Definitions for A Quantum Fluctuation Theorem

QAlgorithms.Papers.Independent.Intakethird100.P15.CommonTraceDomain

For an index type $\iota$, a complex Hilbert space $E$, a Hilbert basis $(b_i)_{i\in\iota}$, a surjective complex linear isometry $U:E\to E$, an arbitrary energy function $\varepsilon:\iota\to\mathbb R$, $\beta\in\mathbb R$, and $z\in\mathbb C$, define the family

$$q_w(i,j)= e^{w(\varepsilon_j-\varepsilon_i)} \left|\langle b_j,U b_i\rangle\right|^2 e^{-\beta\varepsilon_i} \qquad ((i,j)\in\iota\times\iota).$$

The stated condition means that all three complex families $q_z$, $q_{-z-\beta}$, and $q_{-\overline z-\beta}$ are summable: their nets of finite partial sums converge in $\mathbb C$. There is no positivity assumption on $\beta$ or restriction on $\varepsilon$. If $\iota$ is empty, all three summability conditions hold.

QAlgorithms.Papers.Independent.Intakethird100.P15.CyclicDrive

For a complex Hilbert basis $b=(b_i)_{i\in\iota}$ of $E$ and a measurable space $\Omega$, a CyclicDrive consists of positive real numbers $P$ and $\hbar$, a natural number $c$, spectral data $H_t=(\mu_t,\varepsilon_t,T_t)$ for every real $t$, surjective complex linear isometries $V_{n,r}:E\to E$ for every natural number $n$ and $r\in\{0,\ldots,n\}$, and a surjective complex linear isometry $U:E\to E$, satisfying the following requirements: each $\mu_t$ is a measure on $\Omega$, each $\varepsilon_t:\Omega\to\mathbb R$ is measurable, and each $T_t:E\to L^2(\Omega,\mu_t;\mathbb C)$ is a surjective complex linear isometry with $\operatorname{Im}(T_tb_i)=0$ $\mu_t$-almost everywhere for each $i$; the entire spectral data satisfy $H_{t+P}=H_t$ and $H_{-t}=H_t$ for every real $t$; writing $\Delta_n=cP/(n+1)$ and $t_{n,r}=(r+\tfrac12)\Delta_n$, one has, for all $n,r$ and all $x,y\in E$,

$$\langle x,V_{n,r}y\rangle = \int_\Omega \overline{(T_{t_{n,r}}x)(s)} \,e^{\,i(\Delta_n/\hbar)\varepsilon_{t_{n,r}}(s)} (T_{t_{n,r}}y)(s)\,d\mu_{t_{n,r}}(s);$$

and, for every $x\in E$,

$$V_{n,n}\circ\cdots\circ V_{n,0}(x)\longrightarrow Ux \qquad(n\to\infty).$$

The convergence is pointwise in $x$ in the topology of $E$. The value $n=0$ gives a single slice, and $c=0$ is allowed, in which case all slice times and durations are zero.

QAlgorithms.Papers.Independent.Intakethird100.P15.HasFiniteHeatTraces

For any index type $\iota$ and function $e:\iota\to\mathbb R$, HasFiniteHeatTraces means that, for every real $t>0$, the family $(e^{-t e_i})_{i\in\iota}$ is summable in $\mathbb R$. There is no requirement at $t=0$ or at negative $t$, and the condition holds vacuously for an empty index type.

QAlgorithms.Papers.Independent.Intakethird100.P15.IsInitialEnergyBasis

For a complex Hilbert basis $b=(b_i)_{i\in\iota}$ of $E$, a cyclic drive $d$ relative to $b$ on a measurable space $\Omega$, and any function $e:\iota\to\mathbb R$, let $(\mu_0,\varepsilon_0,T_0)$ denote the drive’s spectral data at time zero: $\varepsilon_0$ is measurable, $T_0:E\to L^2(\Omega,\mu_0;\mathbb C)$ is a surjective complex linear isometry, and each $T_0b_i$ is almost everywhere real. Here the drive also supplies positive period and $\hbar$, a natural cycle count, periodic and reflection symmetric spectral data, spectral slice isometries, and their pointwise limiting evolution. IsInitialEnergyBasis means that, for every $i\in\iota$,

$$\varepsilon_0(s)(T_0b_i)(s)=e_i(T_0b_i)(s) \quad\text{for }\mu_0\text{-almost every }s.$$

The exceptional null set may depend on $i$; no distinctness or ordering of the numbers $e_i$ is required.

