Quantum state expansion

It is experimentally demonstrated that an arbitrary quantum state of a single spin 1/2: ↓ + ↑ b a can be converted into a superposition of the two ferromagnetic states of a spin cluster: ↓↓ ↓↓ + ↑↑ ↑↑ b a . The physical system is a cluster of seven dipolar-coupled nuclear spins of single-labeled C-benzene molecules in a liquidcrystalline matrix. In this complex system, the pseudopure ground state and the required controlled unitary transformations have been implemented. The experimental scheme can be considered as an explicit model of quantum measurement. PACS: 03.65.-w, 03.67.Mn, 42.50.Dv, 76.60.-k Theory of quantum measurements, which describes a boundary between quantum and classical worlds [1,2], is the least established part of quantum theory. Different approaches to this problem lead to different interpretations of quantum mechanics. A serious difficulty in exploring this subject is that practical measuring devices are too complex to allow a detail analysis of their dynamics. It may be helpful to consider some simple and explicit models of quantum measurement, using systems with controllable quantum dynamics. One of possible experimental models is proposed and studied in this work. Let us consider a system of 1 + N spins 1/2 (qubits) in the initial state N N b a 0 0 0 0 ) 1 0 ( ψ 1 2 1 0 0 in − + = ... , 1 2 2 = + b a , (1) where the qubit notations k k ↑ ≡ 0 and k k ↓ ≡ 1 are used. In this state, the 0-th qubit is in some arbitrary state, defined by two complex coefficients a and b , while the qubits 1 to are in the ground state. Quantum logic circuit [3] N 1 , 0 2 , 1 1 , 2 , 1 CNOT CNOT CNOT CNOT U N N N N ... − − − = (2) is a chain of unitary controlled-not gates , which flip the target qubit n when the control qubit m is in the state n m CNOT , m 1 and do not change the qubit n when the qubit m is in the state m 0 . If the 0-th qubit is in the state 0 1 , it flips the qubit 1, the qubit 1 flips the qubit 2, and so on. A wave of flipped qubits, triggered by the 0-th qubit, propagates until it covers the entire system. As a result, the circuit (2) converts the initial state (1) into the final state N N N N b a U 1 1 1 1 0 0 0 0 ψ ψ 1 1 0 1 1 0 in out − − + = = ... ... . (3)