So far we have dealt with quantum systems in a well-defined state, namely, systems that can be fully described by a single state vector https://www.w3.org/1998/Math/MathML" display="inline"> | ψ ⟩ . Such states are called pure . Here we will learn how to describe situations where knowing the state vector is not enough because the system is under the influence of another, typically larger, system which we know little about. This kind of interaction degrades the “quantumness” of the system we are interested to describe. The mathematics used to described this phenomenon is also suitable for establishing the degree of entanglement present in a quantum system, being it in a pure state or not.
So far we have dealt with quantum systems in a well-defined state, namely, systems that can be fully described by a single state vector https://www.w3.org/1998/Math/MathML" display="inline"> | ψ ⟩ . Such states are called pure . Here we will learn how to describe situations where knowing the state vector is not enough because the system is under the influence of another, typically larger, system which we know little about. This kind of interaction degrades the “quantumness” of the system we are interested to describe. The mathematics used to described this phenomenon is also suitable for establishing the degree of entanglement present in a quantum system, being it in a pure state or not.
There is more information in a quantum state than in the classical initial values https://www.w3.org/1998/Math/MathML" display="inline"> ( x 0 , p 0 ) that determine the complete 1-dimensional motion of a point particle for a given force. The initial wave function https://www.w3.org/1998/Math/MathML" display="inline"> Ψ ( x , 0 ) requires an infinite number of parameters, which can be expressed in observational terms as the two basic expectation values https://www.w3.org/1998/Math/MathML" display="inline"> ⟨ x ^ ⟩ and https://www.w3.org/1998/Math/MathML" display="inline"> ⟨ p ^ ⟩ together with the two variances https://www.w3.org/1998/Math/MathML" display="inline"> ( σ x 2 , σ p 2 ) and the covariance 5.1 https://www.w3.org/1998/Math/MathML" display="block"> C x p = 1 2 ⟨ x ^ p ^ + p ^ x ^ ⟩ − ⟨ x ^ ⟩ ⟨ p ^ ⟩ ,
The formalism for quantum systems developed so far applies to what are called pure states. A system in a pure state is completely specified by the state vector. A complete set of experimental tests will determine the system state fully: we have maximal knowledge of the system. For example, for a spin system, we can find a particular orientation of an SG machine such that the state is in its + or − port. This also means that the state is an eigenvector of some operator, or is always a linear combination of the computational basis states.
All computers are quantum computers but some are more quantum than others. Universal classical computers rely on the transistor, a quantum semiconductor, for their operation but use simple binary digital logic based on bits, 1 and 0. Quantum computers, those we are interested in here, utilize the qbits https://www.w3.org/1998/Math/MathML" display="inline"> | 1 〉 and https://www.w3.org/1998/Math/MathML" display="inline"> | 0 〉 , which are quantum states, to realize their quantum logic. These qbits are used in conjunction with superposition probability amplitudes, such as https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 + , https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 − , https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 + , and https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 − , to interact with optical components such as beam splitters, interferometers, and polarization rotators to perform quantum logical operations. In this chapter, a basic description of quantum logic, and quantum optical configurations, utilized in workable quantum computers is given.
In quantum mechanics, the physics of probability amplitudes, that exists in the imaginary and complex mathematical realm, is brought to the measurable realm by the iconic Born rule https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 that convert the superposition probability amplitude https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 into a measurable interferometric probability distribution. When Born first introduced his rule, in 1926, he did so in the absence of a physical derivation. In fact, Born never offered a derivation for his most crucial rule. In this chapter a transparent interferometric derivation of Born's rule is given. Furthermore, it is indicated that far from a ‘collapse’ the https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 → | ψ 〉 | ψ 〉 ∗ transition, for interferometric configurations, clearly involves a mathematical expansion . The https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 → | ψ 〉 | ψ 〉 ∗ transition for quantum entanglement is also discussed.
If we include all three spatial dimensions, the time-dependent wave function https://www.w3.org/1998/Math/MathML" display="inline"> Ψ ( x , y , z , t ) depends on four coordinates. A wave function of this form determines the probability density https://www.w3.org/1998/Math/MathML" display="inline"> ρ t ( x , y , z ) = | Ψ ( x , y , z , t ) | 2 for measurements of the spatial position https://www.w3.org/1998/Math/MathML" display="inline"> ( x , y , z ) . The product https://www.w3.org/1998/Math/MathML" display="inline"> | Ψ ( x , y , z , t ) | 2 d x d y d z therefore equals the probability of finding a particle in a small cubic volume with side lengths https://www.w3.org/1998/Math/MathML" display="inline"> ( d x , d y , d z ) around https://www.w3.org/1998/Math/MathML" display="inline"> ( x , y , z ) at time t . The normalization condition for this probability density is given by 4.1 https://www.w3.org/1998/Math/MathML" display="block"> ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ | Ψ ( x , y , z , t ) | 2 d x d y d z = 1 for all t .