Quantum Concepts Explained

This comprehensive guide provides in-depth explanations of fundamental quantum mechanics concepts that underpin quantum computing. Each section includes mathematical formulations, worked examples, and physical intuition to build a solid understanding of quantum phenomena.

Table of Contents

1. Quantum Superposition
2. Quantum Entanglement
3. Quantum Measurement
4. Quantum Interference
5. Quantum Gates & Unitary Operations
6. Decoherence

Quantum Superposition

Introduction: Beyond Classical States

Classical bits exist in definite states: either 0 or 1 at any given moment. Quantum systems, however, can exist in superposition - a coherent combination of multiple states simultaneously. This is not uncertainty about which state the system is in (as in classical probability), but rather a fundamentally quantum phenomenon where the system genuinely occupies all states simultaneously until measured.

Mathematical Formulation

A quantum bit (qubit) is described by a state vector in a two-dimensional complex vector space (Hilbert space). The general state of a qubit is written using Dirac notation:

|ψ⟩ = α|0⟩ + β|1⟩

Where:

• |0⟩ and |1⟩ are the basis states (computational basis), analogous to classical 0 and 1
• α and β are complex probability amplitudes
• The coefficients satisfy the normalization condition: |α|² + |β|² = 1

The probability of measuring the qubit in state |0⟩ is |α|², and in state |1⟩ is |β|². The phases of α and β are crucial - they determine how quantum states interfere.

Matrix Representation

In the computational basis, we can represent states as column vectors:

|0⟩ = [1]    |1⟩ = [0]
      [0]          [1]

Therefore, a general qubit state is:

|ψ⟩ = α[1] + β[0] = [α]
       [0]     [1]   [β]

The Bloch Sphere

Any single-qubit pure state can be represented as a point on the surface of a Bloch sphere, a unit sphere in three-dimensional space. Using angles θ (polar) and φ (azimuthal), we can write:

|ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩

Where:

• θ ∈ [0, π] determines how much |0⟩ vs |1⟩
• φ ∈ [0, 2π) determines the relative phase

On the Bloch sphere:

• North pole (θ=0): |0⟩
• South pole (θ=π): |1⟩
• Equator (θ=π/2): Equal superpositions
  • |+⟩ = (|0⟩ + |1⟩)/√2 at φ=0
  • |−⟩ = (|0⟩ − |1⟩)/√2 at φ=π
  • |+i⟩ = (|0⟩ + i|1⟩)/√2 at φ=π/2
  • |−i⟩ = (|0⟩ − i|1⟩)/√2 at φ=3π/2

Worked Example: Creating and Analyzing Superposition

Problem: A qubit starts in state |0⟩. We apply a Hadamard gate H. What is the resulting state and what are the measurement probabilities?

Solution:

The Hadamard gate is defined by its action:

H|0⟩ = (|0⟩ + |1⟩)/√2 = |+⟩
H|1⟩ = (|0⟩ − |1⟩)/√2 = |−⟩

Or in matrix form:

H = (1/√2)[1   1]
           [1  -1]

Applying H to |0⟩ = [1, 0]ᵀ:

H|0⟩ = (1/√2)[1   1][1] = (1/√2)[1] = (1/√2)|0⟩ + (1/√2)|1⟩
              [1  -1][0]          [1]

So α = β = 1/√2.

Measurement probabilities:

• P(0) = |α|² = (1/√2)² = 1/2
• P(1) = |β|² = (1/√2)² = 1/2

The qubit is in a perfect 50-50 superposition. Each measurement will give 0 or 1 randomly, but the state before measurement genuinely contains both possibilities.

Physical Implementations

Different quantum systems realize superposition in different ways:

1. Photon Polarization: Horizontal (|H⟩) and vertical (|V⟩) polarization states can be superposed to create diagonal or circular polarizations.

2. Electron Spin: Spin-up (|↑⟩) and spin-down (|↓⟩) along any axis can be in superposition along orthogonal axes.

3. Superconducting Qubits: Current flowing clockwise and counterclockwise in a superconducting loop can superpose.

4. Trapped Ions: Different electronic energy levels of an ion can be placed in coherent superposition.

Superposition in Quantum Algorithms

Superposition enables quantum parallelism - the ability to process multiple inputs simultaneously:

• Deutsch-Jozsa algorithm: Evaluates a function on all 2ⁿ inputs simultaneously using n qubits in superposition
• Grover’s algorithm: Searches an unsorted database by placing the index in superposition
• Quantum Fourier Transform: Exploits superposition to achieve exponential speedup over classical FFT

The power comes not just from superposition alone, but from how quantum gates manipulate and interfere these superposed amplitudes.

