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Simple Quantum Systems: The Qubit (2-Level Systems)
Simple Quantum Systems: The Qubit (2-Level Systems)
A qubit, or quantum bit, is the fundamental unit of quantum information. Unlike a classical bit, which can only be in one of two states (0 or 1), a qubit can exist in a superposition of these states. It is a two-level quantum mechanical system.
Here are some physical implementations of qubits:
Polarized Light: The polarization of a single photon can be used to represent a qubit. For instance, horizontal polarization can represent the state ∣0⟩ and vertical polarization can represent the state ∣1⟩. A photon can also be in a superposition of these polarizations (e.g., diagonally polarized).
Electron Spin in a Magnetic Field: An electron possesses an intrinsic angular momentum called spin. When placed in a magnetic field, the electron's spin can align either with the field (spin up, representing ∣0⟩) or against the field (spin down, representing ∣1⟩). It can also exist in a superposition of these spin states.
Mach-Zehnder Interferometer (Time-Bin Qubit): In this setup, a single photon can take two possible paths. If the photon takes the "early" path, it can represent ∣0⟩, and if it takes the "late" path, it can represent ∣1⟩. By manipulating the photon's path with beam splitters and phase shifters, it can be put into a superposition of these time bins.
Superconducting Charge Qubit: These qubits are based on small superconducting circuits. The two levels can be represented by the presence (e.g., for ∣1⟩) or absence (e.g., for ∣0⟩) of a Cooper pair (a pair of electrons) on a small superconducting island, or by different quantized charge states.
Single Qubit (Pure State)
Single Qubit (Pure State)
A pure state of a qubit can be represented by a state vector, denoted as a "ket" ∣ψ⟩. This vector resides in a 2-dimensional complex Hilbert space.
If we have an orthonormal basis for this space, say {∣0⟩,∣1⟩}, then any pure state ∣ψ⟩ can be written as a linear combination (superposition) of these basis states:
α0 and α1 are complex numbers called probability amplitudes.
The probability of measuring the qubit in state ∣0⟩ is p0=∣α0∣^2. The probability of measuring the qubit in state ∣1⟩ is p1=∣α1∣^2.
Normalization Condition: For the probabilities to sum to 1, the state vector must be normalized:
Each pure state ∣ψ⟩ corresponds to a one-dimensional subspace (a ray) in the Hilbert space. This means that e^iγ∣ψ⟩ represents the same physical state as ∣ψ⟩, where e^iγ is a complex number with modulus 1 (a global phase factor).
Parameterization of α0 and α1:
Parameterization of α0 and α1:
Observable Consequence of the Global Phase Factor
Observable Consequence of the Global Phase Factor
The global phase factor e^iγ has no observable consequence. This is because in quantum mechanics, we don't directly observe the state vector ∣ψ⟩. Instead, we observe physical quantities like position, momentum, or spin. These quantities are represented by Hermitian operators called observables.
If M is an observable, its expectation value (the average value we would get from many measurements on identically prepared systems) for a system in state ∣ψ⟩ is given by the Born rule:
Since the expectation values for all observables are the same for ∣ψ⟩ and e^iγ∣ψ⟩, these two state vectors are physically indistinguishable.
Density Matrix and Global Phase:
Density Matrix and Global Phase:
The Bloch Sphere
The Bloch Sphere
Since the global phase factor e^iγ is unobservable, we can effectively set γ=0 for the purpose of describing the distinct physical state. The state vector then becomes:
∣ψ⟩=cos(θ/2)∣0⟩+e^iϕsin(θ/2)∣1⟩
This state can be represented as a point on the surface of a unit sphere called the Bloch sphere. The parameters θ and ϕ are spherical coordinates:
θ: The angle made with the positive z-axis (polar angle). 0≤θ≤π.
ϕ: The angle made with the positive x-axis in the xy-plane (azimuthal angle). 0≤ϕ<2π.
The Cartesian coordinates of this point on the Bloch sphere are given by the Bloch vector r:
r=(x,y,z) where:
x=sinθcosϕ
y=sinθsinϕ
z=cosθ
The Bloch vector has unit length: x^2+y^2+z^2 = 1 (check).
Properties of r (Different Cases):
Properties of r (Different Cases):
Density Matrix ρ and its Relation to the Bloch Vector
Density Matrix ρ and its Relation to the Bloch Vector
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