QuantumLab Beginner

1 · Vectors — column form & Dirac notation

A quantum state is just a list of complex numbers (a vector). For a single qubit the list has length 2.

The "ket" |ψ⟩ is a column vector. Its conjugate-transpose is the "bra" ⟨ψ| = (α*, β*). The inner product ⟨φ|ψ⟩ is a single complex number. A state is normalised when ⟨ψ|ψ⟩ = |α|² + |β|² = 1.

2 · Complex numbers

A complex number is z = a + bi where i² = −1. We use them because quantum amplitudes carry a phase as well as a magnitude.

Born-rule probabilities use |amplitude|², so an overall phase e^iθ on a state is unobservable — only relative phases matter.

3 · Operators & matrices

An operator is a linear map that takes a vector and returns another vector. On a single qubit it is a 2×2 matrix.

Composition is just matrix multiplication. A·B means "apply B first, then A". Order matters: in general AB ≠ BA.

4 · Hermitian operators & eigenvalues

The adjoint A^† ("A-dagger") is the conjugate-transpose. A matrix is Hermitian when A^† = A. Hermitian operators represent every observable in QM (energy, spin, position, …).

Example. The Pauli-Z operator is Hermitian. Its eigenvalues are ±1; its eigenvectors are |0⟩ (eigenvalue +1) and |1⟩ (eigenvalue −1). Measuring "Z" returns one of those two real numbers — never anything else.

5 · Unitary operators

A matrix U is unitary when U^†U = I. Geometrically, U preserves lengths: ⟨Uψ|Uψ⟩ = ⟨ψ|ψ⟩. So unitary evolution always keeps a normalised state normalised — total probability stays 1.

Every quantum gate (X, Y, Z, H, CNOT, …) is unitary. That's why quantum computation is reversible — the only step that isn't is measurement.

① State vector lives in a Hilbert space

The complete state of an isolated quantum system is described by a unit vector |ψ⟩ in a complex Hilbert space ℋ (a complex inner-product space that's closed under limits).

For one qubit, ℋ = ℂ² (two complex amplitudes). For n qubits, ℋ = ℂ²ⁿ — the Hilbert space grows exponentially in qubit count. That exponential room is the resource quantum computers exploit.

② Time evolution is unitary

An isolated (no-measurement) quantum system evolves through a unitary operator U. Discretely:

Continuously, the Schrödinger equation:

Here H is the system's Hamiltonian (a Hermitian operator — the observable for energy). Quantum gates are nothing more than carefully chosen unitary U's applied for finite time.

③ Observables are Hermitian operators

Every physically measurable quantity (energy, position, spin, …) is represented by a Hermitian operator A on ℋ. The only possible outcomes of a measurement are the eigenvalues a_i of A.

Example: measuring "spin along Z" uses the Pauli-Z operator with eigenvalues ±1 and eigenvectors |0⟩, |1⟩ — these are the only readings the apparatus can ever produce.

④ Born rule

The probability of measuring outcome a_i on state |ψ⟩ is the squared modulus of the projection of |ψ⟩ onto the corresponding eigenvector:

Probabilities of all possible outcomes sum to 1 because |ψ⟩ is normalised. This is the only non-deterministic ingredient in QM — everything else (states, gates, evolution) is fully deterministic.

⑤ Collapse — projective measurement

Immediately after a measurement that returned outcome a_i, the system's state collapses to the corresponding (normalised) eigenvector:

A second measurement of the same observable will return a_i with certainty (probability 1) — the random part is "spent". Collapse is what makes quantum mechanics non-reversible at the moment of measurement, and it's exactly the rule that lets entanglement look like spooky action at a distance.

Definition

For an ensemble where the system is in pure state |ψᵢ⟩ with probability pᵢ:

Properties

Pure state special case

A pure state |ψ⟩ has density matrix:

Bloch-sphere representation (single qubit)

Any single-qubit density matrix can be written as:

where σ⃗ = (X, Y, Z) are the Pauli matrices. |r⃗| = 1 → pure state on the Bloch-sphere surface; |r⃗| < 1 → mixed state inside the ball; r⃗ = 0 → maximally mixed ρ = I/2 (no information).

