rank one operator and R s being the sum of the positive operators from Eq. (S.26) we have that Eq. (S.22) holds if and only if ψ|I J ⊗ I K ⊗ s (JK) |ψ = λ JK |I I| ∀J, K. (S.27) From the identity I j ⊗ I = J∈Irr(U * j ⊗U ) I J ⊗ I m (j) J (we remind that I m (j) J has rank one), we obtain |ψ |I = j J∈Irr(U * j ⊗U ) p j d j |I J |I m (j) J = J |I J |φ J (S.28) |φ J : = j∈j JJ p j d j |I m (j) J . (S.29) Using Eqs. (S.26), (S.28) into λ = 1 d 2 I| ψ|R s |ψ |I we obtain λ = J λ JJ λ JJ = d J d 2 φ J |s (JJ) |φ J (S.30) where the λ JK 's were defined in Eq. (S.27). It is now easy to show that we can assume R s = J I J ⊗ I J ⊗ s (J) , (S.31) where s (J) := j,j ∈j JJ s (J) jj |I m (j) J I m (j ) J |. Indeed, let

We analyze quantum algorithms for cloning of a quantum measurement. Our aim is to mimic two uses of a device performing an unknown von Neumann measurement with a single use of the device. When the unknown device has to be used before the bipartite state to be measured is available we talk about 1 → 2 learning of the measurement, otherwise the task is called 1 → 2 cloning of a measurement. We perform the optimization for both learning and cloning for arbitrary dimension of the Hilbert space. For 1 → 2 cloning we also propose a simple quantum network that realizes the optimal strategy.