QAlgorithms.Papers.Independent.Intakethird100.P15.IsSpectralStep

For a complex Hilbert basis $b=(b_i)_{i\in\iota}$ of $E$, spectral data $H=(\mu,\varepsilon,T)$ consisting of a measure on a measurable space $\Omega$, a measurable real energy function, and a surjective complex linear isometry $T:E\to L^2(\Omega,\mu;\mathbb C)$ taking each $b_i$ to an almost everywhere real function, arbitrary real numbers $\tau,\hbar$, and a surjective complex linear isometry $V:E\to E$, IsSpectralStep means that, for every $x,y\in E$,

$$\langle x,Vy\rangle = \int_\Omega \overline{(Tx)(s)} \,e^{\,i(\tau/\hbar)\varepsilon(s)} (Ty)(s)\,d\mu(s).$$

Neither $\tau$ nor $\hbar$ is required to be positive or nonzero. Division by zero is defined to be zero, so when $\hbar=0$ the exponential factor is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P15.RealSpectralHamiltonian

For a complex Hilbert basis $b=(b_i)_{i\in\iota}$ of $E$ and a measurable space $\Omega$, a RealSpectralHamiltonian consists of a measure $\mu$ on $\Omega$, a measurable function $\varepsilon:\Omega\to\mathbb R$, and a surjective complex linear isometry $T:E\to L^2(\Omega,\mu;\mathbb C)$, such that, for every $i\in\iota$, $\operatorname{Im}(Tb_i)(x)=0$ for $\mu$-almost every $x$. The exceptional null set may depend on $i$. No finiteness condition on $\mu$ or boundedness condition on $\varepsilon$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P15.bathObservedEnergy

For arbitrary types $B$ and $S$, functions $E_B:B\to\mathbb R$ and $O:S\to\mathbb R$, and a pair $p=(a,s)\in B\times S$, this defines the real number

$$E_{\mathrm{obs}}(a,s)=E_B(a)+O(s).$$

No finiteness, countability, boundedness, or positivity assumptions are imposed. If either type is empty, there is no pair $p$ to which the definition can be applied.

QAlgorithms.Papers.Independent.Intakethird100.P15.bathPreparation

For arbitrary types $B,S$, functions $E_B:B\to\mathbb R$ and $O:S\to\mathbb R$, an arbitrary $\beta\in\mathbb R$, and $p=(a,s)\in B\times S$, put

$$Z_B(\beta)=\sum_{a'\in B}e^{-\beta E_B(a')}, \qquad Z_S(\beta)=\sum_{s'\in S}e^{-\beta O(s')}.$$

This defines

$$w(a,s)= \frac{e^{-\beta E_B(a)}}{Z_B(\beta)} \frac{e^{-\beta O(s)}}{Z_S(\beta)}.$$

Both sums are total unordered real sums, taking value $0$ when their families are not summable; real division by $0$ also takes value $0$. Consequently, a zero denominator makes the displayed product $0$. No summability, normalization, or positivity hypothesis on $\beta$ is included. If $B$ or $S$ is empty, no pair $p$ exists.

QAlgorithms.Papers.Independent.Intakethird100.P15.chronologicalEvolution

For a complex inner product space $E$ and any finite list $L=[V_0,\ldots,V_{m-1}]$ of surjective complex linear isometries $E\to E$, chronologicalEvolution is the surjective complex linear isometry

$$V_{m-1}\circ\cdots\circ V_0.$$

Thus the maps act in their listed order, starting with $V_0$; the empty list gives the identity map.

QAlgorithms.Papers.Independent.Intakethird100.P15.generatingTrace

For an index type $\iota$, a complex Hilbert space $E$, a Hilbert basis $(b_i)_{i\in\iota}$, a surjective complex linear isometry $U:E\to E$, an arbitrary function $\varepsilon:\iota\to\mathbb R$, an arbitrary real number $\beta$, and $z\in\mathbb C$, this defines

$$\mathcal T_{b,U,\varepsilon,\beta}(z) = \sum_{(i,j)\in\iota\times\iota} e^{z(\varepsilon_j-\varepsilon_i)} \left|\langle b_j,U b_i\rangle\right|^2 e^{-\beta\varepsilon_i}.$$

The sum is the unordered infinite sum in $\mathbb C$. No summability hypothesis is imposed: its value is $0$ when the family is not summable, and an empty index set also gives $0$. The expression is not divided by a heat trace.

QAlgorithms.Papers.Independent.Intakethird100.P15.generatingTrace_kmsSymmetry

Let $E$ be a complex Hilbert space, $\Omega$ a measurable space, and $b=(b_i)_{i\in\iota}$ a Hilbert basis with $\iota$ countable and nonempty. The cyclic drive has $T>0$, $\hbar>0$, $C\in\mathbb N$, and spectral data $H_t=(\mu_t,h_t,F_t)$, where $h_t$ is measurable and $F_t:E\to L^2(\mu_t;\mathbb C)$ is a complex linear isometric bijection with every $F_tb_i$ real almost everywhere; $H_{t+T}=H_t$ and $H_{-t}=H_t$. Its slice isometries $V_{n,r}$, for $0\le r\le n$, obey

$$\langle x,V_{n,r}y\rangle = \int \overline{F_{t_{n,r}}x(s)} e^{\,i(\Delta_n/\hbar)h_{t_{n,r}}(s)} F_{t_{n,r}}y(s)\,d\mu_{t_{n,r}}(s), \quad \Delta_n=\frac{CT}{n+1},\quad t_{n,r}=(r+\tfrac12)\Delta_n,$$

for all $x,y\in E$, and their chronological products converge on every vector to the supplied evolution $U$. Let $\kappa$ also be countable and nonempty, let $c=(c_j)_{j\in\kappa}$ be another Hilbert basis of $E$, and assume $\operatorname{Im}\langle b_i,c_j\rangle=0$ for every $i,j$. For an arbitrary $\varepsilon:\kappa\to\mathbb R$, $\beta>0$, and $z\in\mathbb C$, define

$$q_w(a,b)=e^{w(\varepsilon_b-\varepsilon_a)} |\langle c_b,Uc_a\rangle|^2e^{-\beta\varepsilon_a}, \qquad \mathcal T(w)=\sum_{a,b}q_w(a,b).$$

Assume that $q_z$, $q_{-z-\beta}$, and $q_{-\overline z-\beta}$ are all summable. Then

$$\mathcal T(z)=\mathcal T(-z-\beta) \quad\text{and}\quad \mathcal T(z)=\overline{\mathcal T(-\overline z-\beta)}.$$

The energy function $\varepsilon$ is arbitrary, and $C=0$ is permitted.

QAlgorithms.Papers.Independent.Intakethird100.P15.heatTrace

For any index type $\iota$, any function $e:\iota\to\mathbb R$, and any real $t$, heatTrace is the real-valued total sum

$$Z_e(t)=\sum_{i\in\iota}e^{-t e_i}.$$

No summability or positivity assumption on $t$ is imposed. If the family is not summable in $\mathbb R$, this total sum is defined to be zero; an empty index type also gives zero.

QAlgorithms.Papers.Independent.Intakethird100.P15.preparedWorkLaw

For a complex Hilbert basis $b=(b_i)_{i\in\iota}$ of $E$, a surjective complex linear isometry $U:E\to E$, and arbitrary real-valued functions $e,w:\iota\to\mathbb R$, preparedWorkLaw is the measure on $\mathbb R$

$$\sum_{(a,g)\in\iota\times\iota} \max\!\left\{w_a\,|\langle b_g,Ub_a\rangle|^2,0\right\} \,\delta_{e_g-e_a},$$

where the coefficients are regarded as nonnegative extended real numbers and $\delta_x$ is the unit point mass at $x$. No nonnegativity or normalization condition on $w$, and no summability condition, is imposed; negative coefficients before conversion contribute zero. The resulting measure is not required to have total mass one or finite total mass.

QAlgorithms.Papers.Independent.Intakethird100.P15.qTerm

For a complex Hilbert basis $b=(b_i)_{i\in\iota}$ of $E$, a surjective complex linear isometry $U:E\to E$, an arbitrary function $e:\iota\to\mathbb R$, a real number $\beta$, a complex number $z$, and an ordered pair $(a,g)\in\iota\times\iota$, qTerm is the complex number

$$e^{\,z(e_g-e_a)} \,|\langle b_g,Ub_a\rangle|^2 \,e^{-\beta e_a},$$

with the real factors embedded in $\mathbb C$. No positivity assumption on $\beta$, summability assumption, or normalization factor is included.

QAlgorithms.Papers.Independent.Intakethird100.P15.sliceTime

For any real number $P$, natural numbers $c,n$, and an index $r\in\{0,\ldots,n\}$, sliceTime is the real number

$$\left(r+\frac12\right)\frac{cP}{n+1}.$$

There is no positivity requirement on $P$, and $c=0$ and $n=0$ are included. For $c=0$ the value is zero; for $n=0$ the sole value is $cP/2$.

QAlgorithms.Papers.Independent.Intakethird100.P15.spectralGibbsWeight

For any index type $\iota$, function $e:\iota\to\mathbb R$, real number $\beta$, and index $i\in\iota$, spectralGibbsWeight is

$$w_i=\frac{e^{-\beta e_i}}{\displaystyle\sum_{j\in\iota}e^{-\beta e_j}},$$

where the denominator is the real-valued total sum, defined to be zero if the family is not summable. No positivity assumption on $\beta$ or summability assumption is imposed. If the denominator is zero, the quotient is defined to be zero.

QAlgorithms.Papers.Independent.Intakethird100.P15.workLaw

For a complex Hilbert basis $b=(b_i)_{i\in\iota}$ of $E$, a surjective complex linear isometry $U:E\to E$, a function $e:\iota\to\mathbb R$, and any real $\beta$, workLaw is the measure

$$\sum_{(a,g)\in\iota\times\iota} \max\!\left\{ \frac{e^{-\beta e_a}}{Z_e(\beta)} |\langle b_g,Ub_a\rangle|^2,0 \right\}\delta_{e_g-e_a}, \qquad Z_e(\beta)=\sum_{j\in\iota}e^{-\beta e_j}.$$

The denominator is the real-valued total sum, which is zero for a nonsummable family, and division by zero gives zero. No positivity assumption on $\beta$ or finite heat trace assumption is required; if $Z_e(\beta)=0$, this definition gives the zero measure.