Quantum Entanglement

Introduction: “Spooky Action at a Distance”

Entanglement is perhaps the most counterintuitive feature of quantum mechanics. When two or more quantum systems become entangled, their states are correlated in ways that cannot be explained by classical physics. Measuring one particle instantaneously affects the other, regardless of the distance separating them - what Einstein famously called “spooky action at a distance.”

Crucially, entanglement is not simply correlation. Classical systems can be correlated (e.g., gloves in separate boxes are correlated - if one is left-handed, the other is right-handed). Quantum entanglement produces correlations that violate Bell inequalities, proving they cannot be explained by any local hidden variable theory.

Mathematical Formulation

Consider two qubits. A general two-qubit state has the form:

|ψ⟩ = α₀₀|00⟩ + α₀₁|01⟩ + α₁₀|10⟩ + α₁₁|11⟩

Where |ij⟩ denotes qubit 1 in state i and qubit 2 in state j, and the normalization condition requires:

|α₀₀|² + |α₀₁|² + |α₁₀|² + |α₁₁|² = 1

A state is separable (not entangled) if it can be written as a product:

|ψ⟩ = (a|0⟩ + b|1⟩) ⊗ (c|0⟩ + d|1⟩) = ac|00⟩ + ad|01⟩ + bc|10⟩ + bd|11⟩

If a state cannot be written in this product form, it is entangled.

Bell States: Maximally Entangled Pairs

The four Bell states form an orthonormal basis of maximally entangled two-qubit states:

|Φ⁺⟩ = (|00⟩ + |11⟩)/√2    (Bell state / EPR pair)
|Φ⁻⟩ = (|00⟩ − |11⟩)/√2
|Ψ⁺⟩ = (|01⟩ + |10⟩)/√2
|Ψ⁻⟩ = (|01⟩ − |10⟩)/√2    (Singlet state)

These states are “maximally entangled” because measuring one qubit immediately determines the other’s state with certainty, yet each individual qubit is completely random (maximally mixed).

Worked Example: Analyzing an EPR Pair

Problem: Consider the Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2. What happens when we measure qubit 1 in the computational basis? What is the state of qubit 2 before and after this measurement?

Solution:

Before measurement: The joint state is |Φ⁺⟩ = (|00⟩ + |11⟩)/√2.

To find the state of qubit 2 alone, we compute the reduced density matrix by tracing out qubit 1:

ρ₂ = Tr₁(|Φ⁺⟩⟨Φ⁺|)
   = (1/2)(|0⟩⟨0| + |1⟩⟨1|)
   = (1/2)I

This is the maximally mixed state - qubit 2 has no definite state and is completely random (50% |0⟩, 50% |1⟩).

After measurement:

• If qubit 1 is measured and gives result 0, the joint state collapses to |00⟩, so qubit 2 is definitely in |0⟩
• If qubit 1 gives result 1, the state collapses to |11⟩, so qubit 2 is definitely in |1⟩
• Each outcome occurs with probability 1/2

Key insight: Before measurement, qubit 2 has no definite state. After measuring qubit 1, qubit 2 instantaneously acquires a definite state, perfectly correlated with qubit 1’s measurement result. This happens even if the qubits are light-years apart!

Bell’s Theorem and Non-Locality

Bell’s theorem (1964) proves that no theory based on local hidden variables can reproduce all predictions of quantum mechanics. The proof uses Bell inequalities - mathematical constraints that any local realistic theory must satisfy, but which quantum mechanics violates.

The CHSH inequality (Clauser-Horne-Shimony-Holt) provides an experimentally testable form:

|⟨A₁B₁⟩ + ⟨A₁B₂⟩ + ⟨A₂B₁⟩ − ⟨A₂B₂⟩| ≤ 2

Where:

• Alice measures observable A₁ or A₂ on her qubit
• Bob measures observable B₁ or B₂ on his qubit
• ⟨AᵢBⱼ⟩ is the correlation between measurements (each giving ±1)

Classical bound: Any local hidden variable theory gives values ≤ 2

Quantum mechanics: For entangled states with optimal measurement angles, the left side reaches Tsirelson’s bound of 2√2 ≈ 2.828, clearly violating the classical limit.

Worked Example: CHSH Violation

Problem: Consider the singlet state |Ψ⁻⟩ = (|01⟩ − |10⟩)/√2. Alice and Bob measure spin along angles a₁=0°, a₂=45° and b₁=22.5°, b₂=67.5° respectively. Calculate the CHSH quantity.

Solution:

For the singlet state and these specific angles, quantum mechanics predicts:

⟨A₁B₁⟩ = −cos(2×22.5°) = −cos(45°) = −1/√2
⟨A₁B₂⟩ = −cos(2×67.5°) = −cos(135°) = +1/√2
⟨A₂B₁⟩ = −cos(2×22.5°) = −cos(45°) = −1/√2
⟨A₂B₂⟩ = −cos(2×22.5°) = +cos(45°) = +1/√2

Therefore:

|⟨A₁B₁⟩ + ⟨A₁B₂⟩ + ⟨A₂B₁⟩ − ⟨A₂B₂⟩|
= |−1/√2 + 1/√2 − 1/√2 − 1/√2|
= |−2/√2|
= 2√2 ≈ 2.828

This violates the classical bound of 2, proving the correlations cannot be explained by any local realistic theory!

Creating Entanglement with Gates

Starting from |00⟩, we can create a Bell state using two gates:

1. Apply Hadamard to qubit 1: H ⊗ I |00⟩ = (|0⟩ + |1⟩)/√2 ⊗ |0⟩ = (|00⟩ + |10⟩)/√2
2. Apply CNOT with qubit 1 as control: CNOT (|00⟩ + |10⟩)/√2 = (|00⟩ + |11⟩)/√2 = |Φ⁺⟩

This simple circuit creates maximum entanglement from a separable state.

Applications of Entanglement

1. Quantum Teleportation: Transfer a quantum state using an EPR pair and two classical bits of communication

2. Quantum Key Distribution (QKD): Use entangled photons to create provably secure cryptographic keys (E91 protocol)

3. Dense Coding: Transmit two classical bits of information by sending one qubit and sharing an EPR pair

4. Quantum Computing: Entanglement is a resource for quantum algorithms - separable states can be efficiently simulated classically

5. Quantum Error Correction: Encode logical qubits in entangled states of multiple physical qubits

Schmidt Decomposition

Any pure state of a bipartite system can be written in Schmidt form:

|ψ⟩ = Σᵢ √λᵢ |iₐ⟩ ⊗ |iᵦ⟩

Where {|iₐ⟩} and {|iᵦ⟩} are orthonormal bases for subsystems A and B, and λᵢ ≥ 0 are Schmidt coefficients with Σλᵢ = 1.

The number of non-zero Schmidt coefficients is the Schmidt rank or Schmidt number:

• Schmidt rank = 1: State is separable (not entangled)
• Schmidt rank > 1: State is entangled
• Schmidt rank = 2 with λ₁ = λ₂ = 1/2: Maximally entangled

For the Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2, the Schmidt decomposition is already explicit with λ₁ = λ₂ = 1/2, confirming maximal entanglement.

Quantum Measurement

Introduction: The Measurement Problem

Measurement in quantum mechanics is fundamentally different from classical observation. In classical physics, measurement reveals a pre-existing property without disturbing the system. In quantum mechanics, measurement causes an irreversible change: the wave function collapses from a superposition to a definite state.

This raises profound questions (the “measurement problem”): What constitutes a measurement? When does collapse occur? Why does measurement produce definite outcomes from superposed states? While interpretations vary, the mathematical framework provides unambiguous predictions.

The Born Rule

Given a quantum state |ψ⟩ and an observable represented by operator Â with eigenstates {|aᵢ⟩} and eigenvalues {aᵢ}, the Born rule states:

Probability of measuring outcome aᵢ:

P(aᵢ) = |⟨aᵢ|ψ⟩|²

State after measurement (if outcome aᵢ obtained):

|ψ⟩ → |aᵢ⟩

Expected value of the observable:

⟨Â⟩ = ⟨ψ|Â|ψ⟩ = Σᵢ aᵢ P(aᵢ)

Projection Operators

Measurement can be described using projection operators (or projectors):

Pᵢ = |aᵢ⟩⟨aᵢ|

Properties:

• Pᵢ² = Pᵢ (idempotent)
• Pᵢ†= Pᵢ (Hermitian)
• PᵢPⱼ = δᵢⱼPᵢ (orthogonal for i≠j)
• Σᵢ Pᵢ = I (completeness)

The probability of outcome i when measuring state |ψ⟩:

P(i) = ⟨ψ|Pᵢ|ψ⟩ = Tr(Pᵢ|ψ⟩⟨ψ|)

The post-measurement state (unnormalized):

|ψ'⟩ = Pᵢ|ψ⟩

Normalized:

|ψ'⟩ = Pᵢ|ψ⟩ / √P(i)

