This reduces |ψ〉 to the post measurement state of |ψ’〉
〉
Postulate 4 – Composite of Quantum States
The composite state space H of a group of states {|ψ1〉, |ψ2〉, |ψ3〉,…, |ψn〉} is the tensor product of these states. If the subsystems are in the states |ψi〉, then the joint state of the entire composite system is
〉 system is in a decomposeable state. Otherwise, the system is in an entangled state.
For example, the state (|00 |11)
The property of entanglement is crucial in many of the important algorithms that portray the major benefits of QC, including superdense coding and quantum
teleportation [NC00, KLM07].
2.1.5. Measurement and Decoherence in Quantum Mechanics
Measurement in quantum mechanics cause any state held at superposition to collapse to an observable result. Decoherence is the phenomenon by which a quantum state held at superposition tends to “decay” into the basis states, resulting in a loss of the quantum state information (similar to noise in classical computing and communication theory).
In the following discussion we show that the measurement of an observable of a quantum system comprised of a single qubit will always yield an eigenvalue of the Hermitian matrix A representing the measurement operator.
We first need to prove the following fundamental Theorem:
Theorem 2.1: The eigenvalues λi,λj,.... of a Hermitian operator A are real, and the eigenkets ζi,ζj,.... of A corresponding to different eigenvalues λi,λj,.... are orthogonal.
Proof: Following [Saku94], by definition of eigenvalues and eigenvectors A|ζi〉=λi|ζi〉 (2.20) and because of the Hermitian properties,
〈ζj |A=λ*j〈ζj | (2.21)
By left multiplication of both sides of equation (2.14) by〈ζj|, right
multiplication of both sides of equation (2) by |ζj〉, and subtracting the two results, we get
If the eigenvalues are different, then since they are real, the first multiplicand of (2.22) cannot disappear. Thus we obtain the orthogonality property
0
| 〉=
〈ζj ζi , (ζj ≠ζj) (2.23) This completes the proof.
Before the measurement of the observable A, we can represent our system |ψ〉
(not necessarily just the single qubit of the question) by the following linear combination:
We now multiply on the left with 〈λ| and by applying the orthogonality property from theorem 2.1, we find that for all λ, and consequently, equation (2.17) becomes
∑
〉〈 〉When the measurement is performed, the system is “thrown to” one of the eigenstates of the observable A, say |λ〉. This is what Dirac meant in his 1958 book regarding the postulate that “a measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured.” In this sense, the result of the measurement yields one of the eigenvalues of the observable being measured. Since the observables are the quantum analogue of dynamical variables (e.g. position, spin, linear and angular momentum, energy, polarization), the theorem stating that the eigenvalues of the Hermitian operator are real is crucial. In other words, we cannot observe “complex” outcomes of measurements.
We note, of course, that if |ψ〉 is already in the state |λ〉, the measurement by A yields the result λ with certainty.
However, postulate 3 of quantum mechanics states that the probability of the system being thrown to or collapsed to a particular eigenstate | λ〉 is
Pλ = 〈λ|ψ〉2 (2.27) This postulate insures us that the probability conforms to the requirements that each eigenvalue of A has a nonnegative probability. It also provides
∑
=1λ λ
P (2.28)
Since we deal here with a single qubit, the Hermitian observable A has two eigenvalues, λ1 and λ2 each with a probability as shown in (2.24). It should be noted that it is possible to have the original qubit already prepared in the eigenstate | ζ1〉. In this
case, the same observable A will result in the eigenvalue λ1 with certainty. This is similar to the case of measurements in repeated tests [MM05].
The measurement problem is one the key concepts of quantum mechanics. An interesting discussion relates to the famous Schrödinger’s Cat Paradox [Schr35]. The single quantum bit has the basis states of ALIVE and DEAD, and while the cat is in the box, its state |ψ〉 is the superposition of
〉 A common superposition with 50/50 chance for the cat being alive is
)
| 2(|
|ψ0〉= 1 ALIVE〉+ DEAD〉 (2.30)
The measurement observable A of this qubit is performed by simply opening the box in which the “superpositioned” cat is enclosed. When the box is opened and a measurement is made, the system is “thrown to” (or collapses to) only one of the eigenvalues of the observable A – ALIVE or DEAD. Clearly, the observable A has the form
This observable was the basis of a huge debate at the heart of quantum physics:
“What mechanism converts the probabilities of live/dead into such a sharp outcome?”
Briefly, there were the Copenhagen Interpretation (QC deals only with the probabilities of the observable quantities – all other quantities are just meta-physical), the Quantum Decoherence (the macroscopic nature of the measurement apparatus
allows physicists to distinguish the fuzzy boundary between quantum world and actual world) and other philosophies (non-scientific like “conscious collapse” etc.) [Bohm01, BCS04, Hirv04, MM05, Omne99].
A very detailed and modern discussion of the measurement problem and
decoherence appears in Schlosshauer [Schl05]. Another extensive discussion appears in Chapter 3 of the original class notes from 1997 of John Preskill of Caltech [Pres97].
2.2. Reversible Logic
Quantum computation is reversible in nature. In this section we present the foundation of reversible logic. We show how reversible logic can be regarded from the point of view of group symmetry. Efficient reversible circuits that minimize garbage inputs and outputs are built in the form of cascades of gates that are defined in Section 2.2.3. We also discuss simple implementations of common logic functions that may be obtained directly from Toffoli and Fredkin gates.
2.2.1. Defining Reversible Logic
Definition 2.9: A gate or a circuit is logically reversible if it maps each input