① State postulate (reformulated)

The state of an isolated quantum system is described by a density operator ρ on Hilbert space ℋ, satisfying:

Generalises the pure-state postulate (|ψ⟩ ∈ ℋ with ⟨ψ|ψ⟩ = 1) by allowing classical mixtures and entanglement-induced mixedness. Pure states are the special case where ρ has rank 1.

② Evolution postulate (reformulated)

The density operator of an isolated system evolves by unitary conjugation:

Continuously, the von Neumann equation — the density-matrix analog of Schrödinger's equation:

For a pure state ρ = |ψ⟩⟨ψ|, this reduces to the standard Schrödinger evolution. The von Neumann equation also extends naturally to open systems (the Lindblad master equation), where dynamics aren't unitary because the system exchanges information with its environment.

③ Observable postulate (reformulated)

Every measurable quantity is represented by a Hermitian operator A with spectral decomposition:

Each eigenvalue aᵢ ∈ ℝ is a possible measurement outcome; Pᵢ projects onto the corresponding eigenspace.

Identical to the pure-state version — observables don't depend on how the state is described.

④ Born rule (reformulated)

The probability of measuring outcome aᵢ when the system is in state ρ is given by a trace:

The expectation value of any observable A is:

For a pure state, this reproduces the original rule: P(aᵢ) = ⟨ψ|Pᵢ|ψ⟩ = |⟨aᵢ|ψ⟩|² (just unfold ρ = |ψ⟩⟨ψ| and use cyclicity of trace). The trace formula extends linearly to mixed states for free.

⑤ Collapse postulate (reformulated)

If a projective measurement of A returns outcome aᵢ, the post-measurement state is:

The denominator normalises so that Tr(ρ′) = 1.

For a pure state ρ = |ψ⟩⟨ψ| this reduces to ρ′ = |aᵢ⟩⟨aᵢ| — the familiar collapse to the eigenvector. For a mixed state, only the part of the ensemble consistent with the observed outcome survives. A second measurement of the same observable returns aᵢ with certainty.

Timeline · 1900 → 1928

Try it · Bloch sphere toggle

Try it · Rotate the qubit

Try it · Probability slider

Try it · Bell pair

Press “Measure A” — both will collapse to the same value.

Try it · CHSH experiment

A Bell pair |Φ⁺⟩ is shared between Alice and Bob. Each trial they independently pick a measurement angle at random — Alice from {a₀ = 0°, a₁ = 90°}, Bob from {b₀ = 45°, b₁ = 135°}. We tally same-vs-different outcomes per setting pair and compute the correlation E(aᵢ,bⱼ) = (#same − #diff) / N.

Quantum prediction: |S| → 2√2 ≈ 2.828. Classical max: |S| ≤ 2.

Try it · Repeated measurement

Counts — zeros: 0   ones: 0

Gate library

Try it · Circuit builder

Pick a gate, then click a wire to place it.

|00⟩  100.00%

Try it · CNOT phase kickback

Setup: control starts in |+⟩ (after H on |0⟩). Pick a target state, then run CNOT. Watch what happens to each qubit.

1 · Bit vs Qubit

A classical bit is one of two definite values. A qubit is a complex unit vector in a 2-dimensional Hilbert space — a continuum of possibilities. But here's the catch: you can only extract one classical bit per qubit measurement.

⚠ Holevo bound (1973). A single qubit transmits at most 1 cbit of classical info. The exponentially-large state space is real, but most of it is inaccessible to a measurement.

2 · Density matrix — the universal state representation

Kets |ψ⟩ describe pure states only — perfectly known wavefunctions. Real systems are often mixed: probabilistic ensembles of pure states (from noise, tracing out an environment, or a partial measurement). The density matrix ρ handles both uniformly.

• Pure state: ρ = |ψ⟩⟨ψ|,   Tr(ρ²) = 1
• Mixed state: ρ = Σ pᵢ |ψᵢ⟩⟨ψᵢ|,   Tr(ρ²) < 1
• Maximally mixed: ρ = I/d,   "no information" — measurements look uniform

For a composite system AB in a joint state ρ_AB, Alice's local view is obtained by the partial trace ρ_A = Tr_B(ρ_AB). For an entangled pure state ρ_AB = |ψ⟩⟨ψ|, the partial trace ρ_A is generally mixed — the entanglement appears as local randomness.

3 · Von Neumann entropy

The quantum analog of Shannon entropy:

where λᵢ are the eigenvalues of ρ. Properties:

• Pure state: S(|ψ⟩⟨ψ|) = 0 — perfect knowledge.
• Maximally mixed: S(I/d) = log₂ d — total ignorance.
• Bell pair |Φ⁺⟩: S(ρ_AB) = 0 (joint pure) but S(ρ_A) = 1 ebit. That gap is the entanglement.
• Subadditivity: S(ρ_AB) ≤ S(ρ_A) + S(ρ_B) — quantum correlations reduce joint entropy below the sum of parts (the opposite is impossible classically).

3a · Worked example — density matrices & von Neumann entropy

Three states, three density matrices, three entropies. The recipe in each case: build ρ, diagonalise to read off its eigenvalues {λᵢ}, then compute S(ρ) = −Σ λᵢ log₂ λᵢ (using the convention 0·log 0 = 0).

A · Pure state |+⟩ = (|0⟩+|1⟩)/√2

For a pure state, ρ = |ψ⟩⟨ψ|. With α = β = 1/√2:

Eigenvalues: λ = {1, 0}.   Tr(ρ²) = 1 → confirms pure.

B · Classically-mixed state — 50/50 ensemble of |0⟩ and |1⟩

Alice flips a fair coin and prepares |0⟩ or |1⟩ accordingly:

Eigenvalues: λ = {½, ½}.   Tr(ρ²) = ½ < 1 → mixed (in fact, maximally mixed).

⚠ Same ρ ≠ same preparation. The ensemble ½|+⟩⟨+| + ½|−⟩⟨−| also gives ρ = I/2. Any two preparations with the same ρ are operationally indistinguishable — no measurement can tell them apart.

C · Entangled Bell state |Φ⁺⟩ = (|00⟩+|11⟩)/√2

The joint state is pure, so S(ρ_AB) = 0. But what does Alice see locally? Trace out Bob:

Eigenvalues of ρ_A: {½, ½}.   Locally Alice sees the maximally mixed state — even though the joint state is perfectly known.

The gap S(ρ_A) − S(ρ_AB) = 1 bit is the entanglement entropy — quantifies "1 ebit" of shared resource. Classically impossible: a joint distribution can never be more certain than its marginals.

Summary

4 · No-cloning theorem (Wootters & Zurek, 1982; Dieks, 1982)

There is no unitary operator U on a 2-qubit system that maps |ψ⟩|0⟩ → |ψ⟩|ψ⟩ for every input state |ψ⟩.

Proof sketch. Suppose U exists. For two states |ψ⟩, |φ⟩, take the inner product of both sides of U|ψ⟩|0⟩ = |ψ⟩|ψ⟩ and U|φ⟩|0⟩ = |φ⟩|φ⟩. Unitarity preserves inner products, so ⟨ψ|φ⟩ = ⟨ψ|φ⟩². The only solutions are 0 and 1 — meaning |ψ⟩ and |φ⟩ must be orthogonal or identical. Cloning works for those, but not for arbitrary states. Quantum information is fundamentally uncopyable.

Consequences: QKD security (Eve cannot copy intercepted qubits), no perfect quantum amplifiers, no faster-than-light signalling via entanglement, quantum money (Wiesner 1969 → BB84 1984).

5 · The quantum-information conversion table

Three resources interconvert: qubits sent over a quantum channel, cbits sent over a classical channel, and ebits shared as entangled Bell pairs.

Together, teleportation + superdense coding say that 1 qubit ≡ 2 cbits + 1 ebit and 2 cbits ≡ 1 qubit + 1 ebit — consistent only because entanglement alone cannot send classical information (no-signalling theorem).

6 · Quantum channels

A quantum channel ℰ is a completely-positive trace-preserving (CPTP) map from density matrices to density matrices. It generalises classical noisy channels. Common examples on a single qubit:

Channels have multiple capacities: the classical capacity (cbits per use), the quantum capacity (qubits per use), the entanglement-assisted capacity, the private capacity. Unlike classical channels, these are different numbers, and computing them exactly remains a major open problem.

7 · Shannon ↔ von Neumann — explained with examples

Shannon's three classical theorems (1948–1959) each have a quantum twin. The easiest way in is one concrete example per theorem — then look at the formula.

The basic swap

① Compression — "how few bits do I need?"

Classical example. A weather sensor outputs "sunny" 90% of days and "rainy" 10%. You don't need 1 bit per day — Shannon says you can compress to:

Real-world example: English text has H ≈ 1.0–1.5 bits per letter (not log₂ 26 ≈ 4.7), which is why zip files shrink novels to ~30% of their size.

Quantum example. A source emits |0⟩ 90% of the time and |+⟩ 10% of the time (non-orthogonal states — you can't even reliably tell them apart). Schumacher's theorem says you can squeeze them into:

Punchline. Same formula, same answer — just replace "probabilities" with "eigenvalues of ρ". When the quantum states happen to be orthogonal (perfectly distinguishable, like |0⟩ and |1⟩), S(ρ) = H(p) exactly, and Schumacher reduces to Shannon.

② Channel capacity — "how fast can I talk without errors?"

Classical example. A noisy phone line flips each bit with probability 10%. Shannon's noisy-coding theorem says the maximum reliable rate is:

Send any rate below 0.53 — error rate can be made arbitrarily small with the right code. Try to send faster — errors are unavoidable. This is why your Wi-Fi slows down when the signal is weak but never silently corrupts files.

Quantum example. Send qubits down a noisy fiber. Now you get two capacity numbers:

Why two? A quantum channel can carry classical info or quantum info, and the rates are different. Always C ≥ Q (classical is easier).

Surprise without classical analogue — superactivation. Take two quantum channels each with zero quantum capacity (Q = 0 — neither alone can send a single qubit). Use them together and you can get Q > 0! (Smith & Yard, 2008.) Like adding two zeros and getting a positive number. Impossible classically.

③ Lossy compression — "how small can I get if I accept some error?"

Classical example. JPEG, MP3, and video codecs throw away detail your senses won't miss. Shannon's rate–distortion theorem (1959) tells you the minimum bits per pixel/sample for a chosen distortion D (e.g. mean-squared error):

That's why a 4K image compresses to ~1 bit per pixel at "visually perfect" quality but lossless PNG needs 8–12×.

Quantum example. Quantum memory is expensive and noisy. If you can tolerate a small fidelity drop (e.g. 1 − F ≤ 0.05), how few qubits do you need to store ρ? Barnum (2000) and Datta–Hsieh–Wilde (2013) give:

Same shape as Shannon, with coherent information I_c replacing mutual information I. Useful for compressing quantum data in noisy memories or simulators — and, like the quantum capacity, generally harder to compute than its classical sibling.

What's genuinely new in the quantum world

Headline. Quantum information is a strict superset of classical. Take any quantum result, restrict to commuting (≈ classical) states, and Shannon's theorems pop right out. Allow non-commuting states, entanglement, and superposition, and you unlock effects — negative entropy, superactivation, ebit-assisted capacity — that have no classical mirror at all.

Try it · Send a qubit through a noisy channel

Click "Send 1 qubit" to watch one travel through.

Try it · build up the codes step by step

Try it · Pick a demo

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

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

|s⟩  = H^⊗n |0⟩^⊗n  =  (1/√N) Σ_x |x⟩       (uniform)
|w⟩  = the marked basis state
|s'⟩ = (1/√(N−1)) Σ_{x ≠ w} |x⟩              (uniform over the rest)