Worked Example: Measuring Superposition in Different Bases

Problem: A qubit is in state |ψ⟩ = (|0⟩ + |1⟩)/√2 = |+⟩. Calculate:

1. Measurement probabilities and post-measurement states in the computational basis {|0⟩, |1⟩}
2. Measurement probabilities and post-measurement states in the Hadamard basis {|+⟩, |−⟩}

Solution:

Part 1: Computational basis measurement

Projectors:

P₀ = |0⟩⟨0| = [1  0]
              [0  0]

P₁ = |1⟩⟨1| = [0  0]
              [0  1]

Probabilities:

P(0) = |⟨0|ψ⟩|² = |⟨0|(|0⟩ + |1⟩)/√2|² = |1/√2|² = 1/2
P(1) = |⟨1|ψ⟩|² = |⟨1|(|0⟩ + |1⟩)/√2|² = |1/√2|² = 1/2

Post-measurement states:

• If outcome 0: |ψ⟩ → |0⟩ with probability 1/2
• If outcome 1: |ψ⟩ → |1⟩ with probability 1/2

Part 2: Hadamard basis measurement

Basis states:

|+⟩ = (|0⟩ + |1⟩)/√2
|−⟩ = (|0⟩ − |1⟩)/√2

Since |ψ⟩ = |+⟩:

P(+) = |⟨+|ψ⟩|² = |⟨+|+⟩|² = |1|² = 1
P(−) = |⟨−|ψ⟩|² = |⟨−|+⟩|² = |0|² = 0

Post-measurement state:

• Outcome +: |ψ⟩ → |+⟩ with probability 1 (certainty!)
• Outcome −: Never occurs

Key insight: The same quantum state gives different measurement statistics depending on the measurement basis. Measuring |+⟩ in its own basis gives a deterministic result, while measuring in the computational basis gives random outcomes.

Measurement in Arbitrary Bases

Any measurement basis can be obtained by rotating from the computational basis. To measure in basis {|φ₀⟩, |φ₁⟩}, we can:

1. Apply unitary U that transforms: U|φ₀⟩ = |0⟩ and U|φ₁⟩ = |1⟩
2. Measure in the computational basis
3. The outcome probabilities are the same as measuring in the original basis

For example, measuring in the Hadamard basis {|+⟩, |−⟩} is equivalent to applying H then measuring in {|0⟩, |1⟩}.

Generalized Measurements (POVMs)

The most general quantum measurement is described by Positive Operator-Valued Measures (POVMs), a set of positive operators {M₁, M₂, …, Mₘ} satisfying:

Σᵢ Mᵢ† Mᵢ = I

The probability of outcome i when measuring state |ψ⟩:

P(i) = ⟨ψ|Mᵢ† Mᵢ|ψ⟩

Post-measurement state:

|ψ'⟩ = Mᵢ|ψ⟩ / √P(i)

POVMs generalize projective measurements and include:

• Unsharp measurements: Don’t completely collapse the state
• Measurements with more outcomes than dimensions: E.g., trine measurement on a qubit (3 outcomes)
• Measurements that extract information while minimizing disturbance

Measurement Postulates Summary

The measurement process follows these postulates:

1. Before measurement: Quantum state is |ψ⟩ = Σᵢ cᵢ|aᵢ⟩ in superposition

2. During measurement: The state probabilistically collapses to one eigenstate |aₖ⟩

3. Outcome probability: P(aₖ) = |⟨aₖ|ψ⟩|² = |cₖ|²

4. After measurement: The system is in definite state |aₖ⟩

5. Observable value: The measured value is the eigenvalue aₖ

6. Disturbance: Repeated immediate measurements give the same result (state has collapsed)

Quantum Interference

Introduction: Waves of Probability

Quantum interference is the phenomenon where probability amplitudes (complex numbers) combine according to wave-like addition rules, leading to constructive or destructive interference. This is fundamentally different from classical probability, where probabilities (positive real numbers) always add.

The canonical demonstration is the double-slit experiment: particles exhibit wave-like interference patterns when both paths are available, but this pattern disappears when “which-path” information is obtained - even without disturbing the particle’s motion.

Mathematical Framework

Consider a quantum system that can reach a final state |f⟩ via multiple paths. If the amplitudes for each path are α₁, α₂, …, αₙ, the total amplitude is:

A_total = α₁ + α₂ + ... + αₙ

The probability of reaching |f⟩ is:

P(f) = |A_total|² = |α₁ + α₂ + ... + αₙ|²

This is NOT equal to |α₁|² + |α₂|² + ... unless the amplitudes have no relative phase. Expanding:

|α₁ + α₂|² = |α₁|² + |α₂|² + 2Re(α₁*α₂)

The cross term 2Re(α₁*α₂) is the interference term:

• Constructive interference: Cross term positive (amplitudes in phase)
• Destructive interference: Cross term negative (amplitudes out of phase)
• No interference: Amplitudes orthogonal (cross term zero)

The Double-Slit Experiment

Setup: Particles (photons, electrons, atoms) pass through two slits and hit a screen.

Classical expectation: Each particle goes through either slit 1 or slit 2. The probability distribution should be:

P_classical(x) = P₁(x) + P₂(x)

Quantum reality: The particle’s wave function passes through both slits in superposition:

|ψ⟩ = (1/√2)(|slit 1⟩ + |slit 2⟩)

At position x on the screen, the amplitudes from the two paths interfere:

ψ(x) = ψ₁(x) + ψ₂(x)
P(x) = |ψ₁(x) + ψ₂(x)|²
     = |ψ₁(x)|² + |ψ₂(x)|² + 2Re(ψ₁*(x)ψ₂(x))

The interference term 2Re(ψ₁*(x)ψ₂(x)) creates the characteristic pattern of bright and dark fringes:

• Bright fringes: Where path lengths differ by integer wavelengths (nλ) → Constructive interference
• Dark fringes: Where path lengths differ by half-integer wavelengths ((n+1/2)λ) → Destructive interference

Which-Path Information Destroys Interference

If we place detectors at the slits to determine which path the particle took, the interference pattern vanishes. Mathematically, this is because the state becomes:

ρ = (1/2)|slit 1⟩⟨slit 1| + (1/2)|slit 2⟩⟨slit 2|

This is a mixed state (statistical mixture), not a coherent superposition. The density matrix formalism shows:

Coherent superposition (interference):

ρ_coherent = |ψ⟩⟨ψ| = (1/2)(|1⟩⟨1| + |1⟩⟨2| + |2⟩⟨1| + |2⟩⟨2|)

Off-diagonal terms |1⟩⟨2| and |2⟩⟨1| represent coherence.

Incoherent mixture (no interference):

ρ_mixed = (1/2)|1⟩⟨1| + (1/2)|2⟩⟨2|

No off-diagonal terms → no interference.

The act of measurement (even in principle) destroys the coherence and thus the interference.

Worked Example: Mach-Zehnder Interferometer

Problem: A photon enters a Mach-Zehnder interferometer via beam splitter BS₁, which creates superposition:

|ψ₁⟩ = (1/√2)(|path A⟩ + i|path B⟩)

The paths reflect off mirrors and meet at beam splitter BS₂. If the path lengths are equal, what is the probability of detecting the photon at each output port?

Solution:

At BS₂, each path splits again:

|path A⟩ →(BS₂) (1/√2)(|D₁⟩ + i|D₂⟩)
|path B⟩ →(BS₂) (1/√2)(|D₁⟩ - i|D₂⟩)  [note different phase]

The total state after BS₂:

|ψ_final⟩ = (1/√2)(|path A⟩ + i|path B⟩) →(BS₂)
          = (1/√2)[(1/√2)(|D₁⟩ + i|D₂⟩) + i(1/√2)(|D₁⟩ - i|D₂⟩)]
          = (1/2)|D₁⟩ + (i/2)|D₂⟩ + (i/2)|D₁⟩ + (1/2)|D₂⟩
          = (1+i)/2 |D₁⟩ + (i+1)/2 |D₂⟩

Wait, let me recalculate more carefully. With proper beam splitter transformations:

|ψ_final⟩ = (1/2)[(|D₁⟩ + i|D₂⟩) + i(i|D₁⟩ + |D₂⟩)]
          = (1/2)[|D₁⟩ + i|D₂⟩ - |D₁⟩ + i|D₂⟩]
          = (1/2)[2i|D₂⟩]
          = i|D₂⟩

Detection probabilities:

P(D₁) = |⟨D₁|ψ_final⟩|² = |0|² = 0
P(D₂) = |⟨D₂|ψ_final⟩|² = |i|² = 1

Result: The photon always goes to detector D₂! The amplitude from path A interfering with the amplitude from path B completely destroys the probability of detection at D₁ (destructive interference) and maximizes it at D₂ (constructive interference).

If we block one path or insert a phase shifter, the interference changes and D₁ can detect photons. This demonstrates the exquisite sensitivity of quantum interference to path differences.

Interference in Quantum Algorithms

Quantum algorithms exploit interference to amplify correct answer amplitudes while suppressing incorrect ones:

Deutsch-Jozsa Algorithm:

• Creates superposition over all inputs
• Oracle marks the answer
• Interference between “constant” and “balanced” paths gives deterministic result

Grover’s Algorithm:

• Repeatedly applies Grover operator: Oracle + Diffusion
• Amplitudes of correct items constructively interfere and grow
• Amplitudes of incorrect items destructively interfere and shrink
• After O(√N) iterations, measuring gives the answer with high probability

Quantum Fourier Transform:

• Implements the discrete Fourier transform using interference
• Different frequency components interfere to produce the transformed state
• Central to Shor’s factoring algorithm and quantum phase estimation

The computational power of quantum algorithms derives from orchestrating massive destructive and constructive interference among exponentially many amplitudes.

Quantum Gates & Unitary Operations

Introduction: Quantum Logic

Just as classical computation uses logic gates (AND, OR, NOT) to manipulate bits, quantum computation uses quantum gates to manipulate qubits. However, quantum gates must be reversible and unitary (except for measurement, which is non-unitary).

A unitary operator U satisfies:

U† U = U U† = I

Where U† is the conjugate transpose (Hermitian adjoint). Unitarity preserves:

• Norm: ⟨ψ|ψ⟩ = ⟨ψ'|ψ'⟩
• Inner products: ⟨φ|ψ⟩ = ⟨φ'|ψ'⟩
• Probabilities: Total probability remains 1

Physically, unitary evolution represents coherent dynamics - quantum systems evolve unitarily when isolated from the environment.

Single-Qubit Gates

The most important single-qubit gates:

Pauli Gates:

X = [0  1]   (Bit flip: X|0⟩ = |1⟩, X|1⟩ = |0⟩)
    [1  0]

Y = [0  -i]  (Bit flip + phase flip)
    [i   0]

Z = [1   0]  (Phase flip: Z|0⟩ = |0⟩, Z|1⟩ = -|1⟩)
    [0  -1]

Hadamard Gate:

H = (1/√2)[1   1]   Creates superposition: H|0⟩ = |+⟩, H|1⟩ = |−⟩
           [1  -1]

Phase Gates:

S = [1  0]   (Phase gate: S|1⟩ = i|1⟩)
    [0  i]

T = [1   0        ]   (π/8 gate: T|1⟩ = e^(iπ/4)|1⟩)
    [0  e^(iπ/4)]

Rotation Gates:

R_x(θ) = [cos(θ/2)   -isin(θ/2)]   (Rotation around X-axis)
         [-isin(θ/2)  cos(θ/2)  ]

R_y(θ) = [cos(θ/2)   -sin(θ/2)]    (Rotation around Y-axis)
         [sin(θ/2)    cos(θ/2)]

R_z(θ) = [e^(-iθ/2)    0     ]     (Rotation around Z-axis)
         [0         e^(iθ/2)]

General rotation around axis n = (n_x, n_y, n_z):

R_n(θ) = exp(-iθn·σ/2) = cos(θ/2)I - isin(θ/2)(n_x X + n_y Y + n_z Z)

Bloch Sphere Interpretation

On the Bloch sphere, quantum gates correspond to rotations:

• X gate: 180° rotation around X-axis (equator flip)
• Y gate: 180° rotation around Y-axis
• Z gate: 180° rotation around Z-axis (phase flip)
• H gate: 180° rotation around (X+Z)/√2 axis
• S gate: 90° rotation around Z-axis
• T gate: 45° rotation around Z-axis

Any single-qubit gate can be decomposed into rotations. The Z-Y-Z decomposition:

U = e^(iα) R_z(β) R_y(γ) R_z(δ)

This means any single-qubit gate can be built from three rotation gates.

Two-Qubit Gates

CNOT (Controlled-NOT):

CNOT = [1  0  0  0]   Flips target if control is |1⟩
       [0  1  0  0]
       [0  0  0  1]
       [0  0  1  0]

Action:

CNOT|00⟩ = |00⟩
CNOT|01⟩ = |01⟩
CNOT|10⟩ = |11⟩  (flip!)
CNOT|11⟩ = |10⟩  (flip!)

CNOT creates entanglement:

CNOT (|0⟩ + |1⟩)/√2 ⊗ |0⟩ = (|00⟩ + |11⟩)/√2 = |Φ⁺⟩

Controlled-Z (CZ):

CZ = [1  0  0   0]   Adds phase -1 if both qubits are |1⟩
     [0  1  0   0]
     [0  0  1   0]
     [0  0  0  -1]

CZ is symmetric - control and target are interchangeable.

