Summary of "Single Systems | Understanding Quantum Information & Computation | Lesson 01"

Lesson overview / context

Lecturer: John Watrous (Technical Director, IBM Quantum Education).
Course: Lesson 01 of Unit 1 in the video course “Understanding Quantum Information and Computation.”
Companion resources: Qiskit Textbook and an overview video.
Goal: explain how information is represented and manipulated for a single system (classical then quantum); introduce Dirac (bra–ket) notation; present the simplified quantum description (state vectors + unitary operations) used throughout Unit 1.

Classical information — main ideas and mathematical representation

System and states
- A classical system X has a finite set Σ of classical states (examples: {0,1} for a bit; {1..6} for a die; {♣, ♦, ♥, ♠} for a card suit).
- Deterministic state: a definite element a ∈ Σ.
- Probabilistic (mixed) state: a probability distribution over Σ, represented by a probability vector v whose entries are nonnegative real numbers summing to 1.
Measurement (classical)
- “Looking” returns a single classical state sampled according to the probability vector.
- After measurement the state becomes the corresponding basis vector (certainty for the observed value).
Operations
- Deterministic operations: functions f: Σ → Σ. Represented by matrices whose columns are standard basis vectors |f(a)⟩ (one 1 per column, zeros elsewhere). Applying to a probability vector: v’ = M v.
- Probabilistic operations: represented by stochastic matrices (entries nonnegative; each column sums to 1). Each column is the output probability vector for that input basis state.
- Any probabilistic operation is a random mixture of deterministic operations (convex combination of permutation/one-hot matrices).
Composition
- Sequential operations correspond to matrix multiplication. Order matters: matrix multiplication is generally noncommutative (the rightmost matrix acts first).

Dirac (bra–ket) notation — definitions and rules

Ket |a⟩: a column vector with a 1 in the coordinate for state a and 0s elsewhere (standard basis vector).
Bra ⟨a|: the conjugate transpose of |a⟩ (a row vector). For basis states, ⟨a|b⟩ = 1 if a=b, else 0.
Inner product (bracket): ⟨ψ|φ⟩ is a scalar (overlap).
Outer product: |a⟩⟨b| is the matrix with a 1 in row a, column b and zeros elsewhere (an operator).
Conjugate transpose (“dagger”): for any ket |ψ⟩, ⟨ψ| = (|ψ⟩)† (transpose + complex conjugate of entries).

Quantum information — states, measurement, operations

Quantum states (pure states, simplified description)
- A pure quantum state is a column vector (ket) indexed by classical states, with complex entries (amplitudes) whose squared absolute values sum to 1. Equivalently, a unit vector in a complex Euclidean space.
- Amplitudes are not probabilities: the probability of a measurement outcome is the squared magnitude of the corresponding amplitude.
Measurement (standard/computational basis)
- Outcomes are the classical states a ∈ Σ.
- If the system is in |ψ⟩, probability of outcome a is |⟨a|ψ⟩|².
- On obtaining outcome a, the post-measurement state collapses to |a⟩.
- Different observers with different information can assign different states (classical or quantum) to the same system.
Operations: unitary matrices
- Definition: a square complex matrix U is unitary if U† U = U U† = I (so U−1 = U†).
- Unitaries preserve Euclidean norm and map unit vectors to unit vectors.
- Unitaries are the quantum analogue of stochastic matrices for pure states. Composition is matrix multiplication; the product of unitaries is unitary.
Single-qubit examples
- Basis states: |0⟩ = (1,0)ᵀ, |1⟩ = (0,1)ᵀ.
- ± basis: |+⟩ = (1/√2)(|0⟩ + |1⟩), |−⟩ = (1/√2)(|0⟩ − |1⟩).
- Pauli matrices: X (bit flip / NOT), Y, Z (phase flip). E.g., X|0⟩ = |1⟩, X|1⟩ = |0⟩; Z|0⟩ = |0⟩, Z|1⟩ = −|1⟩.
- Hadamard H: H|0⟩ = |+⟩, H|1⟩ = |−⟩. Applying H toggles between computational and ± bases.
- Phase gates: diagonal unitaries diag(1, e^{iθ}). Notable cases: S (θ = π/2), T (θ = π/4).
- Composite example: H S H produces a unitary whose square equals X — a “square-root of NOT”, a uniquely quantum phenomenon without a classical stochastic analogue.
Measurement and operations interplay
- To apply an operation then measure: compute U|ψ⟩ and measure in the standard basis.
- Because amplitudes can interfere (complex phases), quantum operations can produce effects impossible classically (e.g., nontrivial square roots of classical operations).

Two mathematical descriptions of quantum information (scope)

Simplified model (covered in this unit)
- Pure states as kets (vectors), operations as unitary matrices, measurements as projective/standard-basis measurements. Sufficient for most basic quantum algorithms.
General model (covered later)
- Density matrices (positive semidefinite, trace 1) for mixed states, general measurements (POVMs), and completely positive trace-preserving (CPTP) maps to model noise. The simplified model is a special case of the general model.

Practical rules / step-by-step recipes

Represent a classical probabilistic state over Σ as a probability vector v: nonnegative entries summing to 1.
Deterministic classical function f: Σ → Σ: represent as matrix M = Σ_a |f(a)⟩⟨a| and apply via v’ = M v.
Probabilistic classical operation: use a stochastic matrix (each column is a probability vector); apply v’ = M v.
Use bra–ket arithmetic:
- Inner product ⟨φ|ψ⟩ gives a scalar (overlap).
- Outer product |φ⟩⟨ψ| is an operator.
- Conjugate transpose: ⟨ψ| = (|ψ⟩)†.
Represent a pure quantum state as a normalized ket |ψ⟩ (Σ |amplitude|² = 1).
Standard-basis measurement on |ψ⟩:
- Prob(outcome a) = |⟨a|ψ⟩|².
- Post-measurement state (if outcome a) = |a⟩.
Represent quantum operations as unitary matrices U; apply as |ψ’⟩ = U |ψ⟩.
Compose operations by matrix multiplication; apply in right-to-left order (the rightmost matrix acts first).
To test unitarity: verify U† U = I, or check that U preserves norms.

Notable comparisons and takeaways

Parallels
- Classical probability vectors ↔ quantum state vectors.
- Stochastic matrices ↔ unitary matrices (both preserve normalization in their respective settings).
- Measurement collapse has an analogue in both frameworks.
Crucial differences
- Quantum amplitudes are complex and phases cause interference; measurement probabilities come from squared magnitudes of amplitudes.
- The simplified vector/unitary model does not capture noise or general mixed states — use density matrices and CPTP maps for those.
Practical note
- Order matters for compositions in both classical and quantum cases because matrix multiplication is noncommutative.
- Dirac notation is a compact and flexible tool for linear-algebraic calculations in both classical and quantum contexts.

Examples emphasized (to know / recognize)

Classical
- Bit with probability vector (3/4, 1/4).
- Deterministic 2×2 matrices for identity, constant-0, NOT, constant-1.
Quantum
- Qubit states: |0⟩, |1⟩, |+⟩, |−⟩.
- Measurement probabilities and post-measurement collapse to basis states.
- Single-qubit unitaries: X, Y, Z, H, S, T, and composed operations like H S H producing a square-root-of-NOT.