By Dominik Janzing
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Additional info for Computer Science Approach to Quantum Control
Given some definition of macroscopic observables we consider macro-realism as the statement that for all macroscopic observables A the trace norm of the commutator [ρ, A] is small for all states ρ in nature. The consideration of the trace-norm is justified by the observation  that for any pure state ρ, the expression [A, ρ] tr coincides with the standard deviation of A up to the factor 2. Even though we do not have a definition for macroscopic observables, we have argued above2 that all observables of the form a := 1 n n aj j=1 are macroscopic, where aj is the 1-qubit operator a acting on qubit j.
However, it may be illustrated by the following: A shop sells batteries and collects those which are used up for recycling purposes. After almost all the full batteries are sold, the shop contains almost only empty batteries. Unfortunately, the owner is sloppy and he forgets to keep the full and empty batteries separately. If only one full battery is among the empty, its energy is lost from the point of view of a lazy shop-owner who does not want to search all his shelves for the full one. However, it is not lost from the fundamental point of view.
1. Start with the state |ψ0 := |0 ⊗n . 2. Perform a Hadamard gate on the first qubit and obtain √ |ψ1 := 1/ 2(|0 + |1 ) ⊗ |0 ⊗n−1 . 3. Perform a C-NOT controlled by qubit 1 with qubit 2 as target. This prepares a cat state on 2 qubits. 4. Given a cat state on a 2k -qubit cluster obtain a cat state on a 2k+1 qubit cluster by applying 2k C-NOT gates. Each of them is controlled by one qubit in the 2k cluster and has an arbitrary target qubit in the remaining set. To show that O(log2 n) is optimal we proved in  that every state prepared by a quantum circuit with depth d from a product state satisfies [ρ, a] 1 ≤ 2 d 2 .
Computer Science Approach to Quantum Control by Dominik Janzing