Maximally mixed state I/2ⁿ

Every mixed-state construction in this catalog has the maximally mixed state as a limiting or reference case: it is the infinite-temperature limit of any Gibbs state, the reduced state of half a maximally entangled pair, and the fixed point of complete depolarization — making it the natural 'zero information' baseline against which every other state's purity and entanglement are measured.

TakesNothingWhat joins here

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Returns2 qubitsWhat joins here

Returns a 2-qubit register a next stage can take. This is the end that joins.

12 entries meet this end. Named below.

Prepares a state. Nothing goes in, and what comes out is a register another stage can take. See all 13 →

Fits after this 12

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.

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 $\sum_x \frac{1}{2^n}|x\rangle\langle x| = I/2^n$ .
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.
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.

TakesNothingWhat joins here

Nothing goes in, and that is what this entry is: a pipeline starts here. There is no upstream to choose — what you pick is what comes after.

Nothing in the Atlas meets this end.

Returns2 qubitsWhat joins here

Returns a 2-qubit register a next stage can take. This is the end that joins.

12 entries meet this end. Named below.

Prepares a state. Nothing goes in, and what comes out is a register another stage can take. See all 13 →

Fits after this 12

Qubits: n (n=1,2 shown)
Purity: Tr(ρ²) = 1/2ⁿ
Entropy: S(ρ) = n bits (maximal)
State: ρ = I/2ⁿ
Invariance: UρU† = ρ for all unitary U
T→∞ limit of: Any Gibbs state e^{-βH}/Z

Uniform mixture over {|00⟩,|01⟩,|10⟩,|11⟩} on q[0], q[1]
≡ partial trace of Bell pair on q[0], q[1]

Each basis string (n=2)25%

Quantum vs classical

Classical baseline

Use a classical state-vector or matrix simulation at the same width, precision, and measurement objective.

Quantum claim

The quantum record demonstrates a state or operator behavior; it does not make classical simulation or communication costs disappear.

How to compare

Compare fidelity, samples, gate depth, noise, memory, and the cost of preparing and reading the state.

Nobody has reviewed this record for gaps yet.

FrameworkNative

maximally_mixed_state.py

import numpy as np
from qiskit import QuantumCircuit
from qiskit.quantum_info import DensityMatrix, Statevector, partial_trace

def maximally_mixed(n: int) -> DensityMatrix:
    return DensityMatrix(np.eye(2 ** n) / 2 ** n)

rho2 = maximally_mixed(2)
print("Trace:", np.trace(rho2.data))                  # 1.0
print("Eigenvalues:", np.linalg.eigvalsh(rho2.data))  # [0.25, 0.25, 0.25, 0.25]

# Construction 2: partial trace of a Bell pair reproduces I/2 for one qubit
qc = QuantumCircuit(2)
qc.h(0)
qc.cx(0, 1)
bell = Statevector.from_instruction(qc)
reduced = partial_trace(bell, [1])  # trace out qubit 1
print("Reduced state of one Bell-pair qubit:\n", np.real(reduced.data))  # [[0.5, 0], [0, 0.5]]


FINAL_CIRCUIT = qc

Where the map uses this

This record is an instance of an object the map names, so these are the processes that consume or produce one. None of them is about this record in particular.

State you can prepare 5 of 34 processes

Quantum linear solve hands one back
Prepare an input state hands one back
Amplify a success branch hands one back
Estimate an observable takes one
Minimise the objective over the parameters hands one back

Related entries

Single-qubit thermal (Gibbs) state
Operators
Werner state (two-qubit)
Operators
Bell state measurement
Basic circuits