Part VIII · Formalism & Frontiers — 22

State Tomography

You can prepare a state, evolve it, send it through a channel — but how do you ever find out which state you actually have? A single measurement barely whispers. The answer is statistics: rebuild the whole state from many shots.

↩ before you start · keep these handy

·From Ch. 4: a qubit is a point in the Bloch ball, its coordinates the averages ⟨X⟩, ⟨Y⟩, ⟨Z⟩.

·From Ch. 3: one measurement gives one bit and collapses the state; you can’t peek twice.

·From Ch. 15: a single qubit only yields one bit — the continuum is real but not readable in one shot.

🔑 symbol decoder · every new mark, in plain words

⟨X⟩,⟨Y⟩,⟨Z⟩the average result of measuring along each axis — between −1 and +1. ρ = ½(I + r·σ)the density matrix written from the Bloch vector r = (⟨X⟩,⟨Y⟩,⟨Z⟩). σthe trio of Pauli matrices (X, Y, Z) — the three measurement directions. 1/√Nhow fast the error shrinks with N shots — 4× the shots halves the error. 4ₙhow the number of measurement settings grows for n qubits — why real devices are hard to characterise.

feel

One shot tells you almost nothing

Measure a qubit once and you get a single bit — 0 or 1 — and the state is gone (collapse, chapter 03; no-cloning, chapter 7a). From that one bit you could never name the continuous angles that defined it. But prepare the same state many times and measure along different axes, and the frequencies of outcomes slowly betray it. Tomography is detective work: each axis gives one coordinate of the Bloch arrow, and enough shots sharpen the estimate until the hidden state stands fully reconstructed.

📸 everyday picture

To capture a sculpture’s shape you don’t need a magic 3-D scanner — just photograph it from the front, side and top, and the three silhouettes pin it down. Tomography (the word literally means “imaging by slices”) does the same to a qubit: measuring along X, Y and Z gives three “shadows,” one Bloch coordinate each. The catch a sculptor doesn’t face — each quantum photo destroys the subject, so you need a fresh identical copy for every single shot.

recapMany shots along three axes turn outcome frequencies into the three Bloch coordinates.

play

Reconstruct the hidden state

A hidden qubit (faint arrow) is waiting. Fire batches of measurements along X, Y and Z; each batch sharpens that coordinate of your estimate (solid arrow). Watch it home in on the truth — and notice it takes many shots to get there.

▸ Bloch tomographyρ = ½(I + r·σ)

shown in the X–Z plane · ⟨Y⟩ in the gauge →

              ⟨{{ a.name }}⟩ = {{ a.est }}{{ a.shots }} shots

total shots{{ total }}
reconstruction error{{ error }}
{{ desc }}

recapThe estimate homes in on the hidden arrow as shots pile up — slowly, like 1/√N.

math

Three expectations fix the state

Any single-qubit density matrix is pinned down by just three numbers — the expectation values of the three Pauli measurements — which are the components of the Bloch vector:

ρ = ½(I + ⟨X⟩X + ⟨Y⟩Y + ⟨Z⟩Z)
⟨Z⟩ = P(0) − P(1),  estimated from Z-basis counts
⟨X⟩, ⟨Y⟩ the same, in the X- and Y-bases
statistical error of each shrinks like 1/√N — the cost of certainty

Three measurement settings, many repetitions each, and the Bloch vector emerges. For n qubits it generalises — though the number of settings grows as 4ⁿ, which is exactly why characterising a real device is hard. Crucially, every estimate needs a fresh copy: you cannot measure the same qubit three ways, because the first measurement already destroyed it.

✎ worked example · estimating ⟨Z⟩ from 100 shots

measure 100 fresh copies in the Z basis: say 75 give 0, 25 give 1.
⟨Z⟩ = P(0) − P(1) = 0.75 − 0.25 = +0.5 — one Bloch coordinate found.
repeat in X and Y for ⟨X⟩, ⟨Y⟩; together they fill r = (⟨X⟩,⟨Y⟩,⟨Z⟩).
uncertainty on each ≈ 1/√100 = 0.1; want 0.01? You need 10,000 shots. ✓

recapThree Pauli expectations fix ρ = ½(I + r·σ); each needs many fresh copies, error falling like 1/√N.

⚠ common misconception

“Just measure the qubit and read off its state.” A single qubit can never reveal its own state — one measurement yields one bit and collapses the rest. Tomography only works because you have many identical copies to spend, one per shot. The continuous state is real but only knowable in aggregate, echoing the Holevo lesson of chapter 15.

This is how every claim in this course is checked in a real lab: prepare, measure thousands of times, reconstruct. Next we turn from reading states to two algorithms that bridge the gap between the first quantum win and the most famous one.

✓ you can now

✓estimate ⟨X⟩, ⟨Y⟩, ⟨Z⟩ from measurement counts and assemble ρ = ½(I + r·σ)

✓explain why tomography needs many identical copies, and how error falls like 1/√N

✓say why characterising n qubits is hard — the settings grow as 4ₙ

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