SWAP:

SWAP = [1  0  0  0]   Exchanges two qubits
       [0  0  1  0]
       [0  1  0  0]
       [0  0  0  1]

Controlled-Phase:

CP(θ) = [1  0  0      0    ]
        [0  1  0      0    ]
        [0  0  1      0    ]
        [0  0  0  e^(iθ)]

Universal Gate Sets

A set of gates is universal if any unitary operation can be approximated to arbitrary precision using gates from this set.

Common universal sets:

1. {H, T, CNOT} - Clifford+T set (standard for fault-tolerant quantum computing)
2. {Rotation gates, CNOT} - All single-qubit rotations + CNOT
3. {U3, CNOT} - General single-qubit unitary + CNOT

Solovay-Kitaev theorem: Any single-qubit gate can be approximated to precision ε using O(log^c(1/ε)) gates from a universal set, where c ≈ 2.

Worked Example: Gate Decomposition

Problem: Decompose the SWAP gate into CNOT gates.

Solution:

The SWAP gate can be built from three CNOTs:

SWAP = CNOT₁₂ · CNOT₂₁ · CNOT₁₂

Where CNOT₁₂ means control=qubit 1, target=qubit 2.

Let’s verify by tracing the action on basis states:

For |10⟩:

|10⟩ →(CNOT₁₂) |11⟩ →(CNOT₂₁) |01⟩ →(CNOT₁₂) |01⟩ ✓

For |01⟩:

|01⟩ →(CNOT₁₂) |01⟩ →(CNOT₂₁) |11⟩ →(CNOT₁₂) |10⟩ ✓

For |00⟩ and |11⟩:

|00⟩ →(CNOT₁₂) |00⟩ →(CNOT₂₁) |00⟩ →(CNOT₁₂) |00⟩ ✓
|11⟩ →(CNOT₁₂) |10⟩ →(CNOT₂₁) |10⟩ →(CNOT₁₂) |11⟩ ✓

This decomposition is crucial for hardware with limited connectivity - SWAP operations are needed to route qubits for interaction.

Gate Fidelity

Real quantum gates are imperfect. Gate fidelity measures how close an actual gate U_actual is to the ideal U_ideal:

F = |⟨ψ_ideal|ψ_actual⟩|² = |⟨ψ|U_ideal† U_actual|ψ⟩|²

Averaged over all input states, the average gate fidelity is:

F_avg = ∫ dψ |⟨ψ|U_ideal† U_actual|ψ⟩|²

For a d-dimensional system:

F_avg = (Tr(U_ideal† U_actual) + d) / (d(d+1))

State-of-the-art gate fidelities:

• Single-qubit gates: > 99.99% (trapped ions), > 99.9% (superconducting)
• Two-qubit gates: > 99.9% (trapped ions), > 99% (superconducting)

The error per gate is ε = 1 - F. For fault-tolerant quantum computing, we need ε < 10^-3 to 10^-4 depending on the error correction code.

Decoherence & Quantum-to-Classical Transition

Introduction: Fragility of Quantum States

Quantum superposition and entanglement are fragile. Interaction with the environment causes decoherence - the irreversible loss of quantum coherence that transforms pure quantum states into classical-like mixtures. Decoherence is the primary obstacle to building large-scale quantum computers and explains why we don’t observe macroscopic superpositions in everyday life.

Mathematical Description

A closed quantum system evolves unitarily according to the Schrödinger equation:

iℏ ∂|ψ⟩/∂t = H|ψ⟩

But real systems interact with environments (photons, phonons, stray fields, etc.). The combined system+environment state evolves as:

|ψ_system⟩ ⊗ |E₀⟩ → Σᵢ cᵢ|ψᵢ⟩ ⊗ |Eᵢ⟩

The system becomes entangled with the environment. Different system states correlate with different (orthogonal) environment states.

To describe just the system, we trace out the environment, obtaining the reduced density matrix:

ρ_system = Tr_env(ρ_total)

Initially, if the system is in pure state |ψ⟩:

ρ(0) = |ψ⟩⟨ψ|   (pure state, Tr(ρ²) = 1)

After decoherence:

ρ(t) = Σᵢ pᵢ|ψᵢ⟩⟨ψᵢ|   (mixed state, Tr(ρ²) < 1)

The coherence (off-diagonal elements in some basis) decays exponentially.

Decoherence Time Scales

T₁ - Energy relaxation time: Characterizes amplitude damping - loss of energy to the environment. A qubit in |1⟩ spontaneously decays to |0⟩ with rate Γ₁ = 1/T₁:

ρ₁₁(t) = ρ₁₁(0) e^(-t/T₁)   (Population of excited state)

T₂ - Dephasing time: Characterizes phase relaxation - loss of relative phase information. The off-diagonal elements (coherences) decay:

ρ₀₁(t) = ρ₀₁(0) e^(-t/T₂)   (Coherence)

Generally, T₂ ≤ 2T₁ because energy relaxation causes dephasing, but additional “pure dephasing” can make T₂ shorter. The pure dephasing time T_φ satisfies:

1/T₂ = 1/(2T₁) + 1/T_φ

Simple Decoherence Model

Consider a qubit in superposition coupled to a single environmental qubit:

Initial state:

|ψ⟩ ⊗ |E₀⟩ = (α|0⟩ + β|1⟩) ⊗ |E₀⟩

After interaction (dephasing):

→ α|0⟩⊗|E₀⟩ + β|1⟩⊗|E₁⟩

Where |E₀⟩ and |E₁⟩ are orthogonal environment states (⟨E₀|E₁⟩ = 0).

The reduced density matrix of the system:

ρ = Tr_E[|ψ⟩⟨ψ|]
  = |α|²|0⟩⟨0| + |β|²|1⟩⟨1| + αβ*|0⟩⟨1|⟨E₀|E₁⟩ + α*β|1⟩⟨0|⟨E₁|E₀⟩
  = |α|²|0⟩⟨0| + |β|²|1⟩⟨1|   (since ⟨E₀|E₁⟩ = 0)

The coherence terms (off-diagonal) have vanished! The superposition has decohered into a classical mixture.

Decoherence vs. Measurement

Decoherence is often called “measurement by the environment.” The key difference:

• Measurement: Correlation with a macroscopic apparatus that records information
• Decoherence: Correlation with many untrackable environmental degrees of freedom

Both create entanglement between system and surroundings, but measurement implies accessible information while decoherence spreads information irreversibly into the environment.

Protecting Against Decoherence

Strategies to combat decoherence:

1. Isolation: Minimize coupling to environment (vacuum, low temperature, shielding)

2. Fast operations: Complete gates before decoherence (gates faster than T₂)

3. Decoherence-free subspaces: Encode information in states unaffected by particular decoherence mechanisms

4. Dynamical decoupling: Apply pulse sequences (spin echoes) to reverse dephasing

5. Quantum error correction: Actively detect and correct errors caused by decoherence

6. Topological protection: Use topological qubits with intrinsic decoherence resistance

The Quantum-to-Classical Transition

Decoherence explains why macroscopic objects behave classically:

For a macroscopic object in superposition of two positions separated by distance d, the decoherence time due to scattering of air molecules or photons is:

τ_D ~ ℏ / (λ_d² d² n k_B T)

Where n is particle density, λ_d is particle wavelength, T is temperature.

For a dust grain (d ~ 10 μm) at room temperature:

τ_D ~ 10^(-40) seconds   (essentially instantaneous!)

This ultra-fast decoherence makes macroscopic superpositions impossible to observe - they collapse to classical states before we can detect them.

For microscopic quantum systems (atoms, photons, qubits):

τ_D ~ milliseconds to seconds   (observable quantum behavior)

Decoherence solves the measurement problem: We don’t need a special collapse postulate. The “collapse” emerges from entanglement with the environment, which happens continuously and irreversibly for macroscopic systems but can be controlled and corrected for microscopic quantum computers.

Summary

These fundamental quantum concepts - superposition, entanglement, measurement, interference, unitary operations, and decoherence - form the foundation of quantum mechanics and quantum computing:

• Superposition enables quantum parallelism and probabilistic computation
• Entanglement provides correlations stronger than classical physics allows, essential for quantum protocols
• Measurement collapses superpositions and extracts classical information from quantum states
• Interference allows amplification of correct answers and suppression of wrong ones in quantum algorithms
• Unitary gates manipulate quantum states coherently, implementing quantum logic
• Decoherence describes the fragility of quantum states and the quantum-to-classical transition

Understanding these concepts mathematically and physically is essential for developing quantum algorithms, designing quantum hardware, and advancing quantum information science.

For deeper exploration of specific terms and advanced topics, see our Quantum Glossary and Quantum Dictionary.

This guide provides foundational knowledge for quantum computing. The mathematical rigor and worked examples aim to bridge intuition and formal understanding.