Quantum speed limits for an open system in contact with a thermal bath

TL;DR

This paper establishes fundamental bounds on the speed of quantum evolution for open systems interacting with thermal baths, applicable beyond weak coupling and without relying on Markovian assumptions.

Contribution

It introduces rigorous, expectation-value-based bounds on quantum speed limits for open systems that do not depend on the Markov approximation.

Findings

Bounds are valid for strong and weak system-bath coupling regimes.

The bounds are expressed in terms of few-body observables.

Results extend the understanding of quantum dynamics beyond traditional approximations.

Abstract

We prove fundamental rigorous bounds on the speed of quantum evolution for a quantum system coupled to a thermal bath. The bounds are formulated in terms of expectation values of few-body observables derived from the system-bath Hamiltonian. They do not rely on the Markov approximation and, as a consequence, are applicable beyond the limit of weak system-bath coupling.

Equations

$$\begin{array}[]{c}\displaystyle H_{t}=H^{S}_{t}+H^{\mathrm{int}}_{t}+H^{B}.\end{array}$$

$$\begin{array}[]{c}\displaystyle i\partial_{t}\rho_{t}=[H_{t},\rho_{t}].\end{array}$$

$$\begin{array}[]{c}\displaystyle\rho_{0}=\rho^{S}_{\rm in}\,\rho^{B}_{\beta},\end{array}$$

$$\begin{array}[]{c}\displaystyle\rho^{B}_{\beta}=e^{-\beta H^{B}}/Z_{B},\qquad Z_{B}={\rm tr}\,e^{-\beta H^{B}}.\end{array}$$

$$\begin{array}[]{c}\displaystyle D(\rho_{1},\rho_{2})=1-{\rm tr}(\sqrt{\rho_{1}}\sqrt{\rho_{2}}).\end{array}$$

$$D({\rho }_0,{\rho }_t{)}^2$$

$$e^{-\beta H^B/2}{H}_{t^{\prime }}^{\mathrm{int}}{e}^{\beta H^B/2}.$$

$${\rho }_{\mathrm{in}}^S=1,\beta \to 0.$$

$${\rm tr}\big{(}-[A,\sqrt{\rho}]^{2}\big{)}\leq 2\,{\rm tr}(A-c)^{2}\rho$$

$$D(\rho_{0},\rho_{t})^{2}\leqslant 2\,t\int_{0}^{t}{\rm tr}\Big{(}(H^{\mathrm{int}}_{t^{\prime}}+H^{S}_{t^{\prime}}-c_{t^{\prime}})^{2}\,\rho^{S}_{\rm in}\,\rho^{B}_{\beta}\Big{)}\,dt^{\prime}.$$

$$D({\rho }_0,{\rho }_t{)}^2⩽$$

$$\displaystyle+2\,t\int_{0}^{t}d{t^{\prime}}\,{\rm tr}\big{(}-[H^{S}_{t^{\prime}}+H^{\rm int}_{t^{\prime}},\sqrt{\rho^{S}_{\rm in}}]^{2}\,\rho^{B}_{\beta}\big{)}$$

$$D(\rho_{0},\rho_{t})^{2}\leqslant\frac{t}{4}\int_{0}^{t}d{t^{\prime}}\,{\rm tr}\big{(}-[H^{S}_{t^{\prime}}+H^{\rm int}_{t^{\prime}},\,\beta H^{B}-\log\rho^{S}_{\rm in}]^{2}\,\rho^{S}_{\rm in}\,\rho^{B}_{\beta}\big{)}.$$

$$\begin{array}[]{c}\displaystyle H^{B}=\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k},\end{array}$$

$$e^{-\beta H^B/2}{b}_k{e}^{\beta H^B/2}=e^{\beta {\omega }_k/2}{b}_k,\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}{e}^{-\beta H^B/2}{b}_k^{†}{e}^{\beta H^B/2}=e^{-\beta {\omega }_k/2}{b}_k^{†},$$

$$\begin{array}[]{c}\displaystyle H^{\mathrm{int}}_{t}=\frac{1}{\sqrt{N}}\sum_{k}\left(J_{k}S^{\dagger}_{k}b_{k}+J_{k}^{*}S_{k}b_{k}^{\dagger}\right),\end{array}$$

$$D({\rho }_0,{\rho }_t{)}^2$$

$$+\sum _k\frac{|J_k{|}^2}{N(e^{\beta {\omega }_k}-1)}(e^{\beta {\omega }_k}{S}_k^{†}{S}_k+S_k{S}_k^{†}){\rho }_{\mathrm{in}}^S$$

$$\displaystyle-\sum_{k}\frac{2\,|J_{k}|^{2}\,\,e^{\beta\omega_{k}/2}}{N(e^{\beta\omega_{k}}-1)}S_{k}^{\dagger}\sqrt{\rho^{S}_{\rm in}}\,S_{k}\sqrt{\rho^{S}_{\rm in}}\Bigg{)}.$$

$$H^S=\frac{\Delta }{2}{\sigma }^z,\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}{H}^{\mathrm{int}}=\frac{\gamma }{\sqrt{N}}{\sigma }_x\sum _k{g}_k\left(b_k+b_k^{†}\right),\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}{H}^B=\sum _{k=1}^N{\omega }_k{b}_k^{†}{b}_k,$$

$$\begin{array}[]{c}\displaystyle N^{-1}\sum_{k}|g_{k}|^{2}f(\omega_{k})\xrightarrow{N\rightarrow\infty}\int_{0}^{\infty}d\omega J(\omega)f(\omega)\end{array}$$

$$\begin{array}[]{c}\displaystyle J(\omega)=\omega e^{-\alpha\omega},\end{array}$$

$$\begin{array}[]{c}\displaystyle\rho^{S}_{\rm in}=\frac{1}{2}(1+\mathbf{p}\bm{\sigma}),\quad\mathbf{p}=p\mathbf{n},\quad|\mathbf{n}|=1,\quad p\leq 1.\end{array}$$

$$\begin{gathered} f\left(\frac{1}{2}(1+\mathbf{p}\mathbf{\sigma })\right)=A+B\mathbf{n}\mathbf{\sigma },\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\mathrm{where} \\ A=\frac{1}{2}\left(f\left(\frac{1+p}{2}\right)+f\left(\frac{1-p}{2}\right)\right),B=\frac{1}{2}\left(f\left(\frac{1+p}{2}\right)-f\left(\frac{1-p}{2}\right)\right). \end{gathered}$$

$$\begin{array}[]{c}\displaystyle K_{\beta}=\beta^{2}\int\limits_{0}^{\infty}J(\omega)\frac{e^{\beta\omega}+1}{e^{\beta\omega}-1}d\omega,\end{array}$$

$$\begin{array}[]{c}\displaystyle I_{\beta}=\beta^{4}\int\limits_{0}^{\infty}\omega^{2}J(\omega)\frac{e^{\beta\omega}+1}{e^{\beta\omega}-1}d\omega,\end{array}$$

$$\begin{array}[]{c}\displaystyle L_{\beta}=\beta^{2}\int\limits_{0}^{\infty}J(\omega)\frac{e^{\frac{\beta\omega}{2}}}{e^{\beta\omega}-1}d\omega.\end{array}$$

$$\begin{array}[]{c}\displaystyle\frac{D(\rho_{0},\rho_{t})^{2}}{t^{2}}\leqslant\frac{1-p_{z}^{2}}{2}\Delta^{2}+2\frac{\gamma^{2}}{\beta^{2}}K_{\beta}.\end{array}$$

$$\begin{array}[]{c}\displaystyle\frac{D(\rho_{0},\rho_{t})^{2}}{t^{2}}\leqslant\frac{\gamma^{2}}{2\beta^{2}}I_{\beta}+4\Delta^{2}B_{p}^{2}(1-n^{2}_{z})+16B_{p}^{2}(1-n^{2}_{x})\frac{\gamma^{2}}{\beta^{2}}K_{\beta}\end{array}$$

$$\begin{array}[]{c}\displaystyle B_{p}^{2}=\frac{1}{4}\left(1-\sqrt{1-p^{2}}\right).\end{array}$$

$$\begin{array}[]{c}\displaystyle\frac{D(\rho_{0},\rho_{t})^{2}}{t^{2}}\leqslant\frac{\gamma^{2}}{4\beta^{2}}I_{\beta}+\frac{1}{4}\Delta^{2}\tilde{B_{p}}^{2}(1-n^{2}_{z})+\tilde{B_{p}}^{2}(1-n^{2}_{x})\frac{\gamma^{2}}{\beta^{2}}K_{\beta}\end{array}$$

$$\begin{array}[]{c}\displaystyle\tilde{B_{p}}=\ln\sqrt{\frac{1+p}{1-p}}.\end{array}$$

Full text

1 Skolkovo Institute of Science and Technology,

Skolkovo Innovation Center 3, Moscow 143026, Russia

2 Faculty of Physics and Earth Sciences, Leipzig University,

Linnéstrasse 5, Leipzig, 04103 Germany

3 Department of Mathematical Methods for Quantum Technologies, Steklov Mathematical Institute of Russian Academy of Sciences,

Gubkina str., 8, Moscow 119991, Russia

1 Introduction

A Quantum Speed Limit (QSL) is an upper bound on the speed of evolution of a quantum state. The first QSL was put forward by Mandelstam and Tamm (MT) [1] (see also [2]), who enquired into a rigorous formulation of the time-energy uncertainty relation. A large variety of QSLs have been discovered since then, as reviewed e.g. in [3, 4, 5]. QSLs have conceptual importance as rigorous analogs of the quantum uncertainty principle in the time-energy domain. They emerge in various branches of quantum science, including optimal control theory [6], quantum resource theory [7], and abstract quantum information theory [8]. QSLs are deeply interrelated with the orthogonality catastrophe, adiabatic conditions, and adiabatic quantum computations [9, 10, 11, 12, 13, 14, 15]. They bound ultimate performance of classical and quantum computers [16, 17, 18, 19], heat engines [20, 21], thermometers [22, 23] and even batteries [24, 25].

QSLs can be particularly useful when dealing with many-body dynamics: They promise simple estimates on the rate of change of a quantum state where addressing this change exactly is not feasible. Importantly, to fulfill this promise, a QSL should be at least finite in the thermodynamic limit. Unfortunately, this is often not the case for the MT QSL and several other popular QSLs, as observed e.g. in [26]. However, several more recent QSLs do meet this requirement [27, 10, 28, 29, 30, 31].

The goal of the present paper is to derive practical QSLs for an open quantum system coupled to a large thermal bath. We will detail what can be considered to be ‘‘practical’’ in Section 2. At this point, we just stipulate that a practical QSL should, at least, (i) be finite in the limit of an infinitely large bath and (ii) avoid expectation values of nonlocal observables.

The above goal has been pursued previously in refs. [32, 33]. In these articles, practical QSLs have been derived in the setting where the open system dynamics is described by the Lindblad equation. The latter has its applicability limits, most importantly – the assumption of Markovianity that is usually justified in the limit of a weak system-bath coupling [34, 35, 36]. To the best of our knowledge, there are no analogous results beyond these applicability limits. This gap is filled in the present paper. QSLs we derive are universally valid for arbitrary system-bath couplings.

The rest of the paper is structured as follows. After setting the stage in Section 2, we present a QSL in Section 3 that closely parallels the QSL by Mondal, Datta, and Sazim (MDS) [27]. This QSL remains finite in the limit of a large bath; however, it involves, in general, nonlocal observables and thus is not practical as such. We propose three ways to reduce it to more practical bounds. The most straightforward one is presented in Section 4. In Section 5 we develop a more refined approach that explicitly exploits the special structure of the thermal state of the bath. In Section 6 we derive a practical QSL for a special case of noninteracting baths. In Section 7 we illustrate and compare the performance of the derived QSLs in the spin-boson model. The proofs of all these bounds are relegated to Section 8. In Appendix A we present improvements of the thermal adiabatic condition derived in ref. [37] and a QSL derived in ref. [28]. These improvements are based on mathematical insights gained in the present work.

2 Setup

We consider the dynamics of a quantum system with the Hamiltonian $H^{S}_{t}$ that is coupled to the external bath with the Hamiltonian $H^{B}$ via the coupling term $H^{\mathrm{int}}_{t}$. The total Hamiltonian of the system and the bath is therefore

[TABLE]

Note that the system Hamiltonian and the system-bath coupling can be time-dependent, while the bath Hamiltonian does not change with time.

The quantum state of the system and the bath is described by the density matrix $\rho_{t}$ that evolves according to the von Neumann equation,

[TABLE]

We assume that at $t=0$ this density matrix is initialized in a tensor product state,

[TABLE]

[TABLE]

Here the initial state of the bath, $\rho^{B}_{\beta}$, is thermal, while the initial state of the system, $\rho^{S}_{\rm in}$, is arbitrary (mixed or pure).

Thermal baths are typically large. Therefore, we require that a practical QSL does not diverge in the thermodynamic limit of the bath.

The second practical requirement is to avoid expectation values of nonlocal111Here locality is understood as the absence of progressively many-body operators in the thermodynamic limit. In other words, a local operator is a few-body operator, and a nonlocal operator is a non-few-body operator. This should not be confused with the geometrical notion of locality that is not used in the present paper. For example, in a lattice spin system of $L$ sites a product of two spin operators at different sites is local (irrespective of the distance between the sites), while a product of $L/2$ spin operators is nonlocal. All Hamiltonians are assumed to be few-body. operators of the bath, as they can be, in general, neither calculated theoretically nor measured experimentally.222Of course, these expectation values should be calculated with respect to the initial state $\rho_{0}$ given by eq. (3), not the time-evolving state $\rho_{t}$. This is an almost trivial remark, considering our motivation to estimate the speed of evolution of a quantum state without solving a prohibitively complex many-body von Neumann equation (2) (see also a related discussion in ref. [33]). Note, however, that there are different agendas where a QSL can involve expectation values with respect to $\rho_{t}$. In particular, this is often the case in control theory, where a desired $\rho_{t}$ is given and one wishes to design a Hamiltonian that ensures this $\rho_{t}$ is attained (see e.g. ref. [26]). We do not consider such cases here. Such expectation values can emerge from the commutation of $\sqrt{\rho^{B}_{\beta}}$ with bath operators, as discussed below in relation to the MDS QSL (5).

At the same time, we do not consider the thermodynamic limit for the system. We have in mind mostly systems of moderate sizes, such as single qubits or qudits. For this reason, we regard arbitrary operations over $\rho^{S}_{\rm in}$ feasible and thus practical. In particular, we will employ the square root $\sqrt{\rho^{S}_{\rm in}}$ and commutators between $\sqrt{\rho^{S}_{\rm in}}$ and operators pertaining to the system.

To quantify the speed of evolution of a quantum state, one needs to define a metric on the space of quantum states. A number of such metrics are known [38]. We choose the Hellinger distance defined as [39]

[TABLE]

It is well suited for quantum state discrimination and relates by two-sided inequalities to other popular measures of quantum state distinguishability [40, 39, 38].

3 Mondal-Datta-Sazim QSL

The first QSL we introduce reads

[TABLE]

Its structure parallels the structure of the QSL by Mondal, Datta and Sazim [27] (where a different metric is employed). We will therefore refer to the bound (5) as MDS QSL.

Remarkably, the MDS QSL does not contain the bath Hamiltonian (it commutes with $\rho_{0}$ and therefore drops from eq. (5)). For this reason, the MDS QSL remains finite in the thermodynamic limit of the bath, under natural assumptions on the system-bath coupling. This will be shown explicitly in the next section.

However, when expanding the square of the commutator in the r.h.s. of eq. (5), one obtains a term {\rm tr}\Big{(}\sqrt{\rho^{S}_{\rm in}}\,H^{\mathrm{int}}_{t^{\prime}}\sqrt{\rho^{S}_{\rm in}}e^{-\beta H^{B}/2}H^{\mathrm{int}}_{t^{\prime}}e^{\beta H^{B}/2}\rho^{B}_{\beta}\Big{)} that contains the operator

[TABLE]

In general, this is a highly nonlocal operator [41] whose thermal expectation value can be neither effectively calculated nor measured in a realistic experiment. For this reason, the MDS QSL (5) should be deemed impractical in the considered setting.

Our strategy is to derive bounds based on the MDS QSL (5) that lack the above drawback. In the following three sections, we pursue three different approaches to this task.

There are two special cases when the operator (6) is local. The first one is when the bath is noninteracting, which is considered in Section 6. The second one is when the bath temperature is infinite, $\beta\rightarrow 0$. This is an instructive limit that will be considered when comparing QSLs derived in what follows.

For systems with finite Hilbert spaces, it is useful to additionally check the obtained QSLs in a trivial limit

[TABLE]

If the bath also has a finite Hilbert space, the von Neumann equation (2) has a trivial solution $\rho_{t}=\rho_{0}\sim\mathbb{1}$ in this limit, and thus $D(\rho_{0},\rho_{t})=0$. Obviously, this result is reproduced by the MDS QSL (5). Although for baths with infinite Hilbert spaces the limit can be more subtle mathematically, we still expect the same result on physical grounds.

Note that it is the full quantum state of the system and the bath whose evolution is bounded by the inequality (5). All other QSLs presented in what follows share this feature. Due to the contractivity of the Hellinger distance with respect to the partial trace, $D({\rm tr}_{B}\rho_{0},{\rm tr}_{B}\rho_{t})\leq D(\rho_{0},\rho_{t})$ [39], the same QSLs hold for the reduced state of the system alone.

4 A simple relaxation of MDS QSL

Since nonlocal operators in the MDS QSL (5) appear due to the commutator, a straightforward way to avoid them is to get rid of the commutator altogether. This can be done by using a simple inequality

[TABLE]

valid for an arbitrary real number $c$, quantum state $\rho$ and self-adjoint operator $A$. Applying this inequality to the MDS QSL (5) we get

[TABLE]

A real function $c_{t}$ is, in general, arbitrary and can be chosen to optimize the bound. Alternatively, one can simply take $c_{t}={\rm tr}(H^{\mathrm{int}}_{t}+H^{S}_{t})\rho_{0}$ – we expect that normally this will be a good choice. This is corroborated by considering a specific example in Section 7, where this choice of $c_{t}$ is employed.

Let us discuss the properties of the QSL (9). First of all, this simple bound lacks any nonlocal operators and, instead, contains only expectation values of few-body observables. Second, as long as the interaction Hamiltonian does not diverge with the bath size (which we assume), the bound is manifestly finite in the limit of infinitely large bath. Thus the QSL (9) fulfills our practicability requirements.

An advantage of the QSL (9) is its simple form. However, this simplicity comes at a cost: the bound can be quite loose, particularly at high bath temperatures. For example, this is evident in the trivial limit (7): the correct result $D(\rho_{0},\rho_{t})=0$ is not, in general, captured by the QSL (9). Still, in other cases the QSL (9) performs quite well, as will be exemplified in Section 7.

5 QSLs explicitly depending on temperature

Here we present two QSLs that exploit the special structure of the bath thermal state and explicitly depend on temperature. They read

[TABLE]

and

[TABLE]

In eq. (10), $\{\dots,\dots\}$ stands for the anticommutator.

We remind that the proofs are presented in Sect. 8. A technique developed in refs. [37, 28] is employed there. This technique allows one to replace commutators with $\sqrt{\rho^{B}_{\beta}}$ in the MDS QSL (5) by commutators with $H^{B}$. This way the above bounds get rid of nonlocal operators, as desired.

Observe that in the trivial limit (7) both inequalities (10) and (11) reproduce the correct result $D(\rho_{0},\rho_{t})=0$, in contrast to the QSL (9).

6 QSL for noninteracting baths

The bath is called noninteracting when it is a collection of noninteracting bosons or fermions,

[TABLE]

where $b_{k}$ is an annihilation operator of the $k$’th bosonic or fermionic mode and $\omega_{k}$ is the energy of this mode. In this case, the operator (6) remains a few-body one and can be calculated explicitly. Namely, thanks to the equalities

[TABLE]

the calculation is reduced to substituting $b_{k}$ by $e^{\beta\omega_{k}/2}\,b_{k}$ and $b_{k}^{\dagger}$ by $e^{-\beta\omega_{k}/2}\,b_{k}^{\dagger}$ in $H^{\mathrm{int}}_{t^{\prime}}$.

As an illustration, we perform this calculation for a bosonic bath and a linear system-bath coupling of the form

[TABLE]

where $N$ is the number of modes, $S_{k}$ are operators acting in the system Hilbert space and $J_{k}$ are possibly complex coupling strengths. In general, $J_{k}$ and $S_{k}$ can depend on time, however, we suppress the subscript $t$ to lighten notations.

In this special case, the MDS QSL (5) reduces to

[TABLE]

One can easily check that in the trivial limit (7) this reduces to $D(\rho_{0},\rho_{t})=0$ as expected.

7 Example: spin-boson model

A system with a noninteracting bath can serve as an instructive example for assessing and comparing the performance of QSLs derived in the previous sections. We perform such a comparison for a spin-boson model with

[TABLE]

where $b_{k}$ are boson operators, $\sigma^{\alpha},\alpha=x,y,z$ are Pauli matrices, $g_{k}$ are constants with the dimension of energy and $\gamma$ is the overall dimensionless interaction strength. We consider the thermodynamic limit $N\rightarrow\infty$ for the bath and introduce the spectral density $J(\omega)$ that, by definition, satisfies

[TABLE]

for an arbitrary smooth function $f(\omega)$. For concrete calculations, we choose a regularized Ohmic spectral density of the form

[TABLE]

where $\alpha$ is a constant responsible for the high-energy cutoff.

We consider a general initial state of the spin given by

[TABLE]

Note that for an arbitrary function $f$

[TABLE]

We introduce the following notations for integrals that will appear in the QSLs:

[TABLE]

[TABLE]

[TABLE]

Since the Hamiltonian (16) does not depend on time, it is convenient to compare the bounds on $D(\rho_{0},\rho_{t})^{2}/t^{2}$. The MDS QSL (9) reduces to

[TABLE]

The QSL (10) reduces to

[TABLE]

with

[TABLE]

The QSL (11) reduces to

[TABLE]

with

[TABLE]

Finally, the QSL (15) reads

[TABLE]

For the specific spectral density (18) one can calculate $K_{\beta}$, $I_{\beta}$ and $L_{\beta}$ explicitly:

[TABLE]

where $\psi^{(m)}(z)$ is the polygamma function of order $m$,

[TABLE]

and $\Gamma(z)$ is the Euler gamma function.

Using the above formulae, we inspect how four derived QSLs perform in the spin-boson model. Several illustrative plots are presented in Fig. 1. Of course, the QSL (15) is never outperformed by QSLs (9), (10), (11). This is because the former is just the MDS QSL (5) specified for the case of noninteracting bosonic bath, while the later are different relaxations of the MDS QSL. We emphasize, however, that for interacting baths the QSL (15) is not available, and one is left with the three remaining ones. Fig. 1 illustrates that each of them can outperform the other two for certain bath temperatures and system initial states.

One can see from Fig. 1 that for pure initial states of the system, the QSL (9) typically outperforms QSLs (10) and (11). Furthermore, it turns out to coincide with the QSL (15) for initial polarizations along $y$- or $z$-axes. This can be easily understood from eqs. (24), (29). However, this superiority of the QSL (9) is abruptly lost as soon as the initial state ceases to be pure and the bath temperature is sufficiently high, which is particularly evident by comparing plots (c) and (f) in Fig. 1. Under the later conditions it is the QSL (11) which typically works better.

8 Proofs

8.1 Proof of eq. (5)

To derive the eq. (5) we start from the von Neumann equation for the square root of density matrix $\sqrt{\rho_{t}}$,

[TABLE]

From the definition (4) of the Hellinger distance one gets

[TABLE]

or, equivalently:

[TABLE]

Thus, using 32 and the cyclic property of the trace one obtains

[TABLE]

Here $H^{B}$ drops from the commutator since it commutes with $\rho_{0}$ given by eqs. (3)

Finally, we apply to eq. (35) the Cauchy-Bunyakovsky-Schwarz inequality in the form

[TABLE]

and obtain eq. (5).

8.2 Lemma

The proof of eqs. (10) and (11) employs the following auxiliary

Lemma. For arbitrary real $x$ and $y$

[TABLE]

To prove the Lemma, we use the Hermite-Hadamard inequality [42]

[TABLE]

valid for any convex function $f(t)$. For $f(t)=e^{-t}$, it takes the form

[TABLE]

This inequality leads to eq. (37) after simple algebraic manipulations. The range of eq. (37) is immediately expanded to $y\leq x$ on symmetry grounds.

8.3 Proof of eq. (10)

Note that

[TABLE]

Multiplying (40) by its conjugate, taking the trace and applying the inequality

[TABLE]

we get

[TABLE]

Then we plug this inequality in the MDS QSL (5). The second term in this inequality directly leads to the second term in eq. (10). To estimate the first term, we employ a technique similar to that in [37, 28]. We expand the trace in the product eigenbasis $|\mu,m\rangle=|\mu\rangle|m\rangle$, where $|\mu\rangle$ are eigenvectors of $\rho^{S}_{\rm in}$ and $|m\rangle$ are eigenvectors of $H^{B}$:

[TABLE]

where $w_{\mu}$ and $w_{\nu}$ are eigenvalues of $\rho^{S}_{\rm in}$. For the function $e^{-\beta E/2}$ the inequality (37) reads

[TABLE]

Using (44) we estimate (43) as

[TABLE]

Combining eq. (45) with eqs. (42) and (5) completes the proof of the bound (10).

8.4 Proof of eq. (11)

We rewrite the integrand in eq. (5) as

[TABLE]

Let us introduce a new variable

[TABLE]

Applying the inequality (37) to

[TABLE]

we get

[TABLE]

Plugging this into eq. (46), we get

[TABLE]

Plugging this inequality to the MDS QSL (5), we obtain the bound (11).

9 Summary

We have proved quantum speed limits (9), (10), (11) and (15) for open systems in contact with thermal baths. In contrast to prior knowledge, they do not rest on Markovian or other approximations and are applicable irrespective of the strength and time dependence of the system-bath couplings. Importantly, these limits are practical when applied to many-body baths, i.e. they do not diverge in the thermodynamic limit and contain only local quantities that can be calculated or measured. The derived QSLs complement each other in different circumstances. In particular, the QSL (15) is the best for noninteracting baths but is not applicable otherwise. The remaining three QSLs can be applied to interacting baths: the QSL (9) is typically the best for pure system initial states or low bath temperatures, while QSLs (10) and (11) perform better for mixed initial system states and sufficiently large bath temperatures.

10 Acknowledgements

This work was supported by the Russian Science Foundation under the grant No 17-71-20158.

Appendix A Improvements of the thermal QSL [28] and thermal adiabatic theorem [37]

While proving the bounds presented in this paper, we have employed and improved the technique used previously in refs. [37, 28]. The improvements allow us to tighten the results of this prior work.

The main improvement is the usage of the Lemma from Section 8.2. A function

[TABLE]

plays an important role in refs. [37, 28]. It has been bounded in [37, 28] by the inequality

[TABLE]

In fact, according to the Lemma (37), this inequality can be replaced by a tighter one,

[TABLE]

In effect, the resulting inequalities in refs. [37, 28] turns out to be twice tighter as the original ones.

A second improvement is the usage of the Cauchy inequality in the integral form (36) instead of its weaker versions in refs. [37, 28].

As a result, one replaces the adiabatic condition (S43) in [37] by

[TABLE]

The notations here are defined in ref. [37]. The adiabatic condition (13) in [37] is improved accordingly.

Analogously, one replaces the QSL (30) from ref. [28] by its stronger version:

[TABLE]

The reader is referred to ref. [28] for the explanation of the setup and notations in this formula. The QSLs (8) and (21) in ref. [28] are modified accordingly.