After k Grover iterations:
  G^k |s⟩ = sin((2k+1)θ) |w⟩ + cos((2k+1)θ) |s'⟩

k_opt ≈ (π/(4θ)) − 1/2  ≈  (π/4)·√N   for large N

|ψ₀⟩ = |0⟩^⊗n ⊗ |1⟩

|ψ₁⟩ = (1/√2ⁿ) Σ_{x∈{0,1}ⁿ} |x⟩  ⊗  |−⟩

|ψ₂⟩ = (1/√2ⁿ) Σ_x (−1)^f(x) |x⟩  ⊗  |−⟩

|ψ₃⟩ = (1/2ⁿ) Σ_y [ Σ_x (−1)^(f(x) ⊕ x·y) ] |y⟩  ⊗  |−⟩

amplitude(|0…0⟩) = (1/2ⁿ) Σ_x (−1)^f(x)
                  = ±1  if f is constant   (all signs agree)
                  =  0  if f is balanced   (signs cancel)

|ψ₂⟩ = (1/√2ⁿ) Σ_x (−1)^(s·x) |x⟩  ⊗  |−⟩

|ψ₃⟩ = (1/2ⁿ) Σ_y [ Σ_x (−1)^(s·x ⊕ x·y) ] |y⟩  ⊗  |−⟩
     = (1/2ⁿ) Σ_y [ Σ_x (−1)^((s⊕y)·x) ] |y⟩  ⊗  |−⟩

amplitude(|y⟩) =  1   if y = s
               =  0   otherwise

(1/√2ⁿ) Σ_x |x⟩|0⟩

(1/√2ⁿ) Σ_x |x⟩|f(x)⟩

(1/√2) ( |x₀⟩ + |x₀ ⊕ s⟩ )

(1/√2ⁿ⁺¹) Σ_y [ (−1)^(x₀·y) + (−1)^((x₀⊕s)·y) ] |y⟩
   = (1/√2ⁿ⁺¹) Σ_y (−1)^(x₀·y) [ 1 + (−1)^(s·y) ] |y⟩

P(y) = 1/2^(n−1)  for all y with y·s = 0
       0          otherwise

|ψ⟩ ⊗ |Φ⁺⟩
  = (α|0⟩ + β|1⟩) ⊗ (|00⟩ + |11⟩)/√2
  = (1/√2) [ α|0⟩(|00⟩+|11⟩) + β|1⟩(|00⟩+|11⟩) ]
  = (1/√2) [ α|000⟩ + α|011⟩ + β|100⟩ + β|111⟩ ]

→ (1/√2) [ α|000⟩ + α|011⟩ + β|110⟩ + β|101⟩ ]

→ (1/2) [ α(|0⟩+|1⟩)|00⟩ + α(|0⟩+|1⟩)|11⟩
        + β(|0⟩−|1⟩)|10⟩ + β(|0⟩−|1⟩)|01⟩ ]

= (1/2) [ |00⟩ (α|0⟩ + β|1⟩)
        + |01⟩ (α|1⟩ + β|0⟩)
        + |10⟩ (α|0⟩ − β|1⟩)
        + |11⟩ (α|1⟩ − β|0⟩) ]

m₁ m₂ = 00  →  Bob has α|0⟩ + β|1⟩   = |ψ⟩            → apply I
m₁ m₂ = 01  →  Bob has α|1⟩ + β|0⟩   = X|ψ⟩           → apply X
m₁ m₂ = 10  →  Bob has α|0⟩ − β|1⟩   = Z|ψ⟩           → apply Z
m₁ m₂ = 11  →  Bob has α|1⟩ − β|0⟩   = XZ|ψ⟩          → apply X then Z

|0⟩ ⊗ (|00⟩ + |11⟩)/√2
  = (1/√2) ( |0⟩|00⟩ + |0⟩|11⟩ )
  = (1/√2) ( |000⟩ + |011⟩ )

(1/√2) ( |000⟩ + |011⟩ )

(1/2) ( |0⟩|00⟩ + |1⟩|00⟩ + |0⟩|11⟩ + |1⟩|11⟩ )
= (1/2) ( |000⟩ + |100⟩ + |011⟩ + |111⟩ )

= (1/2) [ |00⟩|0⟩ + |01⟩|1⟩ + |10⟩|0⟩ + |11⟩|1⟩ ]

m₁ m₂ = 00  →  Bob has |0⟩
m₁ m₂ = 01  →  Bob has |1⟩
m₁ m₂ = 10  →  Bob has |0⟩
m₁ m₂ = 11  →  Bob has |1⟩

