Finite GKSL jump-trajectory branches

Let $d\ge 1$, let $r\ge 0$ be an integer, let $H=H^{\ast }\in M_d(\mathbb{C})$ have units joules, let $\mathrm{\hslash }>0$ have units joule-seconds, and let $L_1,\dots ,L_r\in M_d(\mathbb{C})$ have units ${\mathrm{s}}^{-1\mathrm{/}2}$, with the finite GKSL convention of Finite GKSL generators and units. Put $K=-iH\mathrm{/}\mathrm{\hslash }-\frac{1}{2}{\sum }_{a=1}^r{L}_a^{\ast }{L}_a$ and $R(u)=e^{uK}$ for $u\ge 0$, with the exponential defined by its norm-convergent power series. For a density matrix $\rho \in M_d(\mathbb{C})$, meaning $\rho ={\rho }^{\ast }\ge 0$ and $\mathrm{Tr}\rho =1$, define the no-jump branch at time $t\ge 0$ by $W_{\mathrm{\varnothing },\mathrm{\varnothing }}(t)=R(t)$ and its branch state by $W_{\mathrm{\varnothing },\mathrm{\varnothing }}(t)\rho W_{\mathrm{\varnothing },\mathrm{\varnothing }}(t{)}^{\ast }$.

All time variables $u,t,s_1,\dots ,s_n$ are measured in seconds. For $n\ge 1$, a jump-history label is $\mathbf{a}=(a_1,\dots ,a_n)\in \{1,\dots ,r{\}}^n$ and its jump times are seconds $0<s_1<\cdots <s_n<t$. Define the corresponding branch operator by $W_{\mathbf{a},\mathbf{s}}(t)=R(t-s_n)L_{a_n}R(s_n-s_{n-1})\cdots L_{a_2}R(s_2-s_1)L_{a_1}R(s_1)$ and the unnormalized branch state by $W_{\mathbf{a},\mathbf{s}}(t)\rho W_{\mathbf{a},\mathbf{s}}(t{)}^{\ast }$. Its branch-weight density is the real-valued scalar function $w_{\mathbf{a},\mathbf{s}}(t)=\mathrm{Tr}(W_{\mathbf{a},\mathbf{s}}(t)\rho W_{\mathbf{a},\mathbf{s}}(t{)}^{\ast })$. For $n\ge 1$, let ${\bar{\mathrm{\Delta }}}_n(t)=\{(s_1,\dots ,s_n)\in {\mathbb{R}}^n:0\le s_1\le \cdots \le s_n\le t\}$ and ${\mathrm{\Delta }}_n(t)=\{(s_1,\dots ,s_n)\in {\mathbb{R}}^n:0<s_1<\cdots <s_n<t\}$. Use the same formula for $W_{\mathbf{a},\mathbf{s}}(t)$ on ${\bar{\mathrm{\Delta }}}_n(t)$; it is continuous there because $R(u)=e^{uK}$ is continuous for $u\ge 0$. If $V$ is a finite-dimensional real or complex vector space and $F:{\bar{\mathrm{\Delta }}}_n(t)\to V$ has continuous coordinates in a basis, define ${\int }_{{\mathrm{\Delta }}_n(t)}F(\mathbf{s})d\mathbf{s}$ as exactly the repeated one-variable Riemann integral over the closed limits shown here ${\int }_0^{t}d{s}_n{\int }_0^{s_n}d{s}_{n-1}\cdots {\int }_0^{s_2}F(s_1,\dots ,s_n)d{s}_1$, integrating each real coordinate; for complex coordinates, integrate their real and imaginary parts separately. This value is independent of the basis because a change of basis is a finite linear combination and scalar Riemann integration is linear. For $V=\mathbb{R}$ this defines the scalar integral of the density $w_{\mathbf{a},\mathbf{s}}(t)$; the same convention applies to branch-state integrands with values in $M_d(\mathbb{C})$ and to integrands with values in the finite-dimensional complex vector space of complex-linear maps $M_d(\mathbb{C})\to M_d(\mathbb{C})$. The full recursion is well-defined for these arbitrary continuous integrands by Continuous finite-dimensional ordered-simplex integrals: every inner integral is continuous in all remaining ordered parameters, so the next integral exists. That lemma also proves basis independence, linear-map commutation, the norm bound by the supremum times $t^n\mathrm{/}n!$, and positivity for nonnegative real scalar integrands. Set ${\int }_a^{a}f(s)ds=0$ for a degenerate interval; hence every $n\ge 1$ ordered-simplex integral at $t=0$ is zero. Let ${\mathrm{\Delta }}_0(t)=\{()\}$ and define ${\int }_{{\mathrm{\Delta }}_0(t)}F=F(())$; in the expansion, the $n=0$ contribution is the no-jump branch already defined above. For $n\ge 1$, $W_{\mathbf{a},\mathbf{s}}$ has units ${\mathrm{s}}^{-n\mathrm{/}2}$ and $w_{\mathbf{a},\mathbf{s}}$ has units ${\mathrm{s}}^{-n}$; integrating over the $n$ time variables gives a dimensionless weight. If $r=0$, there are no branches with $n\ge 1$. This definition alone does not assert that the branch weights sum to one.

Finite GKSL generators and units

Finite matrix spectral and state calculus

Every finite-dimensional normed space is Banach

The exponential series of a bounded operator

The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$

A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion

Continuous finite-dimensional ordered-simplex integrals

The definition supplies formal finite jump labels, times, branch operators, and branch weights. The ensemble identity and normalization are proved separately; a branch is not asserted to be a detector record.