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A coherent two-level preparation and the equal statistical mixture are different states

Physical scope

A worked comparison, inside the framework's state space, of two preparations of a two-level system: the pure preparation of the relative-phase superposition (h + i v)/sqrt2 and the equal statistical mixture of the h and v rays. The two assigned states are computed explicitly and shown to be distinct, with different rank and purity: the superposition is pure and the mixture is mixed. Here rank means the number of vectors in a finite basis of the operator's range. In this item, coherent preparation denotes only the pure ray of (h + i v)/sqrt2, whose density matrix in the ordered basis (h,v) has off-diagonal entries (-i/2, i/2); the equal mixture has zero off-diagonal entries. These are descriptions of assigned states only. The item asserts no outcome probability, no measurement model, no apparatus realization and no empirical result; whether any measurement distinguishes the states belongs to the framework's probability law and is not decided here.

Example

Let $H$ be a two-dimensional complex Hilbert space with orthonormal basis $(h, v)$ , and use the state space $D (H)$ and the preparation semantics of The state space of the framework: preparations, pure rays and density operators. Consider the two preparations $superposition: w = \frac{h + i v}{\sqrt{2}}, equal mixture: ρ_{+} = \frac{1}{2} T_{h, h} + \frac{1}{2} T_{v, v},$ where $w$ is a unit vector and $ρ_{+}$ is the statistical mixture of the two basis rays with weights $\frac{1}{2}, \frac{1}{2}$ . The assigned state matrices, in the given order, are: $T_{w, w} = \frac{1}{2} (\begin{array}{cc} 1 & - i \\ i & 1 \end{array}), ρ_{+} = \frac{1}{2} I$ For an operator $A : H \to H$ , write $ran A : = {A x : x \in H}$ and define its rank by $rank A : = \dim_{C} (ran A)$ , using Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis. Relative to the ordered basis $(h, v)$ , call exactly the ray $[w]$ of this displayed superposition the coherent preparation in this example. Its density matrix has off-diagonal entries $(T_{w, w})_{12} = - i / 2$ and $(T_{w, w})_{21} = i / 2$ ; these are the coherence entries in this basis. The mixture has zero off-diagonal entries. These terms refer only to the displayed state data and assert no outcome-statistical or apparatus property.

The two assigned operators are distinct states of the model: $T_{w, w}$ is pure with purity $1$ and rank one, while $ρ_{+}$ is mixed with purity $\frac{1}{2}$ and rank two. In particular the relative phase of the superposition is retained in the assigned state, and the superposition and the mixture are not the same preparation.

Facts & Assumptions

Given: A two-dimensional complex Hilbert space $H$ with orthonormal basis $(h, v)$ ; the unit vector $w = (h + i v) / \sqrt{2}$ ; the mixture $ρ_{+} = \frac{1}{2} T_{h, h} + \frac{1}{2} T_{v, v}$ ; the state space $D (H)$ of The state space of the framework: preparations, pure rays and density operators; and the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), the choice hypothesis of the cited state, mixture and purity suppliers.

[L1]

Inner product and modulus (Real and complex inner product spaces, with the inner product linear in the first argument, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive): $⟨ h, h ⟩ = ⟨ v, v ⟩ = 1$ , $⟨ h, v ⟩ = ⟨ v, h ⟩ = 0$ , $∣ i ∣ = 1$ , and $∥ w ∥^{2} = \frac{1}{2} ∥ h + i v ∥^{2} = 1$ because $∣ 1 ∣^{2} = ∣ i ∣^{2} = 1$ and the two components are orthogonal.

[L2]

Rank-one states and matrices (Rays, unit vectors and rank-one Hilbert projections, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes): for a unit vector $x$ the operator $T_{x, x}$ is the pure state of the ray $[x]$ and the Hilbert orthogonal projection onto the one-dimensional subspace $C x$ ; its matrix in the basis $(h, v)$ has columns $⟨ h, x ⟩ x$ and $⟨ v, x ⟩ x$ . For $h$ and $v$ , these operators are the projections onto $C h$ and $C v$ , with matrices $T_{h, h} = (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array})$ and $T_{v, v} = (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array})$ .

[L3]

Mixtures and purity (The state space is convex, closed under countable mixtures, and complete in the trace norm, Purity, spectral weights and the criterion for a pure state, The state space: density operators, pure and mixed states, and purity, Statistical mixtures, spectral decompositions and the extremal role of pure preparations): finite mixtures with weights summing to one are states; a state is pure exactly when its purity is $1$ and mixed exactly when it has at least two positive eigenvalues.

Verification

technique · direct; compute both matrices in the basis $(h,v)$ and compare purity and rank

1.1givenL1L2L3algebra

[The superposition state] With $w = (h + i v) / \sqrt{2}$ one has $⟨ h, w ⟩ = 1 / \sqrt{2}$ and $⟨ v, w ⟩ = \overline{i} / \sqrt{2} = - i / \sqrt{2}$ by [L1], so the columns of $T_{w, w}$ in the basis $(h, v)$ are $T_{w, w} h = \frac{1}{\sqrt{2}} w = \frac{1}{2} (h + i v)$ and $T_{w, w} v = - \frac{i}{\sqrt{2}} w = - \frac{i}{2} (h + i v) = \frac{1}{2} (v - i h)$ , giving the displayed matrix $\frac{1}{2} (\begin{array}{cc} 1 & - i \\ i & 1 \end{array})$ ; its square is $\frac{1}{4} (\begin{array}{cc} 2 & - 2 i \\ 2 i & 2 \end{array}) = \frac{1}{2} (\begin{array}{cc} 1 & - i \\ i & 1 \end{array}) = T_{w, w}$ , so its purity is $1$ ; by [L2] its range is the one-dimensional subspace $C w$ , hence it has rank one, and by [L3] it is pure.

1.2givenL1L2L3algebra

[The mixture state] By [L2] one has $ρ_{+} = \frac{1}{2} (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}) + \frac{1}{2} (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}) = \frac{1}{2} I$ , a state by [L3]; its range is $H$ , so it has rank two, while its square is $\frac{1}{4} I$ and its purity is $tr (\frac{1}{4} I) = \frac{1}{2}$ ; since $\frac{1}{2} \neq 1$ , it is mixed by [L3].

2.1givenstep 1.1step 1.2algebra∎

[The states are distinct] The difference $T_{w, w} - ρ_{+} = \frac{1}{2} (\begin{array}{cc} 0 & - i \\ i & 0 \end{array})$ is nonzero, so the two preparations are assigned different states; the purities $1$ and $\frac{1}{2}$ , and the ranks one and two, confirm that one is pure and the other mixed. Hence the relative-phase superposition and the equal-weight statistical mixture are distinct states of the model, and the identity of their diagonal entries alone does not make them equal.

Remarks

The comparison is made at the level of the assigned states only. Which measurements distinguish the two states, and by how much, is a statement about the framework's probability law and measurement assumptions, and is not claimed here.
The diagonal entries of the two matrices agree and only the off-diagonal coherence entries defined above differ; this is the state-level expression of the relative phase of the superposition. No claim about interference or detection is made in this item.

A coherent two-level preparation and the equal statistical mixture are different states

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources