Eect of the measurement is that the new state is

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Unformatted text preview: the measurement. Intuitively, a measurement provides the only way of reaching into the Hilbert space to probe the quantum state vector. In general this is done by selecting an orthonormal basis |e0 ￿ , . . . , |ek−1 ￿. The outcome of the measurement is |ej ￿ with probability equal to the square of the length of the projection of the state vector ψ on |ej ￿. A consequence of performing the measurement is that the new state vector is |ej ￿. Thus measurement may be regarded as a probabilistic rule for projecting the state vector onto one of the vectors of the orthonormal measurement basis. Some of you might be puzzled about how a measurement is carried out physically? We will get to that soon when we give more explicit examples of quantum systems. 1.7 Qubits Qubits (pronounced “cue-bit”) or quantum bits are basic building blocks that encompass all fundamental quantum phenomena. They provide a mathemat- 12 CHAPTER 1. INTRODUCTION ically simple framework in which to introduce the basic concepts of quantum physics. Qubits are 2-state quantum systems. For example, if we set k = 2, the electron in the Hydrogen atom can be in the ground state or the first excited state, or any superposition of the two. We shall see more examples of qubits soon. The state of a qubit can be written as a unit (column) vector ( α ) ∈ C2 . β In Dirac notation, this may be written as: | ψ ￿ = α | 0￿ + β | 1￿ with α, β ∈ C and |α|2 + |β |2 = 1. This linear superposition |ψ ￿ = α |0￿ + β |1￿ is part of the private world of the electron. For us to know the electron’s state, we must make a measurement. Making a measurement gives us a single classical bit of information — 0 or 1. The simplest measurement is in the standard basis, and measuring |ψ ￿ in this {|0￿ , |1￿} basis yields 0 with probability |α|2 , and 1 with probability |β |2 . One important aspect of the measurement process is that it alters the state of the qubit: the effect of the measurement is that the new state is exactly the outcome of the measurement. I.e., if the outcome of the measurement of |ψ ￿ = α |0￿ + β |1￿ yields 0, then following the measurement, the qubit is in state |0￿. This implies that you cannot collect any additional information about α, β by repeating the measurement. More generally, we may choose any orthogonal basis {|v ￿ , |w￿} and measure the qubit in that basis. To do this, we rewrite our state in that basis: |ψ ￿ = α￿ |v ￿ + β ￿ |w￿. The outcome is v with probability |α￿ | 2 , and |w￿ with probability |β ￿ | 2 . If the outcome of the measurement on |ψ ￿ yields |v ￿, then as before, the the qubit is then in state |v ￿. Examples of Qubits Atomic Orbitals The electrons within an atom exist in quantized energy levels. Qualitatively these electronic orbits (or “orbitals” as we like to call them) can be thought of as resonating standing waves, in close analogy to the vibrating waves one observe...
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This document was uploaded on 09/22/2013.

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