Von Neumann Entropy: Subadditive Entropy

The explorer above walks through the whole calculation: pick a state, optionally blend in classical noise, and watch the density matrix, its eigenvalue spectrum, and the entropy update together.

1. What von Neumann entropy is

Entropy is one of the deepest ideas in physics: in thermodynamics it measures disorder and multiplicity, in information theory uncertainty. Quantum mechanics fuses the two in one quantity, introduced by John von Neumann in 1932 [1]. For a state with density matrix ρ, the von Neumann entropy is

$$S(\rho) = -\operatorname{Tr}(\rho \log \rho)$$

With the logarithm base 2, entropy is in bits, the convention in quantum information and in the explorer above; with the natural log and a factor of Boltzmann's constant it becomes thermodynamic entropy. It is the direct quantum generalization of Shannon entropy, the uncertainty in a classical probability distribution [2]: a coin certainly heads has zero entropy, a fair coin one full bit.

2. The density matrix connection

Why a density matrix rather than a wavefunction? Because quantum states come in two flavors. A pure state is a fully specified wavefunction |ψ⟩, with ρ = |ψ⟩⟨ψ|. A mixed state is a statistical blend, in |ψ₁⟩ with probability q₁, |ψ₂⟩ with probability q₂, and so on, giving ρ = Σ qₖ|ψₖ⟩⟨ψₖ|. The density matrix handles both in one object [3].

The recipe for computing S(ρ) is short:

Diagonalize ρ. Its eigenvalues p₁, p₂, …, p_d are non-negative and sum to 1, so they behave exactly like probabilities (panel ② in the explorer).
Apply Shannon's formula to the spectrum:

$$S(\rho) = -\sum_i p_i \log_2 p_i$$

That is all the trace formula says: von Neumann entropy is Shannon entropy applied to the density matrix's eigenvalues. Those eigenvalues are basis-independent, so the entropy is an intrinsic property of the state, regardless of how you write ρ.

3. Pure versus mixed states

The two extremes bracket everything:

Pure state: one eigenvalue equals 1, the rest are 0, so S(ρ) = 0. Zero entropy means complete information: the system is in a definite quantum state, even if measurement outcomes are still probabilistic.
Maximally mixed state: ρ = I/d, all d eigenvalues equal 1/d, and S(ρ) = log₂ d, the largest value possible. For n qubits, d = 2ⁿ, so the maximum entropy is exactly n bits.

The mixing slider interpolates between these extremes, ρ(λ) = (1 − λ)|ψ⟩⟨ψ| + λ I/d. As λ grows, the dominant eigenvalue shrinks, the others rise toward 1/d, and the entropy climbs from 0 to its ceiling. Any entropy above zero signals missing information: either genuine ignorance about how the state was prepared, or entanglement with something outside our view.

4. Examples: 1, 2, and 3 qubits

One qubit (d = 2, max entropy 1 bit)

A pure state like |0⟩ or |+⟩ = (|0⟩ + |1⟩)/√2 has S = 0. Even though |+⟩ gives 50/50 outcomes in the computational basis, its entropy is zero: entropy measures uncertainty about the state, not any particular measurement. The maximally mixed qubit ρ = I/2 has eigenvalues {½, ½} and S = 1 bit.

Two qubits (d = 4, max entropy 2 bits)

The product state |00⟩ is pure: S = 0. The Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2 is also pure, global entropy exactly 0, yet its subsystems tell a very different story, as below. The maximally mixed two-qubit state I/4 has four eigenvalues of ¼ and S = 2 bits.

Tracing out qubit B means summing over its basis states. Writing the 4×4 density matrix in 2×2 blocks grouped by the value of qubit A:

$$\rho = \begin{pmatrix} M_{00} & M_{01} \\ M_{10} & M_{11} \end{pmatrix} \implies \rho_A = \operatorname{Tr}_B(\rho) = \begin{pmatrix} \operatorname{tr}(M_{00}) & \operatorname{tr}(M_{01}) \\ \operatorname{tr}(M_{10}) & \operatorname{tr}(M_{11}) \end{pmatrix}$$

Each element of ρ_A is the trace of the corresponding 2×2 block of ρ. The orange-outlined blocks in the explorer are M_{00} and M_{11}; for the states shown, where qubit A carries no coherence with the rest, only these two diagonal blocks contribute.

Three qubits (d = 8, max entropy 3 bits)

Two famous entangled states contrast nicely. The GHZ state (|000⟩ + |111⟩)/√2 and the W state (|001⟩ + |010⟩ + |100⟩)/√3 are both pure, so both have S(ρ) = 0 globally. Trace out two qubits and they differ: a GHZ qubit is maximally mixed (eigenvalues {½, ½}, entropy 1 bit), while a W qubit has eigenvalues {⅔, ⅓} and entropy ≈ 0.918 bits. The two carry entanglement of genuinely different structure [4], and subsystem entropy is one way to see it.

5. Entanglement and subsystem entropy

Here is the most striking feature of von Neumann entropy. Take the Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2. The pair is pure, so its joint density matrix satisfies S(ρ_AB) = 0: complete information about both qubits together. (The two-qubit maximum is 2 bits, reached only by the maximally mixed I/4.) Now trace out one qubit and look at its partner alone:

$$\rho_A = \operatorname{Tr}_B\!\left(|\Phi^+\rangle\langle\Phi^+|\right) = \frac{I}{2} \implies S(\rho_A) = 1\text{ bit}$$

That 1 bit is the most a single qubit can carry, so the subsystem is maximally mixed even though the whole is perfectly known. Classically this is impossible: knowing a composite system completely means knowing its parts. Quantum mechanically the information lives in the correlations between the qubits, not in either alone. The entropy of a reduced state, the entanglement entropy, is therefore a quantitative measure of entanglement for pure states [3], [4]: 0 for a product state, the maximal 1 bit for a Bell state, in between for partially entangled states (try the "tilted" preset and watch panel ④).

This one idea radiates across modern physics: entanglement entropy of black hole horizons, area laws constraining the ground states of local Hamiltonians [5], bounds on how much a quantum error-correcting code can hide in a subsystem, and the capacities of quantum channels [6]. Whenever a quantum system is split into pieces, von Neumann entropy tells you how much information the split conceals.

References

J. von Neumann, Mathematical Foundations of Quantum Mechanics, R. T. Beyer, Trans. Princeton, NJ, USA: Princeton Univ. Press, 1955. (Original work published 1932.)
C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J., vol. 27, no. 3, pp. 379–423, Jul. 1948, doi: 10.1002/j.1538-7305.1948.tb01338.x.
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed. Cambridge, U.K.: Cambridge Univ. Press, 2010.
R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, "Quantum entanglement," Rev. Mod. Phys., vol. 81, no. 2, pp. 865–942, Jun. 2009, doi: 10.1103/RevModPhys.81.865.
J. Eisert, M. Cramer, and M. B. Plenio, "Colloquium: Area laws for the entanglement entropy," Rev. Mod. Phys., vol. 82, no. 1, pp. 277–306, Feb. 2010, doi: 10.1103/RevModPhys.82.277.
M. M. Wilde, Quantum Information Theory, 2nd ed. Cambridge, U.K.: Cambridge Univ. Press, 2017.