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One-bit J-gate teleportation branch identity

Statement

For a normalized qubit $∣ ψ ⟩ = a ∣ 0 ⟩ + c ∣ 1 ⟩$ and real $α$ , prepare $C Z (∣ ψ ⟩ \otimes ∣ + ⟩)$ and project the first qubit in the ordered basis ${∣ b ⟩_{α} : b = 0, 1}$ of J gate and ordered equatorial measurement vectors. The outcome- $b$ branch operator on the second qubit is

$K_{b} = 2^{- 1 / 2} X^{b} J (α) .$

Thus each outcome has probability $1 / 2$ and the normalized output is $X^{b} J (α) ∣ ψ ⟩$ . The operator identity is linear and therefore also holds when the input qubit is entangled with a finite-dimensional reference register.

Facts & Assumptions

Given: A normalized qubit $∣ ψ ⟩ = a ∣ 0 ⟩ + c ∣ 1 ⟩$ , a fresh $∣ + ⟩$ ancilla, and a real angle $α$ .

[F1]

The J-gate definition sets $∣ b ⟩_{α} = J (α)^{†} ∣ b ⟩$ ; its identity lemma gives $∣ b ⟩_{α} = 2^{- 1 / 2} (∣ 0 ⟩ + (- 1)^{b} e^{- i α} ∣ 1 ⟩)$ and proves the pair is orthonormal (J gate and ordered equatorial measurement vectors, J-gate basis and exact phase identities).

[F2]

A finite projective measurement assigns each orthogonal projection its Born outcome weight (Finite projective measurements).

[F3]

A unit representative $ψ$ gives the density operator $∣ ψ ⟩ ⟨ ψ ∣$ ; it is nonnegative with trace one. The outer product has coordinate matrix $u u^{†}$ (Pure state and ray, Ray-to-projector is well-defined and bijective, Density operator, Dirac Notation Conversion, Dirac Notation Coordinate Identity).

[F4]

Trace is the finite sum of diagonal entries in an orthonormal basis and is cyclic for conformable matrices; $∣ u ⟩ ⟨ u ∣$ has matrix $u u^{†}$ , whose trace is the squared norm of $u$ (The basis-independent trace of an endomorphism of a finite-dimensional vector space, For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ , Dirac Notation Coordinate Identity, The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

[F5]

The same two rank-one projectors are a finite POVM with weights $tr (∣ Ψ ⟩ ⟨ Ψ ∣ Π_{b})$ (Finite POVMs and their outcome law).

[F6]

For an instrument branch map $I_{b}$ , its conditional state is $I_{b} (ρ) / p_{b}$ only when $p_{b} > 0$ (Finite instruments and conditional states).

[F7]

The current matrix definitions give $X = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ , $H = 2^{- 1 / 2} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array})$ , $C Z = diag (1, 1, 1, - 1)$ , and $J (α) = H diag (1, e^{i α})$ (The Pauli matrices, Controlled Phase Rotation, Exact one-qubit gates and rotations, J gate and ordered equatorial measurement vectors). Direct multiplication gives $C Z^{*} C Z = I$ .

[F8]

Tensoring equal linear maps with the identity preserves their equality on the ordered product register (Tensor products of finitely many linear maps).

[F9]

The computational basis of $C^{2}$ is orthonormal, so $∣ + ⟩ = 2^{- 1 / 2} (∣ 0 ⟩ + ∣ 1 ⟩)$ is a unit vector (Finite registers and their ordered computational bases, The computational basis is orthonormal).

Proof

1.1F1F7F9algebra

The prepared state expands as $2^{- 1 / 2} (a ∣ 00 ⟩ + a ∣ 01 ⟩ + c ∣ 10 ⟩ - c ∣ 11 ⟩)$ : expand $∣ ψ ⟩ \otimes ∣ + ⟩$ , and the diagonal $C Z$ changes only the $∣ 11 ⟩$ coefficient's sign by [F7] and [F9]. Projecting the first factor onto $∣ b ⟩_{α}$ leaves the second-factor vector $K_{b} ∣ ψ ⟩ = \frac{1}{2} ((a + (- 1)^{b} e^{i α} c) ∣ 0 ⟩ + (a - (- 1)^{b} e^{i α} c) ∣ 1 ⟩),$ by [F1]. Matrix multiplication from [F7] gives $J (α) ∣ ψ ⟩ = 2^{- 1 / 2} ((a + e^{i α} c) ∣ 0 ⟩ + (a - e^{i α} c) ∣ 1 ⟩)$ . For $b = 0$ these coefficients agree; for $b = 1$ , $X$ swaps them. Thus $K_{b} = 2^{- 1 / 2} X^{b} J (α)$ .

2.1step 1.1F1F2F3F4F5F7F9algebra

Let $Ψ = C Z (∣ ψ ⟩ \otimes ∣ + ⟩)$ and $Π_{b} = ∣ b ⟩_{α} ⟨ b ∣ \otimes I$ . The vector $Ψ$ is normalized because $∣ ψ ⟩$ and $∣ + ⟩$ are unit, and direct multiplication of the displayed diagonal $C Z$ matrix gives $C Z^{*} C Z = I$ by [F7] and [F9]. The branch vector is $Π_{b} Ψ = ∣ b ⟩_{α} \otimes K_{b} ∣ ψ ⟩$ by step 1.1 and the unit norm of $∣ b ⟩_{α}$ from [F1]. The projective outcome law [F2] gives $p_{b} = tr (∣ Ψ ⟩ ⟨ Ψ ∣ Π_{b})$ ; [F5] gives the same law as a POVM. Since $Π_{b}^{*} = Π_{b} = Π_{b}^{2}$ , trace cyclicity [F4] and the rank-one coordinate formula [F3] give $p_{b} = tr (∣ Π_{b} Ψ ⟩ ⟨ Π_{b} Ψ ∣) = ∥ Π_{b} Ψ ∥^{2} = ∥ K_{b} ∣ ψ ⟩ ∥^{2}$ . Direct multiplication of the displayed $X$ matrix in [F7] gives $X^{*} X = I$ , and J-gate basis and exact phase identities proves $J (α)$ is unitary, so $∥ K_{b} ∣ ψ ⟩ ∥^{2} = \frac{1}{2} ∥ X^{b} J (α) ∣ ψ ⟩ ∥^{2} = \frac{1}{2} ∥ ψ ∥^{2} = 1 / 2$ . Thus the branch is positive-probability and normalization is defined.

3.1step 1.1step 2.1F6F8algebra∎

Dividing the branch vector in step 1.1 by its norm $1 / \sqrt{2}$ yields $X^{b} J (α) ∣ ψ ⟩$ ; equivalently, the instrument conditional density from [F6] is the rank-one operator of this normalized vector. The map identity in step 1.1 remains an identity after tensoring with the identity on any finite reference register by [F8], so it also holds for entangled inputs.

Depends on

Used by

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Sources

Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415, §20.6, p. 179 (standard reference, not scraped)
Robert Raussendorf and Hans J. Briegel, Quantum computing via measurements only, arXiv:quant-ph/0010033, pp. 1–2 (standard reference, not scraped)
Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel, Measurement-based quantum computation on cluster states, arXiv:quant-ph/0301052, §§II B and II E (standard reference, not scraped)