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Finite Bogoliubov implementability

Statement

Assume AC, $r \geq 1$ , and the finite-mode Fock conventions of Finite-mode quadratures and symplectic data. Let $U, V \in M_{r} (C)$ obey $U U^{*} - V V^{*} = I_{r}, U V^{T} = V U^{T},$ where $^{*}$ is conjugate transpose and $^{T}$ is transpose without conjugation. On the transported Schwartz domain $D_{S} : = U^{- 1} S (R^{r})$ , which contains the finite-particle core (the finite Hermite span under $U$ ), use $a_{j} = (Q_{j} + i P_{j}) / \sqrt{2 ℏ}$ and $a_{j}^{*} = (Q_{j} - i P_{j}) / \sqrt{2 ℏ}$ , and define $a_{j}^{'} = \sum_{k} (U_{j k} a_{k} + V_{j k} a_{k}^{*}) .$ Let $R = (Q_{1}, P_{1}, \dots, Q_{r}, P_{r})^{T}$ , $J$ and $W (z)$ have the interleaved conventions of the definition. Define the real permutation matrix $Π$ by $Π (x_{1}, y_{1}, \dots, x_{r}, y_{r})^{T} = (x_{1}, \dots, x_{r}, y_{1}, \dots, y_{r})^{T},$ and, writing $Q = (Q_{1}, \dots, Q_{r})^{T}$ and $P = (P_{1}, \dots, P_{r})^{T}$ , set $R_{g} = Π R = (\begin{array}{c} Q \\ P \end{array}), J_{0} = Π J Π^{T} = (\begin{array}{cc} 0 & I_{r} \\ - I_{r} & 0 \end{array}), W_{g} (w) = W (Π^{T} w) .$ Then the $a_{j}^{'}$ satisfy $[a_{j}^{'}, (a_{k}^{'})^{*}] = δ_{j k} I$ and $[a_{j}^{'}, a_{k}^{'}] = 0$ on the finite Hermite core. The real matrix $S_{g} = (\begin{array}{cc} Re (U + V) & - Im (U - V) \\ Im (U + V) & Re (U - V) \end{array})$ is symplectic in grouped coordinate order: $S_{g}^{T} J_{0} S_{g} = J_{0}$ . There is a unitary $M$ on finite-mode Fock, obtained by transporting an individual finite symplectic implementer through the Fock–Schrödinger unitary, such that $M^{*} a M = a^{'}$ and $M^{*} R_{g} M = S_{g} R_{g}$ on the transported Schwartz core, and $M^{*} W_{g} (w) M = W_{g} (S_{g}^{T} w) (w \in R^{2 r}) .$ Equivalently, with the interleaved matrix $S_{i} = Π^{T} S_{g} Π$ , $M^{*} W (z) M = W (S_{i}^{T} z)$ for every interleaved label $z$ . The transformed vacuum $M Ω$ has finite mean number $\sum_{j} ∥ a_{j} M Ω ∥^{2} = tr (V V^{*}) .$ No infinite-mode implementability assertion is made.

Statement references

Finite-mode quadratures and symplectic data

Facts & Assumptions

Given: AC, finite $r \geq 1$ , matrices $U, V$ satisfying the two displayed matrix identities, the finite-mode Fock vacuum $Ω$ , and the quadrature and Weyl conventions of the definition.

[A1]

The matrices obey $U U^{*} - V V^{*} = I_{r}$ and $U V^{T} = V U^{T}$ , with AC adopted for the finite-dimensional uniqueness interface (The Axiom of Choice).

[F1]

On the finite Hermite span, $[a_{j}, a_{k}^{*}] = δ_{j k} I$ and all annihilators mutually commute, as do all creators; $a_{j} Ω = 0$ and $a_{j}^{*} Ω = ∣ e_{j} ⟩$ (Creation contraction bounds, Finite-mode Fock to Schrödinger identification).

[F2]

With $Q_{j} = \sqrt{ℏ / 2} (a_{j} + a_{j}^{*})$ and $P_{j} = - i \sqrt{ℏ / 2} (a_{j} - a_{j}^{*})$ , the matrix $S_{g}$ is the unique real coefficient matrix satisfying $a^{'} = \frac{(S_{g} R_{g})_{Q} + i (S_{g} R_{g})_{P}}{\sqrt{2 ℏ}}$ on that core, where $Q = (Q_{1}, \dots, Q_{r})^{T}$ , $P = (P_{1}, \dots, P_{r})^{T}$ and the subscripts denote the first and last $r$ grouped components; this follows by separating real and imaginary parts.

