Maximally mixed state I/2ⁿ

The maximally mixed state on $n$ qubits is

$\rho = \frac{I}{2^n} = \frac{1}{2^n}\sum_{x\in\{0,1\}^n} |x\rangle\langle x|.$

Unitary invariance. For any unitary $U$, $U\rho U^\dagger = U\frac{I}{2^n}U^\dagger = \frac{UU^\dagger}{2^n} = \frac{I}{2^n} = \rho$. This is the only $n$-qubit state with this property (any other state has some eigenbasis with unequal eigenvalues, and a unitary rotating that eigenbasis changes the state), which is why $I/2^n$ represents complete ignorance: no measurement in any basis, and no unitary transformation, can extract or reveal any information from it.

Maximal entropy. The von Neumann entropy $S(\rho) = -\mathrm{Tr}(\rho\log_2\rho) = -\sum_x \frac{1}{2^n}\log_2\frac{1}{2^n} = n$ bits, the maximum possible for an $n$-qubit system (achieved uniquely by $I/2^n$, exactly analogous to a uniform classical distribution maximizing Shannon entropy). Purity $\mathrm{Tr}(\rho^2) = 2^n\cdot(1/2^n)^2 = 1/2^n$ is correspondingly minimal, reflecting that this is the 'most mixed' possible state.

Three equivalent constructions.

1. Classical randomization. Flip $n$ fair, independent classical coins and prepare the corresponding computational basis string $|x\rangle$; averaging the resulting ensemble $\{p_x=1/2^n, |x\rangle\}$ over all $2^n$ outcomes gives exactly $\sum_x \frac{1}{2^n}|x\rangle\langle x| = I/2^n$.
2. Partial trace of a maximally entangled state. Tracing out either half of any maximally entangled $2n$-qubit state (e.g. $n$ Bell pairs) leaves the remaining $n$ qubits in $I/2^n$ — this is the cleanest illustration of how entanglement with an inaccessible system manifests locally as maximal mixedness, even though the global state is pure.
3. Complete depolarization / infinite temperature. $I/2^n$ is the fixed point of the completely depolarizing channel $\mathcal{E}(\rho)=I/2^n$ for all input $\rho$, and is the $\beta\to0$ (infinite-temperature) limit of the Gibbs state $e^{-\beta H}/Z$ for any Hamiltonian $H$ — exactly the limit reached by the single-qubit thermal-state purification construction elsewhere in this catalog as $\theta\to\pi/2$.

Role as a baseline. Because it carries zero distinguishing information, $I/2^n$ is the reference point for every mixedness/entanglement measure used elsewhere in this catalog: it is the $p=0$ endpoint of the Werner-state family, the maximum-entropy endpoint of any thermal state, and the state a qubit decoheres toward under a fully depolarizing noise channel — the worst-case error model in quantum error correction.