The Heisenberg Limit

The standard quantum limit gives $\Delta\theta \sim 1/\sqrt N$. The Heisenberg limit is the ultimate floor allowed by quantum mechanics for $N$ probes interrogated once:

a quadratic improvement in the resource $N$. This lesson derives it and explains why it is the end of the road.

From Fisher information to the limit

We assembled the ingredients in the previous lessons. The GHZ probe has quantum Fisher information $F_Q = N^2$, and the quantum Cramér–Rao bound says

So a single shot of a GHZ probe can in principle reach $\Delta\theta = 1/N$. Contrast the two scalings for, say, $N = 100$ qubits: the SQL gives $\Delta\theta = 1/10$, while the Heisenberg limit gives $\Delta\theta = 1/100$ — a tenfold improvement that grows as $\sqrt N$.

Why $1/N$ and not faster?

Could a cleverer probe beat $1/N$? For a generator $G = \tfrac12\sum_k Z_k$ acting on $N$ qubits, the spread of eigenvalues is bounded: the largest is $+N/2$ and the smallest is $-N/2$. The variance of any state is therefore capped,

so $F_Q = 4\,\mathrm{Var}(G) \le N^2$. The GHZ state, which puts all its weight on the two extreme eigenvalues $\pm N/2$, saturates this bound. No state can do better, so $1/N$ is genuinely the floor — hence the name "Heisenberg" limit, echoing the energy–time uncertainty relation that underlies it.

The advantage in one table

| Strategy | Probe | $F_Q$ | $\Delta\theta$ | | --- | --- | --- | --- | | Classical / independent | $N \times |+\rangle$ | $N$ | $1/\sqrt N$ (SQL) | | Entangled | $|\mathrm{GHZ}_N\rangle$ | $N^2$ | $1/N$ (Heisenberg) |

The gap between the columns is the prize that quantum sensing chases.

Try it

Realise the 2-qubit GHZ-preparation unitary — the $N = 2$ Heisenberg-limited probe builder. The grader compares the full $4\times4$ unitary matrix, so an empty or incorrect circuit will not pass.

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