Finite reduced dynamics, memory and quantum trajectories

Let $d\ge 1$ and let ${\mathrm{\Phi }}_t:M_d(\mathbb{C})\to M_d(\mathbb{C})$ be a family of finite quantum channels indexed by $t\ge 0$, where “finite quantum channel” has the meaning in Finite completely positive trace-preserving maps. The family obeys the semigroup law when ${\mathrm{\Phi }}_0={\mathrm{id}}_{M_d(\mathbb{C})}$ and ${\mathrm{\Phi }}_{t+s}={\mathrm{\Phi }}_t\circ {\mathrm{\Phi }}_s$ for every $s,t\ge 0$.

The family is CP-divisible when for every $0\le s\le t$ there exists a finite quantum channel $V_{t,s}:M_d(\mathbb{C})\to M_d(\mathbb{C})$ such that ${\mathrm{\Phi }}_t=V_{t,s}\circ {\mathrm{\Phi }}_s$. CP-divisibility is this specified factorization property; the definition does not assert that it is a universal meaning of “Markovian”.

Finite completely positive trace-preserving maps

The source literature distinguishes divisibility and time-homogeneous semigroup conditions and discusses several meanings of quantum Markovianity. This item fixes only the map properties used on this page.

Let $T\ge 0$, let $n\ge 0$ be an integer, and let $V$ be a finite-dimensional real or complex vector space with a specified finite ordered basis and its coordinate topology. Put $D_n(T)=\{0\le s_1\le \cdots \le s_n\le T\}$, with $D_0(T)=\{()\}$. Let $F:D_n(T)\to V$ have continuous coordinates. Set $G_0=F$. For $1\le k\le n$, write $y=(s_{k+1},\dots ,s_n)\in D_{n-k}(T)$ and put $b_k(y)=s_{k+1}$ if $k<n$, and $b_n(())=T$. Define recursively

integrating real coordinates, and real and imaginary parts of complex coordinates. Every integral exists and every $G_k$ is continuous on its remaining parameter domain. Write $I_n(F)=G_n(())$; equivalently this is the iterated integral with limits $0\le s_1\le \cdots \le s_n\le T$. Degenerate one-variable integrals are zero. For $n=0$, set $I_0(F)=F(())$; for $n\ge 1$ and $T=0$, $I_n(F)=0$.

These integrals are linear, independent of the chosen finite basis, and commute with finite-dimensional linear maps. For any norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$ on $V$, the corresponding one-variable integral satisfies $\mathrm{\parallel }{\int }_a^{b}g(s)ds\mathrm{\parallel }\le {\int }_a^b\mathrm{\parallel }g(s)\mathrm{\parallel }ds$ for continuous $g$ and $a\le b$. If $\mathrm{\parallel }F(s)\mathrm{\parallel }\le C$ on $D_n(T)$, where $C\ge 0$, then

Here the factor for $n=0$ is $1$, including $T=0$. For scalar $F=1$, $I_n(1)=T^n\mathrm{/}n!$ with this convention. For real scalar $F\ge 0$, every partial integral $G_k$ is nonnegative. No differentiability of $F$ or choice principle is assumed.

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 ${\int }_a^{b}f$

The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$

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.

Work with finite-dimensional system and environment spaces $H_S={\mathbb{C}}^d$ and $H_E={\mathbb{C}}^m$, where $d,m\ge 1$, using the ordered tensor product $H_S\otimes H_E$ with the system factor first. A system or environment state is a positive semidefinite Hermitian matrix of trace one; these state matrices are dimensionless. At the initial time, for every allowed system state $\rho$, adopt the joint preparation $\rho \otimes \eta$, where the fixed environment state $\eta$ does not depend on $\rho$. For each $t\ge 0$, choose a declared unitary $U_t$ on $H_S\otimes H_E$, meaning $U_t^{\ast }{U}_t=U_t{U}_t^{\ast }=I$, with $U_0=I$. The reduced system dynamics is the map ${\mathrm{\Phi }}_t(\rho )={\mathrm{Tr}}_E\text{\hspace{0.17em}⁣}(U_t(\rho \otimes \eta )U_t^{\ast })$, where for the fixed product basis the finite partial trace has entries $({\mathrm{Tr}}_{E}R{)}_{\alpha \beta }={\sum }_{j=1}^m{R}_{(\alpha ,j),(\beta ,j)}$. This assignment excludes initially correlated joint states. It imposes no one-parameter group, continuity, or semigroup law on $U_t$.

A time-homogeneous finite GKSL model may instead be selected as a separate approximation: choose a fixed finite-dimensional Hamiltonian matrix $H=H^{\ast }$ in joules, a constant $\mathrm{\hslash }>0$ in joule-seconds, and finitely many jump matrices $L_a$ in ${\mathrm{s}}^{-1\mathrm{/}2}$, and use $T_t=e^{t\mathcal{L}}$ for $t\ge 0$ with $\mathcal{L}(X)=-\frac{i}{\mathrm{\hslash }}[H,X]+{\sum }_a(L_{a}X{L}_a^{\ast }-\frac{1}{2}\{L_a^{\ast }{L}_a,X\})$. Here $\mathcal{L}:M_d(\mathbb{C})\to M_d(\mathbb{C})$ is complex-linear, $[A,B]=AB-BA$, $\{A,B\}=AB+BA$, and $A^{\ast }$ is conjugate transpose. The exponential means $e^{t\mathcal{L}}:={\sum }_{n=0}^{\mathrm{\infty }}{t}^n{\mathcal{L}}^n\mathrm{/}n!$, with ${\mathcal{L}}^0=\mathrm{id}$ and powers given by composition. This series converges absolutely in the operator norm induced by the finite matrix trace norm $\mathrm{\parallel }X{\mathrm{\parallel }}_1=\mathrm{Tr}\sqrt{X^{\ast }X}$, as in the finite GKSL semigroup theorem cited below; its hypotheses are exactly the finite Hermitian $H$, positive $\mathrm{\hslash }$, and finite jump family specified here. Each application must state its rates and physical regime. Selecting this approximation is not a consequence of the finite unitary reduction, and this postulate claims no weak-coupling or reservoir limit.

In both alternatives, the parameter $t$ indexing $U_t$ or $T_t$ denotes elapsed physical time measured in SI seconds. Thus $t\mathcal{L}$ in the GKSL exponential is dimensionless, since the adopted finite GKSL generator has units ${\mathrm{s}}^{-1}$.

Partial trace over a finite tensor factor

Finite completely positive trace-preserving maps

Finite GKSL generators and units

Complete finite Choi-Kraus equivalence

Finite GKSL generates CPTP semigroups

The diagram records only the defined finite-map composition. It supplies no proof of complete positivity or trace preservation.

diagram failed to render:
\rho \arrow[r, "{X\mapsto X\otimes\eta}"] \arrow[drr, "\Phi_t"'] &
\rho\otimes\eta \arrow[r, "{R\mapsto U_t R U_t^*}"] &
U_t(\rho\otimes\eta)U_t^* \arrow[d, "\operatorname{Tr}_E"] \\
& & \Phi_t(\rho)

In the finite product-state model, fix any $t\ge 0$ and define the complex-linear map ${\mathrm{\Phi }}_t:M_d(\mathbb{C})\to M_d(\mathbb{C})$ by ${\mathrm{\Phi }}_t(X)={\mathrm{Tr}}_E\text{\hspace{0.17em}⁣}(U_t(X\otimes \eta )U_t^{\ast })$. Then ${\mathrm{\Phi }}_t$ is a finite quantum channel. The family $t\mapsto {\mathrm{\Phi }}_t$ need not obey the semigroup law and need not be CP-divisible. The terms “semigroup law” and “CP-divisible” have the definitions of Finite channel families, semigroups, and CP-divisibility.

Finite system-environment and Markov model assignments

Finite channel families, semigroups, and CP-divisibility

Partial trace over a finite tensor factor

The partial trace is characterized by the trace pairing and is basis-independent

Finite completely positive trace-preserving maps

Complete finite Choi-Kraus equivalence

The two-qubit recurrence is a conditional prediction of a finite unitary model. It is not a reported observation and does not establish a physical Markov limit.

For the separately selected qubit GKSL model, let $H=\mathrm{\hslash }\omega \mathrm{\mid }1⟩⟨1\mathrm{\mid }$ with $\omega \in \mathbb{R}$ in ${\mathrm{s}}^{-1}$ and $\mathrm{\hslash }>0$ in joule-seconds, and let $L=\sqrt{\gamma }\mathrm{\mid }0⟩⟨1\mathrm{\mid }$ with $\gamma \ge 0$ in ${\mathrm{s}}^{-1}$, so $L$ has units ${\mathrm{s}}^{-1\mathrm{/}2}$. For any initial density matrix $\rho (0)$ in the ordered basis $(\mathrm{\mid }0⟩,\mathrm{\mid }1⟩)$, the solution $\rho (t)=e^{t\mathcal{L}}\rho (0)$ for $t\ge 0$ satisfies ${\rho }_{11}(t)=e^{-\gamma t}{\rho }_{11}(0)$, ${\rho }_{00}(t)={\rho }_{00}(0)+(1-e^{-\gamma t}){\rho }_{11}(0)$, ${\rho }_{01}(t)=e^{(i\omega -\gamma \mathrm{/}2)t}{\rho }_{01}(0)$, and ${\rho }_{10}(t)=e^{(-i\omega -\gamma \mathrm{/}2)t}{\rho }_{10}(0)$.

Finite system-environment and Markov model assignments

Finite GKSL generators and units

Finite GKSL generates CPTP semigroups

Every finite-dimensional normed space is Banach

Finite matrix trace-norm ideal inequalities

The exponential series of a bounded operator

The complex exponential by its power series

$\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential

The parameter $\gamma$ is a model rate. Its numerical value and the regime in which this generator approximates a physical system must be supplied by a specific application; no measurement or statistical estimate is asserted here.

Let $d\ge 1$, let $H=H^{\ast }\in M_d(\mathbb{C})$ have units joules, let $\mathrm{\hslash }>0$ have units joule-seconds, and let $L_1,\dots ,L_r\in M_d(\mathbb{C})$ be a finite family with units ${\mathrm{s}}^{-1\mathrm{/}2}$, where $r\ge 0$. For the selected time-homogeneous finite GKSL model, put $\mathcal{L}(X)=-\frac{i}{\mathrm{\hslash }}[H,X]+{\sum }_{a=1}^r(L_{a}X{L}_a^{\ast }-\frac{1}{2}\{L_a^{\ast }{L}_a,X\})$. With $K=-iH\mathrm{/}\mathrm{\hslash }-\frac{1}{2}{\sum }_a{L}_a^{\ast }{L}_a$ and the finite branch operators of Finite GKSL jump-trajectory branches, each no-jump operator $R(u)=e^{uK}$ is a contraction, each branch state $W_{\mathbf{a},\mathbf{s}}(t)\rho W_{\mathbf{a},\mathbf{s}}(t{)}^{\ast }$ is positive for $\rho \ge 0$, and for every density matrix $\rho$ and $t\ge 0$,

$e^{t\mathcal{L}}(\rho )={\sum }_{n=0}^{\mathrm{\infty }}{\sum }_{\mathbf{a}\in \{1,\dots ,r{\}}^n}{\int }_{0<s_1<\cdots <s_n<t}{W}_{\mathbf{a},\mathbf{s}}(t)\rho W_{\mathbf{a},\mathbf{s}}(t{)}^{\ast }d{s}_1\cdots d{s}_n,$

where the $n=0$ term is $R(t)\rho R(t{)}^{\ast }$ and branches with $n\ge 1$ are absent when $r=0$. The series converges in trace norm, uniformly on each bounded time interval; the sum of all branch weights is one. This is an ensemble representation of the finite GKSL solution, not a unique microscopic trajectory or a claim about a particular detector.

Finite GKSL jump-trajectory branches

Finite GKSL generators and units

Finite GKSL generates CPTP semigroups