Density Matrices & Mixed States
How to describe a qubit when you're not sure exactly what state it's in
By the end of this topic you'll be able to
Up to now we have always assumed we know exactly which state |ψ⟩ our system is in. Binney and Skinner point out how unrealistic this is even in principle: establishing a system's state this precisely requires that a measurement has just been made, collapsing it into a known eigenstate — a procedure that makes no allowance for the experimental error that is endemic in every real laboratory.
For a macroscopic object the assumption collapses entirely. How would one possibly determine the individual quantum states of the ~10²³ atoms in a diamond, given that measuring even one of them disturbs the rest?
So suppose instead that we admit we don't know the system's state, but believe it is in one of a complete set of states {|n⟩}, each with some probability pₙ. Crucially, this is not the same as saying the system is in the definite superposition Σₙ√pₙ|n⟩ — that would be a perfectly good pure state, and we are explicitly admitting we don't have one.
Binney and Skinner call this an impure state, reserving 'pure state' for an ordinary ket |ψ⟩; the field as a whole more often says mixed state, but the idea is identical either way: the object we need is the density operator, ρ ≡ Σₙ pₙ|n⟩⟨n|.
For an ordinary pure state, ρ reduces to the familiar projector |ψ⟩⟨ψ|. For a 50/50 classical toss between |0⟩ and |1⟩ — uncertainty about which was prepared, not a superposition of the two — ρ = ½|0⟩⟨0| + ½|1⟩⟨1| instead, a genuinely different mathematical object from the superposition (|0⟩+|1⟩)/√2 even though a casual glance might confuse them.
The terminology 'impure' is, as Binney and Skinner note, slightly unfortunate: a system described by such a ρ is in a perfectly good quantum state. What's impure is our knowledge of which state that is — not the state itself.
Density matrices become unavoidable, not merely convenient, for a qubit that is entangled with something else. Given the Bell state |Φ⁺⟩=(|00⟩+|11⟩)/√2, there is simply no ket that describes qubit A alone; tracing out B gives ρ_A = I/2, the maximally mixed state, and A looks on its own exactly like a fair coin, even though the full two-qubit system sits in a perfectly definite pure state.
At purity 1 the arrow sits on the surface — a pure state. Drag it down to 0 and the arrow shrinks to the center — the maximally mixed state I/2, indistinguishable from a fair coin no matter what you measure.
That apparent randomness in A is not fundamental uncertainty about A — it is the unavoidable price of describing A while deliberately ignoring B. The same formalism is the natural language for noise: any qubit that interacts with an uncontrolled environment ends up mixed, and the whole theory of decoherence and quantum error correction, coming up next, is written in terms of ρ rather than |ψ⟩.
Try It Yourself
Write the density matrix for the pure state |+⟩ = (|0⟩+|1⟩)/√2, then for a mixed state that's |0⟩ with 70% probability and |1⟩ with 30% probability. Compute Tr(ρ²) for each and explain what the difference means.
- 1Pure state: ρ = |+⟩⟨+| = (1/2)(|0⟩+|1⟩)(⟨0|+⟨1|) = (1/2)(|0⟩⟨0| + |0⟩⟨1| + |1⟩⟨0| + |1⟩⟨1|) = (1/2)[[1,1],[1,1]] in matrix form.
- 2Tr(ρ²) for this pure state: ρ² = (1/4)[[1,1],[1,1]][[1,1],[1,1]] = (1/4)[[2,2],[2,2]] = (1/2)[[1,1],[1,1]] = ρ itself (a property unique to pure-state projectors), so Tr(ρ²) = Tr(ρ) = 1/2+1/2 = 1.
- 3Mixed state: ρ = 0.7|0⟩⟨0| + 0.3|1⟩⟨1| = [[0.7, 0],[0, 0.3]] — diagonal, since there's no coherent superposition, just classical uncertainty about which basis state you have.
- 4Tr(ρ²) for this mixed state: ρ² = [[0.49, 0],[0, 0.09]], so Tr(ρ²) = 0.49 + 0.09 = 0.58.
The pure state gives Tr(ρ²) = 1 exactly; the mixed state gives Tr(ρ²) = 0.58 < 1. Tr(ρ²) = 1 is a general litmus test for purity — any ρ with Tr(ρ²) < 1 is a genuine statistical mixture, not a coherent superposition, no matter how it's written.
Reference
| Pure state density matrix | Rank-1 matrix; Tr(ρ²) = 1 | |
| Mixed state density matrix | Weighted mixture; Tr(ρ²) < 1 | |
| Maximally mixed qubit | Complete uncertainty; looks like a fair coin on measurement | |
| Partial trace | Describes subsystem A alone, ignoring B |
Quick Check
What's the density matrix of qubit A alone, when the full 2-qubit system is in the Bell state |Φ⁺⟩?
What distinguishes a mixed state from a pure superposition state?