00:  Bob has |0⟩  →  I·|0⟩       = |0⟩       ✓
01:  Bob has |1⟩  →  X·|1⟩       = |0⟩       ✓
10:  Bob has |0⟩  →  Z·|0⟩       = |0⟩       ✓
11:  Bob has |1⟩  →  Z·X·|1⟩     = Z·|0⟩ = |0⟩  ✓

|ψ₀⟩ = |0⟩|1⟩

|ψ₁⟩ = |+⟩|−⟩
     = (1/√2) (|0⟩ + |1⟩) ⊗ |−⟩

|ψ₂⟩ = (1/√2) [ (−1)^f(0) |0⟩ + (−1)^f(1) |1⟩ ] ⊗ |−⟩

|ψ₂⟩ = (−1)^f(0) · (1/√2) [ |0⟩ + (−1)^(f(0)⊕f(1)) |1⟩ ] ⊗ |−⟩

if d = 0  (constant):   (1/√2) [ |0⟩ + |1⟩ ] = |+⟩
if d = 1  (balanced):   (1/√2) [ |0⟩ − |1⟩ ] = |−⟩

if constant:  data → |0⟩   →   measurement = 0
if balanced:  data → |1⟩   →   measurement = 1

QFT |x⟩ = (1/√N) Σ_{k=0}^{N−1} e^(2πi·xk/N) |k⟩

QFT |x⟩ = (1/√N) ⊗_{ℓ=1..n} [ |0⟩ + e^(2πi · x · 2^(−ℓ)) |1⟩ ]

Press the button to factor.

a^r − 1 = (a^(r/2) − 1)(a^(r/2) + 1) ≡ 0 (mod N)
⇒ gcd(a^(r/2) ± 1, N) is a non-trivial factor of N

(1/√2^t) Σ_x |x⟩ |a^x mod N⟩

QFT |comb_r⟩  ⟶  peaks at k · 2^t / r,  k = 0, 1, …, r−1

⟨ψ| Ĥ |ψ⟩  ≥  E_0

⟨ψ| Ĥ |ψ⟩ = Σ_k |c_k|² E_k
            ≥ E_0 · Σ_k |c_k|²
            = E_0    (since Σ |c_k|² = 1)

U(γ, β) = Π_{k=1..p} e^(−iβ_k H_M) · e^(−iγ_k H_C)

e^(−i Δt · H(s_k)) ≈ e^(−iβ_k H_M) · e^(−iγ_k H_C)

F_p(γ, β) = ⟨ψ_p(γ, β)| H_C |ψ_p(γ, β)⟩

Pick a paradox

The threat — "harvest now, decrypt later"

A sufficiently large quantum computer running Shor's algorithm would break virtually all the public-key cryptography deployed on the internet today: RSA, Diffie-Hellman, elliptic-curve (ECC). Even though such a computer doesn't exist yet, adversaries are recording encrypted traffic right now to decrypt later when the hardware arrives. Long-lived secrets (medical records, state secrets, source code, banking) are already at risk — measured in 25-year confidentiality windows.

Families of post-quantum algorithms

PQC schemes derive their security from problems no quantum algorithm can efficiently solve. The five main families:

NIST PQC standardisation

Started in 2016, NIST evaluated >80 submissions over four rounds. The first standards were published in August 2024:

⚠ SIKE (isogeny-based) was a NIST round-4 candidate — broken by Castryck & Decru in 2022 using ordinary classical math (a Magma laptop, in under an hour). A reminder that "post-quantum" doesn't mean "post-cryptanalysis".

Real-world deployments

• Cloudflare & Google have deployed hybrid (X25519 + ML-KEM) key exchange on TLS 1.3 since 2023 — over 30% of Chrome's TLS connections in 2025.
• Apple iMessage PQ3 (2024) — full hybrid post-quantum end-to-end encryption.
• Signal Messenger rolled out PQXDH (X25519 + Kyber) in 2023.
• OpenSSH 9.0+ defaults to hybrid sntrup761x25519 key exchange.
• AWS KMS, Cloudflare 1.1.1.1, Microsoft Azure all support PQ-hybrid TLS.

PQC vs QKD — complementary, not competing

Most realistic deployments will combine both, plus hybrid classical+PQC in the transition period (2025-2030+).