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A two-dimensional normalization and phase computation

Example

Let $H = C^{2}$ carry the standard inner product $⟨ z, w ⟩ = z_{1} \overline{w_{1}} + z_{2} \overline{w_{2}}$ (Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space; completeness follows from Every finite-dimensional normed space is Banach), and let $e_{1} = (1, 0)$ , $e_{2} = (0, 1)$ . Take $ψ = (1, i), z = 1 + i, φ : = z ψ,$ and verify on this explicit data the normalization and phase statements of Normalization, ray equivalence and the phase orbits of the unit sphere: the norm of $φ$ is $∣ z ∣$ times the norm of $ψ$ , the normalized representative of $φ$ is the phase multiple $(z / ∣ z ∣)$ of the normalized representative of $ψ$ , the rays of $ψ$ and $φ$ coincide, and the normalized rank-one operators of $ψ$ , $φ$ and $i ψ$ are the same pure state The state space: density operators, pure and mixed states, and purity.

Facts & Assumptions

Given: The Hilbert space $H = C^{2}$ with the standard inner product and basis $(e_{1}, e_{2})$ ; the vectors $ψ = (1, i)$ , $φ = z ψ$ with $z = 1 + i$ , and $i ψ$ ; the ray, normalization and phase notions of State vectors, unit vectors, rays, normalization and phases of a Hilbert space; and the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), the choice hypothesis of the cited rank-one and state-space suppliers.

[L1]

Standard computations in $C$ : $i^{2} = - 1$ , $(1 + i)^{2} = 2 i$ , $(1 + i) (1 + i) - 1$ expansions below are termwise multiplications with $\overline{1 + i} = 1 - i$ , $∣ 1 + i ∣^{2} = 2$ , $∣ i ∣ = 1$ (The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ , Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[L2]

Norm and pairing: $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ with $⟨ z, w ⟩ = z_{1} \overline{w_{1}} + z_{2} \overline{w_{2}}$ , so $∥ (a, b) ∥^{2} = ∣ a ∣^{2} + ∣ b ∣^{2}$ , and $∥ λ v ∥ = ∣ λ ∣ ∥ v ∥$ (The induced length is a norm, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[L3]

Normalization and phases (Normalization, ray equivalence and the phase orbits of the unit sphere): $ν (v) = ∥ v ∥^{- 1} v$ , $[v] = {w : w = t v, t \neq 0}$ , and $ν (t v) = (t / ∣ t ∣) ν (v)$ for $t \neq 0$ ; $[v] = [w]$ if and only if $ν (w) = ζ ν (v)$ for some $ζ \in U (1)$ .

[L4]

Rank-one operators (Rays, unit vectors and rank-one Hilbert projections, The state space: density operators, pure and mixed states, and purity): $T_{v, v} x = ⟨ x, v ⟩ v$ , $T_{t v, t v} = ∣ t ∣^{2} T_{v, v}$ , $T_{v, v} / ∥ v ∥^{2} = T_{ν (v), ν (v)}$ is a pure state, and for unit $u, w$ one has $T_{u, u} = T_{w, w}$ exactly when $w = ζ u$ with $∣ ζ ∣ = 1$ .

Verification

technique · direct; compute the displayed vectors, their normalizations and the associated rank-one operators

1.1givenL1L2algebra

Compute $φ = z ψ = (1 + i) (1, i) = ((1 + i) \cdot 1, (1 + i) \cdot i) = (1 + i, i - 1)$ . By [L2] and [L1], $∥ ψ ∥^{2} = ∣ 1 ∣^{2} + ∣ i ∣^{2} = 2$ and $∥ φ ∥^{2} = ∣ 1 + i ∣^{2} + ∣ i - 1 ∣^{2} = 2 + 2 = 4$ , so $∥ ψ ∥ = \sqrt{2}$ , $∥ φ ∥ = 2$ , and indeed $∥ φ ∥ = ∣ z ∣ ∥ ψ ∥ = \sqrt{2} \cdot \sqrt{2} = 2$ , verifying the homogeneity used in the normalization lemma.

2.1givenstep 1.1L1L2L3algebra

The normalizations are $ν (ψ) = (1 / \sqrt{2}, i / \sqrt{2})$ and $ν (φ) = ((1 + i) / 2, (i - 1) / 2)$ . With $ζ : = (1 + i) / \sqrt{2}$ one has $∣ ζ ∣^{2} = ∣ 1 + i ∣^{2} / 2 = 1$ , so $ζ \in U (1)$ , and $ζ ν (ψ) = \frac{1 + i}{\sqrt{2}} (\frac{1}{\sqrt{2}}, \frac{i}{\sqrt{2}}) = (\frac{1 + i}{2}, \frac{i (1 + i)}{2}) = (\frac{1 + i}{2}, \frac{i - 1}{2}) = ν (φ),$ which is the identity $ν (z ψ) = (z / ∣ z ∣) ν (ψ)$ ; by [L3] this also identifies $[φ] = [ψ]$ , so $ψ$ , $φ$ and $i ψ$ lie in one ray.

3.1givenstep 2.1L1L4algebra

The rank-one operator of the unit vector $u = ν (ψ) = (1 / \sqrt{2}, i / \sqrt{2})$ has matrix in the basis $(e_{1}, e_{2})$ with columns $T_{u, u} e_{1} = ⟨ e_{1}, u ⟩ u = \overline{u_{1}} u = \frac{1}{\sqrt{2}} u = (\frac{1}{2}, \frac{i}{2})$ and $T_{u, u} e_{2} = ⟨ e_{2}, u ⟩ u = \overline{u_{2}} u = \frac{- i}{\sqrt{2}} u = (- \frac{i}{2}, \frac{1}{2})$ , that is $T_{u, u} = \frac{1}{2} (\begin{array}{cc} 1 & - i \\ i & 1 \end{array}) .$ This matrix is self-adjoint, has trace $1$ , and satisfies $T_{u, u}^{2} = T_{u, u}$ because ${(\begin{array}{cc} 1 & - i \\ i & 1 \end{array})}^{2} = 2 (\begin{array}{cc} 1 & - i \\ i & 1 \end{array})$ by [L1]; hence $T_{u, u}$ is a pure state by [L4].

4.1givenstep 1.1step 3.1L1L2L4algebra∎

For the second representative, $T_{φ, φ} = ∣ z ∣^{2} T_{ψ, ψ} = 2 T_{ψ, ψ}$ by [L4], and dividing by $∥ φ ∥^{2} = 4$ from step 1.1 gives $T_{φ, φ} / ∥ φ ∥^{2} = T_{ψ, ψ} / 2 = T_{ψ, ψ} / ∥ ψ ∥^{2} = T_{u, u}$ ; likewise $T_{i ψ, i ψ} = ∣ i ∣^{2} T_{ψ, ψ} = T_{ψ, ψ}$ , so the phase multiple $i ψ$ determines the same pure state. This verifies on explicit data the representative and phase independence of the normalized rank-one operator.

Remarks

The computation checks the definitions and the normalization lemma on a fixed two-dimensional example. It asserts nothing about any physical system, preparation or apparatus, and it records no measured value.
The phase multiple $i ψ$ and the rescaled representative $φ$ give the same pure state, while the relative phase between the two components of a vector is not removed by any of these operations; the effect of relative phase belongs to the framework's probability law on the observable pages and is not examined here.

A two-dimensional normalization and phase computation

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources