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Unformatted text preview: s on a tightly held piece of string. Two such individual levels can be
isolated to conﬁgure the basis states for a qubit. 1.7. QUBITS 13 Figure 1.3: Energy level diagram of an atom. Ground state and ﬁrst excited
state correspond to qubit levels, |0 and |1, respectively. Photon Polarization
Classically, a photon may be described as a traveling electromagnetic wave.
This description can be ﬂeshed out using Maxwell’s equations, but for our
purposes we will focus simply on the fact that an electromagnetic wave has a
polarization which describes the orientation of the electric ﬁeld oscillations (see
Fig. 1.4). So, for a given direction of photon motion, the photon’s polarization
axis might lie anywhere in a 2-d plane perpendicular to that motion. It is thus
natural to pick an orthonormal 2-d basis (such as and , or “vertical” and
“horizontal”) to describe the polarization state (i.e. polarization direction)
of a photon. In a quantum mechanical description, this 2-d nature of the
photon polarization is represented by a qubit, where the amplitude of the
overall polarization state in each basis vector is just the projection of the
polarization in that direction.
The polarization of a photon can be measured by using a polaroid ﬁlm
or a calcite crystal. A suitably oriented polaroid sheet transmits x-polarized
photons and absorbs y-polarized photons. Thus a photon that is in a superposition |φ = α |x + β |y is transmitted with probability |α|2 . If the photon
now encounters another polariod sheet with the same orientation, then it is
transmitted with probability 1. On the other hand, if the second polaroid
sheet has its axes crossed at right angles to the ﬁrst one, then if the photon is
transmitted by the ﬁrst polaroid, then it is deﬁnitely absorbed by the second
sheet. This pair of polarized sheets at right angles thus blocks all the light. A
somewhat counter-intuitive result is now obtained by interposing a third polariod sheet at a 45 degree angle between the ﬁrst two. Now a photon that is
transmitted by the ﬁrst sheet makes it through the next two with probability 14 CHAPTER 1. INTRODUCTION 1/4. Figure 1.4: Using the polarization state of light as the qubit. Horizontal polarization corresponds to qubit state, |ˆ, while vertical polarization corresponds
to qubit state, |ˆ.
y To see this ﬁrst observe that any photon transmitted through the ﬁrst
ﬁlter is in the state, |0. The probability this photon is transmitted through
the second ﬁlter is 1/2 since it is exactly the probability that a qubit in the
state |0 ends up in the state |+ when measured in the |+ , |− basis. We
can repeat this reasoning for the third ﬁlter, except now we have a qubit in
state |+ being measured in the |0 , |1-basis — the chance that the outcome
is |0 is once again 1/2. Spins
Like photon polarization, the spin of a (spin-1/2) particle is a two-state system,
and can be d...
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This document was uploaded on 09/22/2013.
- Fall '13