[F3]

Every interleaved finite symplectic matrix $S_{i}$ has an implementer $M^{*} W (z) M = W (S_{i}^{T} z)$ and $M^{*} R M = S_{i} R$ on the transported Schwartz core; conjugating by the fixed interleaved-to-grouped permutation gives the same covariance for grouped matrices (Finite symplectic implementation by quadratic generators).

[F4]

The Fock–Schrödinger map is unitary, sends $Ω$ to $g_{0}^{\otimes r}$ , and sends the finite-particle span to the finite Hermite span (Finite-mode Fock to Schrödinger identification).

Proof

1.1A1F1givenalgebra

Expand the commutator using [F1]. The only nonzero terms in $[a_{j}^{'}, (a_{k}^{'})^{*}]$ are $\sum_{l} U_{j l} \overline{U_{k l}} - \sum_{l} V_{j l} \overline{V_{k l}} = (U U^{*} - V V^{*})_{j k} = δ_{j k} .$ Likewise $[a_{j}^{'}, a_{k}^{'}] = \sum_{l} (U_{j l} V_{k l} - V_{j l} U_{k l}) = (U V^{T} - V U^{T})_{j k} = 0.$ These are finite sums on the common core, so no unbounded product is used outside its domain.

2.1F1F2step 1.1algebra

Substitute $a = (Q + i P) / \sqrt{2 ℏ}$ and $a^{*} = (Q - i P) / \sqrt{2 ℏ}$ into $U a + V a^{*}$ . Its real quadrature part is $Re (U + V) Q - Im (U - V) P$ and its imaginary part is $Im (U + V) Q + Re (U - V) P$ . These two blocks form $S_{g} R_{g}$ . Since step 1.1 gives the canonical commutators for these transformed quadratures, bilinearity yields $[(S_{g} R_{g})_{j}, (S_{g} R_{g})_{k}] = i ℏ (S_{g} J_{0} S_{g}^{T})_{j k} = i ℏ (J_{0})_{j k} .$ Here $[R_{g, j}, R_{g, k}] = i ℏ (J_{0})_{j k}$ follows from $R_{g} = Π R$ and $J_{0} = Π J Π^{T}$ . Since $ℏ > 0$ , the displayed identity implies $S_{g} J_{0} S_{g}^{T} = J_{0}$ . Taking determinants shows $(\det S_{g})^{2} = 1$ , so $S_{g}$ is invertible. Inverting this identity and using $J_{0}^{- 1} = - J_{0}$ gives $S_{g}^{T} J_{0} S_{g} = J_{0}$ , as required.

3.1F1F3F4step 2.1algebra∎

Use the permutation $Π$ defined in the Statement and set $S_{i} = Π^{T} S_{g} Π$ . Since $Π^{T} Π = I$ and $J_{0} = Π J Π^{T}$ , step 2.1 gives $S_{i}^{T} J S_{i} = J$ , so $S_{i}$ is symplectic in interleaved order. Apply [F3] to $S_{i}$ and transport its unitary to Fock by [F4]. For every grouped label $w$ , the definition of $W_{g}$ gives $M^{*} W_{g} (w) M = W (S_{i}^{T} Π^{T} w) = W (Π^{T} S_{g}^{T} w) = W_{g} (S_{g}^{T} w) .$ Likewise, on the transported Schwartz core, $M^{*} R_{g} M = Π M^{*} R M = Π S_{i} R = S_{g} R_{g} .$ The Weyl covariance holds on all Fock space because each Weyl operator and the implementer are bounded. The complex recombination in step 2.1 gives $M^{*} a M = a^{'}$ . Each generator in [F3] preserves Schwartz space, so $M Ω$ lies in that core. Therefore, using $a_{j} Ω = 0$ and $a_{j}^{'} Ω = \sum_{k} V_{j k} a_{k}^{*} Ω$ , one has $\sum_{j} ∥ a_{j} M Ω ∥^{2} = \sum_{j} ∥ M a_{j}^{'} Ω ∥^{2} = \sum_{j, k} ∣ V_{j k} ∣^{2} = tr (V V^{*}) .$ The equality is finite and proves the claimed finite mean number. No step passes to infinitely many modes.

Finite Bogoliubov implementability

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources