Hypercontractivity of quasi-free quantum semigroups

In order to present our results, we will need to introduce some notation and definitions. Throughout this paper we will be working exclusively with operators acting on finite Hilbert spaces (d-dimensional), which are isomorphic to the algebra of d-dimensional complex matrices ${{\mathcal{M}}_{d}}\cong {{\mathbb{C}}^{d\times d}}$, when equipped with an inner product. We denote the set of d-dimensional Hermitian operators ${{\mathcal{A}}_{d}}=\{X\in {{\mathcal{M}}_{d}},X={{X}^{\dagger }}\}$, as well as the subset of positive definite operators $\mathcal{A}_{d}^{+}=\{X\in {{\mathcal{A}}_{d}},X\gt 0\}$. The set of states will be denoted ${{\mathcal{S}}_{d}}=\{X\in {{\mathcal{A}}_{d}},X\geqslant 0,{\rm tr}\;\left[ X \right]=1\}$, and the full rank states will be analogously denoted $\mathcal{S}_{d}^{+}$. Observables will always be represented by lower case Latin letters ($f,g\in {{\mathcal{A}}_{d}}$), and states by Greek letters ($\rho ,\sigma \in {{\mathcal{S}}_{d}}$). The results presented below are expressed in the framework of non-commutative ${{\mathbb{L}}_{p}}$ spaces [22, 23]. The central property of the ${{\mathbb{L}}_{p}}$ spaces, is that the norm as well as the scalar product is weighted with respect to some full rank reference state $\sigma \in \mathcal{S}_{d}^{+}.$ The ${{\mathbb{L}}_{p}}$-norm with respect to some $\sigma \in \mathcal{S}_{d}^{+}$,is defined for any $f\in {{\mathcal{A}}_{d}}$ as

$$\begin{eqnarray}&&\parallel f{{\parallel }_{p,\sigma }}={\rm tr}\;{{\left[ |\ {{\sigma }^{\frac{1}{2p}}}f{{\sigma }^{\frac{1}{2p}}}\ {{|}^{p}} \right]}^{\frac{1}{p}}}.\end{eqnarray} \tag{ 1 }$$

Similarly, the ${{\mathbb{L}}_{p}}$-inner product for any $f,g\in {{\mathcal{A}}_{d}}$ is given by

$$\begin{eqnarray}&&{{\left\langle f,g \right\rangle }_{\sigma }}={\rm tr}\;\left[ {{\sigma }^{1/2}}{{f}^{\dagger }}{{\sigma }^{1/2}}g \right].\end{eqnarray} \tag{ 2 }$$

The time evolution of an observable (${{f}_{t}}\in {{\mathcal{A}}_{d}}$) will be described by one-parameter semigroups of completely-positive trace preserving maps (cptp-maps), whose generator (Liouvillian) can always be written in standard Lindblad form

$$\begin{eqnarray}&&{{\partial }_{t}}{{f}_{t}}=\mathcal{L}({{f}_{t}})\equiv i[H,{{f}_{t}}]+\mathop{\sum }\limits_{i}L_{i}^{\dagger }{{f}_{t}}{{L}_{i}}-\frac{1}{2}{{\{L_{i}^{\dagger }{{L}_{i}},{{f}_{t}}\}}_{+}},\end{eqnarray} \tag{ 3 }$$

where ${{L}_{i}}\in {{\mathcal{M}}_{d}}$ are Lindblad operators and $H\in {{\mathcal{A}}_{d}}$ is a Hamiltonian operator. We will denote the semigroup generated by $\mathcal{L}$ by ${{T}_{t}}\equiv {\rm exp} (t\mathcal{L})$. A semigroup is said to be primitive if it has a unique full-rank stationary state. We will typically denote the fixed point of the semigroup by σ. The ${{\mathbb{L}}_{p}}$ norm and other weighted forms will always be expressed with respect to the unique fixed point of $\mathcal{L}$.

An important concept for our analysis is the detailed balance of the semigroupʼs generator. A general definition of detailed balance for Markovian generators in W^*-algebras has been in given in [24]. However, since we work on a finite dimensional state space and already assume the generator to be of Lindblad form we follow [25, 26] and work with the definition below.

Definition 1. We say a Liouvillian $\mathcal{L}:{{\mathcal{M}}_{d}}\to {{\mathcal{M}}_{d}}$ satisfies detailed balance (or is reversible) with respect to the state $\sigma \in \mathcal{S}_{d}^{+}$, if for any $f,g\in {{\mathcal{A}}_{d}}$

$$\begin{eqnarray}&&{{\left\langle f,\mathcal{L}(g) \right\rangle }_{\sigma }}={{\left\langle \mathcal{L}(f),g \right\rangle }_{\sigma }}.\end{eqnarray} \tag{ 4 }$$

We are now in a position to state the definition of hypercontractivity, which will be the main object of study in this paper.

Definition 2. Let ${{T}_{t}}:{{\mathcal{M}}_{d}}\to {{\mathcal{M}}_{d}}$ be a primitive semigroup with stationary state σ. We say T_t is hypercontractive if there exist constants $\alpha ,{{t}_{0}}\gt 0$ such that for any $f\in \mathcal{A}_{d}^{+}$

$$\begin{eqnarray}&&\parallel {{T}_{t}}(f){{\parallel }_{p(t),\sigma }}\leqslant \parallel f{{\parallel }_{2,\sigma }},\end{eqnarray} \tag{ 5 }$$

with $p(t)=1+{{{\rm e}}^{2\alpha t}}$, whenever $t\geqslant {{t}_{0}}$.

The optimal (largest) constant α which satisfies equation (5) is related to the log Sobolev constant, which we define below. The log-Sobolev constant is defined in terms of a variational optimization over an entropy functional and the Dirichlet form of $\mathcal{L}$. See [16] for a detailed analysis.

The Dirichlet form of $\mathcal{L}$ is defined as

$$\begin{eqnarray}&&\mathcal{E}(f)=-{{\left\langle f,\mathcal{L}(f) \right\rangle }_{\sigma }},\end{eqnarray} \tag{ 6 }$$

whereas the ${{\mathbb{L}}_{2}}$ relative entropy is given by

$$\begin{eqnarray}\begin{array}{rcl} {\rm Ent}(f) & = & {\rm tr}\;\left[ {{\left( {{\sigma }^{1/4}}f{{\sigma }^{1/4}} \right)}^{2}}{\rm log} \left( {{\sigma }^{1/4}}f{{\sigma }^{1/4}} \right) \right]-\frac{1}{2}{\rm tr}\;\left[ {{\left( {{\sigma }^{1/4}}f{{\sigma }^{1/4}} \right)}^{2}}{\rm log} \left( \sigma \right) \right] \\ {} & {} & -\frac{1}{2}\parallel f\parallel _{2,\sigma }^{2}\;{\rm log} \left( \parallel f\parallel _{2,\sigma }^{2} \right). \\ \end{array}\end{eqnarray} \tag{ 7 }$$

Definition 3. Let $\mathcal{L}:{{\mathcal{M}}_{d}}\to {{\mathcal{M}}_{d}}$ be a primitive Liouvillian with stationary state $\sigma \in \mathcal{S}_{d}^{+}$. We say that $\mathcal{L}$ satisfies a log-Sobolev inequality, if there exists a positive constant $\alpha \gt 0$ such that

$$\begin{eqnarray}&&\alpha {\rm Ent}(f)\leqslant \mathcal{E}(f),\end{eqnarray} \tag{ 8 }$$

for all $f\in \mathcal{A}_{d}^{+}$. We call the largest α for which equation (8) holds the log-Sobolev constant.

In order to rigorously formulate the equivalence between hypercontractivity and log-Sobolev inequalities, we need to invoke a property of quantum semigroups called ${{\mathbb{L}}_{p}}$-regularity. This condition is elaborate to describe, and not very insightful, and we refer the interested reader to [14, 16] for a detailed description and analysis. Unless otherwise states, all of the results in this paper hold without the additional assumption of ${{\mathbb{L}}_{p}}$ regularity, and hence we will not dwell on it further.

Theorem 4. Let $\mathcal{L}:{{\mathcal{M}}_{d}}\to {{\mathcal{M}}_{d}}$ be a primitive reversible Liouvillian with stationary state σ, and let T_t be its associated semigroup. Then:

• 1.  
  if there exists a $\beta \gt 0$ such that for any $t\gt 0$, $\parallel {{T}_{t}}(f){{\parallel }_{p(t),\sigma }}\leqslant \parallel f{{\parallel }_{2,\sigma }}$ for all $f\in \mathcal{A}_{d}^{+}$ and $2\leqslant p(t)\leqslant 1+{{{\rm e}}^{2\beta t}}$. Then $\mathcal{L}$ satisfies a log-Sobolev inequality with $\alpha \geqslant \beta$.
• 2.  
  If $\mathcal{L}$ is ${{\mathbb{L}}_{p}}$-regular, and has a log-Sobolev constant $\alpha \gt 0$, then $\parallel {{T}_{t}}(f){{\parallel }_{p(t),\sigma }}\leqslant \parallel f{{\parallel }_{2,\sigma }}$ for all $f\in \mathcal{A}_{d}^{+}$, and any $t\gt 0$ when $2\leqslant p(t)\leqslant 1+{{{\rm e}}^{2\alpha t}}$.

To prove a logarithmic Sobolev inequality, it therefore suffices to show hypercontractivity and visa versa. In many cases it is easier to prove hypercontractivity directly and deduce from it a bound on the log-Sobolev constant.

One of the main applications of hypercontractivity and log-Sobolev inequalities is that they imply very strong bounds on the mixing time of the semigroup. In particular, if $\mathcal{L}$ is a primitive semigroup with stationary state σ, and spectral gap λ, then the best generic exponential bound that can be obtained for convergence in trace norm is

$$\begin{eqnarray}&&\mathop{{\rm sup} }\limits_{\rho }\parallel {{{\rm e}}^{t\mathcal{L}}}(\rho )-\sigma {{\parallel }_{1}}\leqslant \sqrt{\parallel {{\sigma }^{-1}}\parallel }{{{\rm e}}^{-t\lambda }}.\end{eqnarray} \tag{ 9 }$$

Whereas if the semigroup is ${{\mathbb{L}}_{p}}$ regular and satisfies a log-Sobolev inequality with log-Sobolev constant α, then

$$\begin{eqnarray}&&\mathop{{\rm sup} }\limits_{\rho }\parallel {{{\rm e}}^{t\mathcal{L}}}(\rho )-\sigma {{\parallel }_{1}}\leqslant \sqrt{2{\rm log} (\parallel {{\sigma }^{-1}}\parallel )}{{{\rm e}}^{-t\alpha }}.\end{eqnarray} \tag{ 10 }$$

Here we denote by $\parallel A\parallel$ the operator norm of the matrix A. Hence, given that $\parallel {{\sigma }^{-1}}\parallel \geqslant d$, where d is the size of the full matrix algebra, if α and λ are both independent of d, then the log-Sobolev bound in equation (10) is exponentially tighter than the spectral gap bound of equation (9). An compelling application of the log-Sobolev inequalities is in proving stability of dissipative dynamics, where a log-Sobolev constant is sufficient to guarantee stability, whereas a constant gap does not seem to suffice [1].

We now present a general method which allows to prove hypercontractivity for a certain class of gapped semigroups. This approach was first pioneered in [12] and has been extended to non-commutative ${{\mathbb{L}}_{p}}$-spaces in [6].

Lemma 7. Let ${{T}_{t}}={\rm exp} (t\mathcal{L})$ denote a primitive semigroup with fixed point σ. Suppose that the following conditions are satisfied.

• 1.  
  The matrix space $\mathcal{B}={{\mathcal{M}}_{d}}$ has the following block decomposition

  $$\begin{eqnarray}&&\mathcal{B}=\mathop{\oplus }\limits_{n=0}^{N}{{\mathcal{B}}_{n}},\end{eqnarray} \tag{ 31 }$$

  where each block $\mathcal{B}$_n is invariant under the action of T_t, and ${{\mathcal{B}}_{0}}={\rm span}\{{{\mathbb{1}}_{d}}\}$.
• 2.  
  The spectrum of $\mathcal{L}$ restricted to the block $\mathcal{B}$_n is contained in the interval $(-\infty ,-\lambda n]$, where λ is a constant that is independent of n.
• 3.  
  For all ${{f}_{n}}\in {{\mathcal{B}}_{n}}$, we have a norm bound of the form

  $$\begin{eqnarray}&&\parallel {{f}_{n}}{{\parallel }_{4,\sigma }}\leqslant {{C}^{n}}\parallel {{f}_{n}}{{\parallel }_{2,\sigma }},\end{eqnarray} \tag{ 32 }$$

  where C is a positive finite constant.

Then, whenever $t\gt {{\lambda }^{-1}}{\rm ln} (C)$ the following norm bound holds

$$\begin{eqnarray}&&\parallel {{T}_{t}}{{\parallel }_{2\to 4,\sigma }}\leqslant {{M}_{4}}\;\;\;\;\;\;{\rm with}\;\;\;\;\;\;{{M}_{4}}=\frac{1}{1-C{{{\rm e}}^{-\lambda t}}}.\end{eqnarray} \tag{ 33 }$$

Proof. We can always decompose f as $f={{\sum }_{n}}{{f}_{n}}$, where ${{f}_{n}}\in {{\mathcal{B}}_{n}}$. Then, from the triangle inequality for ${{\mathbb{L}}_{p}}(\sigma )$-norms, we get $\parallel {{T}_{t}}f{{\parallel }_{4,\sigma }}\leqslant \sum _{n=0}^{N}\parallel {{T}_{t}}{{f}_{n}}{{\parallel }_{4,\sigma }}$. Now, applying conditions 2 and 3, we obtain the norm bound

$$\begin{eqnarray}&&\parallel {{T}_{t}}f{{\parallel }_{4,\sigma }}\leqslant \mathop{\sum }\limits_{n=0}^{N}{{C}^{n}}\parallel {{T}_{t}}{{f}_{n}}{{\parallel }_{2,\sigma }}\leqslant \mathop{\sum }\limits_{n=0}^{N}{{C}^{n}}{{{\rm e}}^{-n\lambda t}}\parallel {{f}_{n}}{{\parallel }_{2,\sigma }}.\end{eqnarray} \tag{ 34 }$$

Moreover, since the f_n are supported on disjoint blocks and are thereby orthogonal, we get that $\parallel {{f}_{n}}{{\parallel }_{2,\sigma }}\leqslant \parallel f{{\parallel }_{2,\sigma }}$. The proof is completed by noting that for $t\gt {{\lambda }^{-1}}{\rm ln} (C)$ the sum $\sum _{k=0}^{N}{{C}^{n}}{{{\rm e}}^{-n\lambda t}}\leqslant {{(1-C{{{\rm e}}^{-\lambda t}})}^{-1}}$ constitutes a geometric series for which

$$\begin{eqnarray}&&\parallel {{T}_{t}}f{{\parallel }_{4,\sigma }}\leqslant \frac{1}{1-C{{{\rm e}}^{-\lambda t}}}\parallel f{{\parallel }_{2,\sigma }}.\end{eqnarray} \tag{ 35 }$$

□

The general strategy to prove bounds on the log-Sobolev constant α is now the following. We first find an invariant block decomposition for the generator $\mathcal{L}$, that furthermore has a restriction on the spectrum. Then, we show that a norm bound for all elements in the block holds, that is of the form as stated in the Lemma. From these three conditions we obtain the norm bound $\parallel {{T}_{{{t}_{4}}}}{{\parallel }_{2\to 4,\sigma }}\leqslant {{(1-{{C}^{-\lambda {{t}_{4}}}})}^{-1}}={{M}_{4}}$. We then invoke theorem 5, to obtain a lower bound to α and infer the hypercontractive bound.

We can now estimate the best constant in terms of C and λ, by choosing ${{t}_{4}}={\rm log} (2C)$ so that ${{M}_{4}}=2$. If we plug this into the bound of equation (29) we obtain.

Corollary 8. Let ${{T}_{t}}\;:{{\mathcal{M}}_{d}}\to {{\mathcal{M}}_{d}}$ be a reversible semigroup that satisfies the conditions of lemma 7, and let C and λ denote the corresponding constants. Then the log Sobolev constant is bounded by

$$\begin{eqnarray}&&\alpha \geqslant \frac{\lambda }{{\rm log} \left( {{C}^{4}}{{2}^{8}}{{e}^{2}} \right)}.\end{eqnarray} \tag{ 36 }$$

In this section, we consider the case of a product of N primitive semigroups. A proof of the analogous classical problem was obtained in [6]. There, a special class of quantum systems was considered as well. Below we extend the proof to general reversible primitive Liouvillians. A celebrated classical result in the theory of log-Sobolev inequalities, is that the log-Sobolev constant of a tensor product of stochastic semigroups is equal to the minimal log-Sobolev constant of its constituents. This result is known as the product property for the log-Sobolev constant. With the exception of a set of very specific channels [16, 19–21], the product property has not been shown for the quantum log-Sobolev constant, and based on similar results on the non-multiplicativity of p norms [32–34], it is expected that this property does not hold in general. However, the theorem below gives strict bounds on how much the product property can be violated.

Theorem 9. Let ${{\mathcal{L}}_{k}}:{{\mathcal{M}}_{d}}\to {{\mathcal{M}}_{d}}$ be primitive reversible Liouvillians with respective stationary states σ_k and spectral gaps Λ_k. Define the product Liouvillian $\mathcal{L}:{{\mathcal{M}}_{{{d}^{N}}}}\to {{\mathcal{M}}_{{{d}^{N}}}}$ as

$$\begin{eqnarray}&&\mathcal{L}\equiv \mathop{\sum }\limits_{k=1}^{N}{{\mathcal{L}}_{k}},\end{eqnarray} \tag{ 37 }$$

where by abuse of notation, we have lifted each $\mathcal{L}$_k such that it is acting non-trivially only on the kth subsystem. The fixed point of $\mathcal{L}$ is given by $\sigma ={{\otimes }_{k}}{{\sigma }_{k}}$. Then the log-Sobolev constant α of $\mathcal{L}$ is bounded as

$$\begin{eqnarray}&&\frac{\Lambda }{{\rm log} ({{d}^{4}}\;s)+11}\leqslant \alpha \leqslant \Lambda ,\end{eqnarray} \tag{ 38 }$$

where $\Lambda ={{{\rm min} }_{k}}{{\Lambda }_{k}}$ and $s={{{\rm max} }_{k}}\parallel \sigma _{k}^{-1}\parallel$.

Observe, that this bound on the log-Sobolev constant of the product semigroup has no dependence on N.

Proof. The proof will follow very closely the analogous classical one in [6] (theorem 3.1). Let ${{\{{{\Phi }_{i,k}}\}}_{i=0,\cdots ,{{d}^{2}}-1}}$ be the eigenvectors of $\mathcal{L}$_k, with ${{\Phi }_{0,k}}=\mathbb{1}$ the eigenvector corresponding to the stationary state, and let $\{{{\lambda }_{i,k}}\}$ be its spectrum; i.e. ${{\mathcal{L}}_{k}}({{\Phi }_{i,k}})={{\lambda }_{i,k}}{{\Phi }_{i,k}}$. It is always assured that such a spectral decomposition exists with real non-positive ${{\lambda }_{i,k}}$ since we consider only reversible maps. The right eigenvectors of $\mathcal{L}$ are now given by

$$\begin{eqnarray}&&\mathcal{L}\left( \mathop{\otimes }\limits_{k}{{\Phi }_{{{i}_{k}},k}} \right)=\mathop{\sum }\limits_{k}{{\lambda }_{{{i}_{k}},k}}\left( \mathop{\otimes }\limits_{k}{{\Phi }_{{{i}_{k}},k}} \right).\end{eqnarray} \tag{ 39 }$$

We now define the sets $X\subset \{1,\ldots ,N\}$ and tuples $K\in {{[1,{{d}^{2}}-1]}^{|X|}}$ which given X, is isomorphic to the set of functions $X\to [1,{{d}^{2}}-1]$. We will refer to K as a tuple, when $|X|$ is specified but not X and as a function when X is uniquely determined. Using this notation, we denote the eigenvectors of ${{\mathcal{L}}_{N}}$ by

$$\begin{eqnarray}&&\Phi _{X}^{K}=\mathop{\otimes }\limits_{k\in X}{{\Phi }_{K(k),k}}.\end{eqnarray} \tag{ 40 }$$

Then consider the subspaces ${{\mathcal{B}}_{n}}={\rm span}\{\Phi _{X}^{K}:|X|=n\}$ which induce a natural block decomposition of

$$\begin{eqnarray}&&{{\mathcal{M}}_{{{d}^{N}}}}=\mathcal{B}=\mathop{\oplus }\limits_{n=0}^{N}{{\mathcal{B}}_{n}},\end{eqnarray} \tag{ 41 }$$

since these spaces are spanned by eigenvectors. Moreover, we can bound the spectrum of $\mathcal{L}{{\mid }_{{{\mathcal{B}}_{n}}}}$ by ${{\sum }_{k}}{{\lambda }_{{{i}_{k}},k}}={{\sum }_{k\in X}}{{\lambda }_{{{i}_{k}},k}}\leqslant -n\Lambda$, where $\Lambda ={{{\rm min} }_{k}}{{\Lambda }_{k}}$ denotes the spectral gap of $\mathcal{L}$ and ${{\Lambda }_{k}}=-{{{\rm max} }_{i\ne 0,k}}{{\lambda }_{i,k}}$ denotes the spectral gap of $\mathcal{L}$_k. Thus, we have an invariant block decomposition of $\mathcal{B}$ under $\mathcal{L}$ where the spectrum within each block $\mathcal{B}$_n is contained in $(-\infty ,-n\Lambda ]$.

Let us now proceed to prove the norm bound assumption equation (32) of lemma 7. Recall that the generators $\mathcal{L}$_k are reversible, which implies that their eigenvectors satisfy the orthogonality relation

$$\begin{eqnarray}&&{{\left\langle {{\Phi }_{j,k}},{{\Phi }_{l,k}} \right\rangle }_{{{\sigma }_{k}}}}={{\delta }_{jl}}\end{eqnarray} \tag{ 42 }$$

for the ρ weighted scalar product. We can expand any ${{f}_{n}}\in {{\mathcal{B}}_{n}}$ in terms of eigenfunctions of $\mathcal{L}$ so that ${{f}_{n}}={{\sum }_{K,X}}\alpha _{X}^{K}\Phi _{X}^{K}$, where $\alpha _{X}^{K}=0$, whenever $|X|\ne n$. We therefore have that $\parallel {{f}_{n}}{{\parallel }_{2,\sigma }}=\sqrt{{{\sum }_{K,X}}|\alpha _{X}^{K}{{|}^{2}}}$.

We define the quartic form $Q(a,b,c,d)={\rm tr}\;\left[ {{\sigma }^{1/4}}{{a}^{\dagger }}{{\sigma }^{1/4}}b{{\sigma }^{1/4}}{{c}^{\dagger }}{{\sigma }^{1/4}}d \right]$. With this definition at hand, we have that

$$\begin{eqnarray}&&\parallel {{f}_{n}}\parallel _{4,\sigma }^{4}=\mathop{\sum }\limits_{\{{{K}^{(i)}}\},\{{{X}^{(i)}}\}}{{\alpha }^{*}}_{{{X}^{(1)}}}^{{{K}^{(1)}}}\alpha _{{{X}^{(2)}}}^{{{K}^{(2)}}}{{\alpha }^{*}}_{{{X}^{(3)}}}^{{{K}^{(3)}}}\alpha _{{{X}^{(4)}}}^{{{K}^{(4)}}}\ Q(\Phi _{{{X}^{(1)}}}^{{{K}^{(1)}}},\Phi _{{{X}^{(2)}}}^{{{K}^{(2)}}},\Phi _{{{X}^{(3)}}}^{{{K}^{(3)}}},\Phi _{{{X}^{(4)}}}^{{{K}^{(4)}}}).\end{eqnarray} \tag{ 43 }$$

Before we proceed to give the bound on $\parallel {{f}_{n}}{{\parallel }_{4,\sigma }}\leqslant {{C}^{n}}\parallel {{f}_{n}}{{\parallel }_{2,\sigma }}$ we state two facts about the function $Q(\Phi _{{{X}^{(1)}}}^{{{K}^{(1)}}},\Phi _{{{X}^{(2)}}}^{{{K}^{(2)}}},\Phi _{{{X}^{(3)}}}^{{{K}^{(3)}}},\Phi _{{{X}^{(4)}}}^{{{K}^{(4)}}})$. We have that

• 1.  
  Restricting to the $|{{X}_{i}}|=n$ we can bound

  $$\begin{eqnarray}&&\mathop{{\rm max} }\limits_{\{\Phi _{{{X}^{(i)}}}^{{{K}^{(i)}}}\}}Q(\Phi _{{{X}^{(1)}}}^{{{K}^{(1)}}},\Phi _{{{X}^{(2)}}}^{{{K}^{(2)}}},\Phi _{{{X}^{(3)}}}^{{{K}^{(3)}}},\Phi _{{{X}^{(4)}}}^{{{K}^{(4)}}})\leqslant {{s}^{n}}.\end{eqnarray} \tag{ 44 }$$

  This bound can be derived from the the following identities: first by applying the Cauchy–Schwartz inequality twice we obtain

  $$\begin{eqnarray}&&\begin{array}{l} Q(\Phi _{{{X}^{(1)}}}^{{{K}^{(1)}}},\Phi _{{{X}^{(2)}}}^{{{K}^{(2)}}},\Phi _{{{X}^{(3)}}}^{{{K}^{(3)}}},\Phi _{{{X}^{(4)}}}^{{{K}^{(4)}}}) \\ \quad ={\rm tr}\;\left[ {{\sigma }^{1/4}}\Phi {{_{{{X}^{(1)}}}^{{{K}^{(1)}}}}^{\dagger }}{{\sigma }^{1/4}}\Phi _{{{X}^{(2)}}}^{{{K}^{(2)}}}{{\sigma }^{1/4}}\Phi {{_{{{X}^{(3)}}}^{{{K}^{(3)}}}}^{\dagger }}{{\sigma }^{1/4}}\Phi _{{{X}^{(4)}}}^{{{K}^{(4)}}} \right] \\ \quad \leqslant \sqrt{{\rm tr}\;\left[ {{\sigma }^{1/4}}\Phi {{_{{{X}^{(1)}}}^{{{K}^{(1)}}}}^{\dagger }}{{\sigma }^{1/4}}\Phi _{{{X}^{(1)}}}^{{{K}^{(1)}}}{{\sigma }^{1/4}}\Phi {{_{{{X}^{(2)}}}^{{{K}^{(2)}}}}^{\dagger }}{{\sigma }^{1/4}}\Phi _{{{X}^{(2)}}}^{{{K}^{(2)}}} \right]} \\ \quad \;\;\sqrt{{\rm tr}\;\left[ {{\sigma }^{1/4}}\Phi {{_{{{X}^{(3)}}}^{{{K}^{(3)}}}}^{\dagger }}{{\sigma }^{1/4}}\Phi _{{{X}^{(3)}}}^{{{K}^{(3)}}}{{\sigma }^{1/4}}\Phi {{_{{{X}^{(4)}}}^{{{K}^{(4)}}}}^{\dagger }}{{\sigma }^{1/4}}\Phi _{{{X}^{(4)}}}^{{{K}^{(4)}}} \right]} \\ \quad \leqslant \parallel \Phi _{{{X}^{(1)}}}^{{{K}^{(1)}}}{{\parallel }_{4,\sigma }}\parallel \Phi _{{{X}^{(2)}}}^{{{K}^{(2)}}}{{\parallel }_{4,\sigma }}\parallel \Phi _{{{X}^{(3)}}}^{{{K}^{(3)}}}{{\parallel }_{4,\sigma }}\parallel \Phi _{{{X}^{(4)}}}^{{{K}^{(4)}}}{{\parallel }_{4,\sigma }}. \\ \end{array}\end{eqnarray} \tag{ 45 }$$

  The full expression can now be bounded with the following inequalities. Let us normalize the eigenvectors so that $\parallel \Phi _{X}^{K}{{\parallel }_{2,\sigma }}=1$. Note that $\parallel \Phi _{X}^{K}{{\parallel }_{4,\sigma }}={{\prod }_{k\in X}}\parallel {{\Phi }_{K(k),k}}{{\parallel }_{4,{{\sigma }_{k}}}}$, so that we have

  $$\begin{eqnarray}&&\parallel \Phi _{X}^{K}{{\parallel }_{4,\sigma }}=\mathop{\prod }\limits_{k\in X}\parallel {{\Phi }_{K(k),k}}{{\parallel }_{4,{{\sigma }_{k}}}}\leqslant \mathop{\prod }\limits_{k\in X}\parallel \sigma _{k}^{-1}{{\parallel }^{1/4}}\parallel {{\Phi }_{K(k),k}}{{\parallel }_{4,{{\sigma }_{k}}}}\leqslant {{s}^{|X|/4}}.\end{eqnarray} \tag{ 46 }$$

  Recall that we consider the space where $n=|{{X}^{(j)}}|$, so that from the previous inequality we get

  $$\begin{eqnarray}&&Q(\Phi _{{{X}^{(1)}}}^{{{K}^{(1)}}},\Phi _{{{X}^{(2)}}}^{{{K}^{(2)}}},\Phi _{{{X}^{(3)}}}^{{{K}^{(3)}}},\Phi _{{{X}^{(4)}}}^{{{K}^{(4)}}})\leqslant {{s}^{n}}.\end{eqnarray} \tag{ 47 }$$
• 2.  
  If for the sets ${{X}^{(1)}},{{X}^{(2)}},{{X}^{(3)}},{{X}^{(4)}}$ we can find a site with only a single excitation, the Q form vanishes. That is, if for any $l\in \{1,2,3,4\}$ we have that ${{X}^{(l)}}\varsubsetneq {{\cup }_{j\ne l}}{{X}^{(j)}}$, then $Q(\Phi _{{{X}^{(1)}}}^{{{K}^{(1)}}},\Phi _{{{X}^{(2)}}}^{{{K}^{(2)}}},\Phi _{{{X}^{(3)}}}^{{{K}^{(3)}}},\Phi _{{{X}^{(4)}}}^{{{K}^{(4)}}})=0$.

Since both the eigenvectors Φ^K_X as well as $\sigma ={{\otimes }_{j}}{{\sigma }_{j}}$ are of tensor products form, the trace factorizes and we can write Q as the product of traces over the local Hilbert spaces. In particular if we have one $k\in {{X}^{(l)\backslash {{\cup }_{j\ne l}}{{X}^{(j)}}}}$, this implies that one factor is

$$\begin{eqnarray}&&{\rm t}{{{\rm r}}_{k}}\;\left[ \rho _{k}^{1/4}{{\Phi }_{{{K}^{(l)}}(k),k}}\rho _{k}^{1/4}{{\Phi }_{0,k}}\rho _{k}^{1/4}{{\Phi }_{0,k}}\rho _{k}^{1/4}{{\Phi }_{0,k}} \right]={{\left\langle {{\Phi }_{{{K}^{(l)}}(k),k}},{{\Phi }_{0,k}} \right\rangle }_{{{\rho }_{k}}}}=0.\end{eqnarray} \tag{ 48 }$$

So the total product is given by $Q(\Phi _{{{X}^{(1)}}}^{{{K}^{(1)}}},\Phi _{{{X}^{(2)}}}^{{{K}^{(2)}}},\Phi _{{{X}^{(3)}}}^{{{K}^{(3)}}},\Phi _{{{X}^{(4)}}}^{{{K}^{(4)}}})=0$.

With these properties of the Q-function at hand, we can proceed to bound the norm

$$\begin{eqnarray}\begin{array}{lll} \parallel {{f}_{n}}\parallel _{4,\sigma }^{4} & = & \mathop{\sum }\limits_{\{{{K}^{(i)}}\},\{{{X}^{(i)}}\}}{{\alpha }^{*}}_{{{X}^{(1)}}}^{{{K}^{(1)}}}\alpha _{{{X}^{(2)}}}^{{{K}^{(2)}}}{{\alpha }^{*}}_{{{X}^{(3)}}}^{{{K}^{(3)}}}\alpha _{{{X}^{(4)}}}^{{{K}^{(4)}}}\ Q(\Phi _{{{X}^{(1)}}}^{{{K}^{(1)}}},\Phi _{{{X}^{(2)}}}^{{{K}^{(2)}}},\Phi _{{{X}^{(3)}}}^{{{K}^{(3)}}},\Phi _{{{X}^{(4)}}}^{{{K}^{(4)}}}) \\ {} & \leqslant & {{s}^{n}}{{S}_{n}}, \\ \end{array}\end{eqnarray} \tag{ 49 }$$

where we have defined

$$\begin{eqnarray}&&{{S}_{n}}=\sum _{\{{{K}^{(i)}}\},\{{{X}^{(i)}}\}}^{\prime }|\alpha _{{{X}^{(1)}}}^{{{K}^{(1)}}}||\alpha _{{{X}^{(2)}}}^{{{K}^{(2)}}}||\alpha _{{{X}^{(3)}}}^{{{K}^{(3)}}}||\alpha _{{{X}^{(4)}}}^{{{K}^{(4)}}}|.\end{eqnarray} \tag{ 50 }$$

The primed sum indicates that we constrain the full summation indices on sets ${{X}^{(1)}},\ldots ,{{X}^{(4)}}$ so that at every site there is more than one particle. Let us now proceed to bounding the sum. Taking this into account we can introduce new summing sets by writing ${{X}^{(ij)}}={{X}^{(i)}}\cap {{X}^{(j)}}$ and only summing over sets which satisfy ${{X}^{(i)}}={{\cup }_{i\ne j}}{{X}^{(ij)}}$. That is we write for the sum now

$$\begin{eqnarray}&&{{S}_{n}}=\mathop{\sum }\limits_{{{K}^{(i)}}}\mathop{\sum }\limits_{{{X}^{(ij)}}}|\alpha _{{{X}^{(12)}}\cup {{X}^{(13)}}\cup {{X}^{(14)}}}^{{{K}^{(1)}}}||\alpha _{{{X}^{(21)}}\cup {{X}^{(23)}}\cup {{X}^{(24)}}}^{{{K}^{(2)}}}|\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}&&\qquad |\alpha _{{{X}^{(31)}}\cup {{X}^{(32)}}\cup {{X}^{(34)}}}^{{{K}^{(3)}}}||\alpha _{{{X}^{(41)}}\cup {{X}^{(42)}}\cup {{X}^{(43)}}}^{{{K}^{(4)}}}|.\end{eqnarray} \tag{ 52 }$$

Note, that we have ${{X}^{(ij)}}={{X}^{(ji)}}$ so we can employ Cauchy–Schwartz inequality on any pair of sets. We first choose ${{X}^{(12)}},{{X}^{(34)}}$, which leads to the bound

$$\begin{eqnarray}&&{{S}_{n}}\leqslant \mathop{\sum }\limits_{{{K}^{(i)}}}\mathop{\sum }\limits_{{{X}^{(ij)}}\ne {{X}^{(12)}},{{X}^{(34)}}}\sqrt{\mathop{\sum }\limits_{{{X}^{(12)}}}|\alpha _{{{X}^{(12)}}\cup {{X}^{(13)}}\cup {{X}^{(14)}}}^{{{K}^{(1)}}}{{|}^{2}}}\sqrt{\mathop{\sum }\limits_{{{X}^{(12)}}}|\alpha _{{{X}^{(21)}}\cup {{X}^{(23)}}\cup {{X}^{(24)}}}^{{{K}^{(2)}}}{{|}^{2}}}\end{eqnarray} \tag{ 53 }$$

$$\begin{eqnarray}&&\qquad \sqrt{\mathop{\sum }\limits_{{{X}^{(34)}}}|\alpha _{{{X}^{(31)}}\cup {{X}^{(32)}}\cup {{X}^{(34)}}}^{{{K}^{(3)}}}{{|}^{2}}}\sqrt{\mathop{\sum }\limits_{{{X}^{(34)}}}|\alpha _{{{X}^{(41)}}\cup {{X}^{(42)}}\cup {{X}^{(43)}}}^{{{K}^{(4)}}}{{|}^{2}}}.\end{eqnarray} \tag{ 54 }$$

Proceeding inductively, we obtain the bound

$$\begin{eqnarray}&&{{S}_{n}}\leqslant \mathop{\sum }\limits_{{{K}^{(i)}}}\sqrt{\mathop{\sum }\limits_{{{X}^{(12)}},{{X}^{(13)}},{{X}^{(14)}}}|\alpha _{{{X}^{(12)}}\cup {{X}^{(13)}}\cup {{X}^{(14)}}}^{{{K}^{(1)}}}{{|}^{2}}}\sqrt{\mathop{\sum }\limits_{{{X}^{(21)}},{{X}^{(23)}},{{X}^{(24)}}}|\alpha _{{{X}^{(21)}}\cup {{X}^{(23)}}\cup {{X}^{(24)}}}^{{{K}^{(2)}}}{{|}^{2}}}\end{eqnarray} \tag{ 55 }$$

$$\begin{eqnarray}&&\qquad \qquad \sqrt{\mathop{\sum }\limits_{{{X}^{(31)}},{{X}^{(32)}},{{X}^{(34)}}}|\alpha _{{{X}^{(31)}}\cup {{X}^{(32)}}\cup {{X}^{(34)}}}^{{{K}^{(3)}}}{{|}^{2}}}\sqrt{\mathop{\sum }\limits_{{{X}^{(41)}},{{X}^{(42)}},{{X}^{(43)}}}|\alpha _{{{X}^{(41)}}\cup {{X}^{(42)}}\cup {{X}^{(43)}}}^{{{K}^{(4)}}}{{|}^{2}}}.\end{eqnarray} \tag{ 56 }$$

The constraint that ${{X}^{(i)}}={{\cup }_{i\ne j}}{{X}^{(ij)}}$ implies that there are in total ${{2}^{2n}}$ possible combinations of the sets ${{X}^{(ij)}}$ compatible with a given ${{X}^{(i)}}$, so that we can write

$$\begin{eqnarray}&&{{S}_{n}}\leqslant {{2}^{4n}}\mathop{\prod }\limits_{i=1}^{4}\mathop{\sum }\limits_{{{K}^{(i)}}}\sqrt{\mathop{\sum }\limits_{{{X}^{(i)}}}|\alpha _{{{X}^{(i)}}}^{{{K}^{(i)}}}{{|}^{2}}}.\end{eqnarray} \tag{ 57 }$$

The number of sets K is bounded by d²ⁿ in each sum we can again apply the Cauchy–Schwartz inequality in the summand K in order to obtain the final bound on the sum

$$\begin{eqnarray}&&{{S}_{n}}\leqslant {{2}^{4n}}{{d}^{4n}}\parallel {{f}_{n}}\parallel _{2,\sigma }^{4}.\end{eqnarray} \tag{ 58 }$$

This finally leads to the subspace norm bound of

$$\begin{eqnarray}&&\parallel {{f}_{n}}{{\parallel }_{4,\sigma }}\leqslant {{({{s}^{1/4}}2d)}^{n}}\parallel {{f}_{N}}{{\parallel }_{2,\sigma }}.\end{eqnarray} \tag{ 59 }$$

With the previous discussion at hand, we are now in a position to apply corollary 8, with the spectral gap Λ and the constant $C=2\;{{s}^{1/4}}d$. Hence, we have the following bound on the log-Sobolev constants:

$$\begin{eqnarray}&&\alpha \geqslant \frac{\Lambda }{{\rm log} \left( {{C}^{4}}{{2}^{8}}{{e}^{2}} \right)}=\frac{\Lambda }{{\rm log} \left( {{d}^{4}}\;s{{2}^{12}}{{e}^{2}} \right)}\geqslant \frac{\Lambda }{{\rm log} ({{d}^{4}}\;s)+11}.\end{eqnarray} \tag{ 60 }$$

To complete the upper bound, recall that $\alpha \leqslant \Lambda$ for any reversible semigroup [14, 16]. □

One relevant application of this formalism for product channels is to the case of Davies maps associated to a graph state Hamiltonian [37]. Let $G(V,E)$ be a graph with vertices V and edges E. The associated graph Hamiltonian acting on $N\equiv |V|$ qubits is given by

$$\begin{eqnarray}&&H=\mathop{\sum }\limits_{j\in V}{{S}_{j}}=\mathop{\sum }\limits_{j\in V}{{X}_{j}}\mathop{\prod }\limits_{\{k,j\}\in E}{{Z}_{k}},\end{eqnarray} \tag{ 71 }$$

where X_j and Z_j denote the standard Pauli matrices acting on the jth qubit respectively. The Hamiltonian H is of Pauli stabilizer form, with the ${{S}_{j}}:={{X}_{j}}{{\prod }_{\{k,j\}\in E}}{{Z}_{k}}$ being the commuting stabilizer operators. These Hamiltonians and their unique ground states (graph states) have been studied extensively in the literature (see [37] and references therein). In particular, assuming the graph is a 2D square lattice, the unique ground state has been shown to be universal for measurement based quantum computation [38]. Note that at sufficiently low temperatures the thermal and ground states are close. In this sense, preparing low temperature thermal states equates to preparing the main resource for measurement based quantum computation.

The graph state Hamiltonian is equivalent to a product Hamiltonian ${{H}_{Z}}={{\sum }_{j}}{{Z}_{j}}$ under a unitary transformation U which is geometrically local on the graph G. In particular this means that the Gibbs state associated with H_Z expressed in the new basis, called graph state basis, is also of product form, i.e. $\sigma ={{\otimes }_{j}}{{\sigma }_{j}}$, with ${{\sigma }_{j}}={{(2{\rm cosh} (\beta ))}^{-1}}{\rm exp} (-\beta {{Z}_{j}})$.

The unitary transformation, mapping the computational basis to the graph state basis, can be succinctly described in terms of the graph $G(V,E)$ as

$$\begin{eqnarray}&&U=\left( \mathop{\prod }\limits_{\{k,j\}\in E}C{{Z}_{kj}} \right)\mathop{\otimes }\limits_{i\in V}{{H}_{i}}.\end{eqnarray} \tag{ 72 }$$

H_j is the Hadamard operator associated to site j and $C{{Z}_{kj}}=C{{Z}_{jk}}$ is the controlled phase gate on qubits j and k described by

$$\begin{eqnarray}C{{Z}_{kj}}=\left( \begin{array}{ccccccccccccccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{array} \right),\quad \qquad {{H}_{j}}=\frac{1}{\sqrt{2}}\left( \begin{array}{ccccccccccccccc} 1 & 1 \\ 1 & -1 \\ \end{array} \right).\end{eqnarray} \tag{ 73 }$$

Clearly, the CZ_kj are all commuting, since they are diagonal in the computational basis.

The transformation U yields an orthonormal basis of eigenstates $\left| b \right\rangle$, where $b\in {{\{0,1\}}^{N}}$ is a binary vector when applied to the original computational basis. Since ${{U}^{\dagger }}{{S}_{j}}U={{Z}_{j}}$, each element of the graph basis is an eigenvector of the stabilizer operators with eigenvalues $\pm 1$

$$\begin{eqnarray}&&{{S}_{j}}\left| b \right\rangle =U{{Z}_{j}}{{U}^{\dagger }}U\left| {{b}^{(z)}} \right\rangle ={{\left( -1 \right)}^{{{b}_{j}}}}\left| b \right\rangle .\end{eqnarray} \tag{ 74 }$$

Furthermore, local ${{U}^{\dagger }}{{Z}_{j}}U={{X}_{j}}$ and hence Z_j operators on the graph basis, act as X_j on the computational basis

$$\begin{eqnarray}&&{{Z}_{j}}\left| {{b}_{1}},...,{{b}_{j}},...,{{b}_{N}} \right\rangle =\left| {{b}_{1}},...,(1-{{b}_{j}}),...,{{b}_{N}} \right\rangle .\end{eqnarray} \tag{ 75 }$$

Theorem 10. Let H be a graph state Hamiltonian as in equation (71), and let ${{\mathcal{L}}_{\beta }}$ denote itʼs Davies generator which originates from the couplings ${{\{{{X}_{i}},{{Y}_{i}},{{Z}_{i}}\}}_{i\in V}}$ to a thermal environment, then the log-Sobolev constant is bounded by

$$\begin{eqnarray}&&\frac{G(2)+G(-2)}{2{\rm log} ({{{\rm e}}^{2\beta }}+1)+28}\leqslant \alpha ,\end{eqnarray} \tag{ 76 }$$

where G is the spectral density of the thermal bath (see equation (63)).

Proof. An important property of the Davies generators is that the Gibbs state is the unique stationary state if the Hamiltonian and the system-bath coupling operators have a trivial commutant [37]. By construction, the graph Hamiltonian has a trivial commutant with the set of $\{{{Z}_{j}}\}$, hence we only need to consider couplings to the bath with the local Z_j. Furthermore, since the stationary state is unique and determined through the KMS condition the gap and log-Sobolev constant will only increase by including further couplings. Indeed

$$\begin{eqnarray}&&{{\alpha }_{Z}}=\mathop{{\rm inf} }\limits_{f}\frac{{{\mathcal{E}}_{Z}}(f)}{{\rm Ent}(f)}\leqslant \mathop{{\rm inf} }\limits_{f}\frac{{{\mathcal{E}}_{Z}}(f)+{{\mathcal{E}}_{X}}(f)+{{\mathcal{E}}_{Y}}(f)}{{\rm Ent}(f)}=\alpha .\end{eqnarray} \tag{ 77 }$$

Working in the graph state basis, $({{S}_{j}},{{Z}_{j}})\to ({{Z}_{j}},{{X}_{j}})$ we may exploit the product nature of the evolution and express the coupling operator in the interaction picture in terms of two components, corresponding to the only two available Bohr frequencies

$$\begin{eqnarray}&&{{{\rm e}}^{-{\rm i}tH}}{{X}_{j}}{{{\rm e}}^{{\rm i}tH}}=\mathop{\prod }\limits_{k}{{{\rm e}}^{-{\rm i}t{{Z}_{k}}}}{{X}_{j}}\mathop{\prod }\limits_{k^{\prime} }{{{\rm e}}^{-{\rm i}t{{Z}_{k^{\prime} }}}}\end{eqnarray} \tag{ 78 }$$

$$\begin{eqnarray}&&={{{\rm e}}^{-{\rm i}t{{Z}_{j}}}}{{X}_{j}}{{{\rm e}}^{-{\rm i}t{{Z}_{j}}}}\end{eqnarray} \tag{ 79 }$$

$$\begin{eqnarray}&&={{{\rm e}}^{-2{\rm i}t}}\left| 0 \right\rangle \left\langle 1 \right|+{{{\rm e}}^{2{\rm i}t}}\left| 1 \right\rangle \left\langle 0 \right|.\end{eqnarray} \tag{ 80 }$$

Thus, working purely in the graph state basis, the Lindblad operators are local qubit raising ${{a}^{+}}:=\left| 1 \right\rangle \left\langle 0 \right|$ and lowering ${{a}^{-}}:=\left| 0 \right\rangle \left\langle 1 \right|$ operators. Hence, the semigroup yields a product channel given by

$$\begin{eqnarray}&&\mathcal{L}(f):=\mathop{\sum }\limits_{j\in V}{{\mathcal{L}}_{j}}(f)\end{eqnarray} \tag{ 81 }$$

$$\begin{eqnarray}&&{{\mathcal{L}}_{j}}(f):=G(2)\left( a_{j}^{+}fa_{j}^{-}-\frac{1}{2}\{a_{j}^{+}a_{j}^{-},f\} \right)+G(-2)\left( a_{j}^{-}fa_{j}^{+}-\frac{1}{2}\{a_{j}^{-}a_{j}^{+},f\} \right),\end{eqnarray} \tag{ 82 }$$

where $G(\pm 2)$ are rates derived from the corresponding spectral densities and satisfy $G(-2)/G(2)={{{\rm e}}^{-2\beta }}$ for a thermal bath. Each of the local Liouvillians $\mathcal{L}$_j has a gap ${{\lambda }_{Z}}=\frac{G(2)+G(-2)}{2}$ which is given by the decay rate of off-diagonal elements, and the smallest eigenvalue of the reduced steady state is $\parallel \sigma _{j}^{-1}\parallel ={{{\rm e}}^{2\beta }}+1$. Thus we can use theorem 9 to guarantee a log-Sobolev constant ${{\alpha }_{Z}}\geqslant \frac{G(2)+G(-2)}{2}\frac{1}{{\rm log} ({{{\rm e}}^{2\beta }}+1)+14}$. □

Comment: We may consider the implication of this result for preparing an approximation of a pure graph state. In order to do this we will impose

$$\begin{eqnarray}&&\parallel {{{\rm e}}^{t{{\mathcal{L}}_{\beta }}}}(\rho )-|0\rangle \langle 0|\;{{\parallel }_{1}}\leqslant \parallel {{{\rm e}}^{t\mathcal{L}}}(\rho )-\sigma {{\parallel }_{1}}+\parallel \sigma -|0\rangle \langle 0|\;{{\parallel }_{1}}\leqslant \epsilon /2+\epsilon /2.\end{eqnarray} \tag{ 83 }$$

For normalized states, we have $\parallel \sigma -|0\rangle \langle 0|\;{{\parallel }_{1}}=2-2\langle 0|\sigma |0\rangle$. Using that both the N particle thermal state σ, and $|0\rangle \langle 0|$ are of product form in the graph state basis allows simplifying

$$\begin{eqnarray}&&\langle 0|\sigma |0\rangle =\mathop{\prod }\limits_{j\in V}{\rm tr}\;\left[ {{\langle 0{{|}_{j}}\sigma |0\rangle }_{j}} \right]={{\left( 1-{{({{{\rm e}}^{2\beta }}+1)}^{-1}} \right)}^{N}}\geqslant 1-N{{({{{\rm e}}^{2\beta }}+1)}^{-1}},\end{eqnarray} \tag{ 84 }$$

where the last inequality can be seen as a union bound. Hence, to ensure $\parallel \sigma -|0\rangle \langle 0|\;{{\parallel }_{1}}\leqslant \epsilon /2$, it is sufficient to choose the inverse temperature β such that $\beta \geqslant {\rm log} (4N/\epsilon )/2$. Furthermore, according to equation (10) we may guarantee that $\parallel {{{\rm e}}^{t{{\mathcal{L}}_{\beta }}}}(\rho )-\sigma {{\parallel }_{1}}\leqslant \epsilon /2$ for $\sqrt{2{\rm log} (\parallel {{\sigma }^{-1}}\parallel )}{{{\rm e}}^{-t\alpha }}\leqslant \epsilon /2$. Since $\parallel {{\sigma }^{-1}}\parallel ={{({{{\rm e}}^{2\beta }}+1)}^{N}}\approx {{\left( 4N/\epsilon \right)}^{N}}$ for $\beta \geqslant {\rm log} (4N/\epsilon )/2$, we evaluate this condition to

$$\begin{eqnarray}&&{{{\rm e}}^{2t\alpha }}\geqslant \frac{8N{\rm log} \left( 4N/\epsilon \right)}{{{\epsilon }^{2}}}.\end{eqnarray} \tag{ 85 }$$

Taking the logarithm and making the low temperature approximation ${{\alpha }^{-1}}\approx 4\beta =2{\rm log} \left( 4N/\epsilon \right)$ in this condition, we may extract the time

$$\begin{eqnarray}&&{{t}_{\epsilon }}\approx {\rm log} \left( 4N/\epsilon \right){\rm log} \left( \frac{8N{\rm log} \left( 4N/\epsilon \right)}{{{\epsilon }^{2}}} \right)=O\left( {{{\rm log} }^{2}}\left( N/\epsilon \right) \right)\end{eqnarray} \tag{ 86 }$$

such that for $t\gt {{t}_{\epsilon }}$ the Davies thermalizing evolution can be guaranteed to produce -approximate ground states assuming the temperature has been appropriately chosen, i.e. $\beta =O\left( {\rm log} (N/\epsilon ) \right)$.

As our main example, we now consider a class of semigroups: free fermion systems coupled linearly to a thermal bath. Fermionic hypercontractivity has been considered previously in the context of the Schrödinger semigroup [9, 10]. That treatment is only very loosely related to the analysis below. Free-fermionic systems are endowed with a natural block diagonal structure, which makes them particularly well suited for the strategy outlined in section 2.1.

We consider the Davies generator for a system described by a quadratic fermion Hamiltonian H, which can always be diagonalized as

$$\begin{eqnarray}&&{{H}_{{\rm S}}}=\mathop{\sum }\limits_{k=1}^{N}{{\nu }_{k}}d_{k}^{\dagger }{{d}_{k}}.\end{eqnarray} \tag{ 87 }$$

The N-modes each with energy ${{\nu }_{k}}\geqslant 0$, are taken to obey fermionic anti-commutation relations so that $\{d_{k}^{\dagger },{{d}_{j}}\}={{\delta }_{kj}}$ and $\{{{d}_{k}},{{d}_{j}}\}=0$.

We assume that this Hamiltonian couples weakly to a heat bath through operators linear in the diagonal fermionic operators

$$\begin{eqnarray}&&{{S}_{\alpha }}=\mathop{\sum }\limits_{k=1}^{N}{{s}_{\alpha ,k}}{{d}_{k}}+s_{\alpha ,k}^{*}d_{k}^{\dagger }.\end{eqnarray} \tag{ 88 }$$

By physical considerations, the full interaction Hamiltonian should be fermion parity-conserving and Hermitian. In contrast to the usual form of equation (61), the interaction Hamiltonian V should be of the form

$$\begin{eqnarray}&&V=\mathop{\sum }\limits_{\alpha }i{{S}_{\alpha }}{{B}_{\alpha }},\end{eqnarray} \tag{ 89 }$$

where ${{B}_{\alpha }}$ are operators supported on the bath algebra which are odd in fermionic operators. The anti-commutation of ${{S}_{\alpha }}$ and ${{B}_{\alpha }}$ guarantees the Hermiticity of each term and makes it necessary to abandon tensor product notation when considering joint operators of system and bath. This same anti-commutation relation will compensate the i² factor accompanying second order terms in the interaction Hamiltonian V which give rise to dissipation in the Davies approximation [35, 39].

Assuming that the coupling is weak with respect to the relevant Bohr frequencies in the system, it remains legitimate to assume the usual 'secular approximation' by which the Lindblad operators can be obtained as the Fourier components from the ${{S}_{\alpha }}$ in the interaction picture

$$\begin{eqnarray}&&{{{\rm e}}^{{\rm i}Ht}}{{S}_{\alpha }}{{{\rm e}}^{-{\rm i}Ht}}=\mathop{\sum }\limits_{\omega }{{S}_{\alpha }}(\omega ){{{\rm e}}^{{\rm i}\omega t}}.\end{eqnarray} \tag{ 90 }$$

Note that, the time evolution of the fermionic mode operators ${{d}_{k}}(t)={\rm exp} ({\rm i}Ht){{d}_{k}}{\rm exp} (-{\rm i}Ht)$ can be obtained easily from the Heisenberg equations of motion, i.e. ${{\partial }_{t}}{{d}_{k}}={\rm i}[H,{{d}_{k}}]$ which can be integrated to yield ${{d}_{k}}(t)={\rm exp} (-{\rm i}{{\nu }_{k}}t){{d}_{k}}$. With this at hand we can immediately state the time evolution of the ${{S}_{\alpha }}$ and obtain

$$\begin{eqnarray}&&{{{\rm e}}^{{\rm i}Ht}}{{S}_{\alpha }}{{{\rm e}}^{-{\rm i}Ht}}=\mathop{\sum }\limits_{k=1}^{N}{{s}_{\alpha ,k}}{{d}_{k}}{{{\rm e}}^{-{\rm i}{{\nu }_{k}}t}}+s_{\alpha ,k}^{*}d_{k}^{\dagger }{{{\rm e}}^{{\rm i}{{\nu }_{k}}t}}.\end{eqnarray} \tag{ 91 }$$

Let us consider now two different cases.

• a.  
  Non-degenerate frequencies: Let us first assume that all ν_k are distinct and positive, i.e. we have ${{\nu }_{N}}\gt \ldots \gt {{\nu }_{1}}\gt 0$. If this is the case, the Lindblad operators can be read off from equation (91) directly. We then have that for every Bohr frequency $\omega ={{\nu }_{k}}\gt 0$ the Lindblad operator is given by

  $$\begin{eqnarray}&&{{S}_{\alpha }}({{\nu }_{k}})={{s}_{\alpha ,k}}{{d}_{k}}.\end{eqnarray} \tag{ 92 }$$

  For this case it is now straightforward to state the full Davies generator

  $$\begin{eqnarray}&&{{\partial }_{t}}f=\mathop{\sum }\limits_{k}{{\mathcal{L}}_{k}}(f),\end{eqnarray} \tag{ 93 }$$

  where we have defined the constant

  $$\begin{eqnarray}&&{{g}_{k}}=\mathop{\sum }\limits_{\alpha }{{G}^{\alpha }}({{\nu }_{k}})|{{s}_{\alpha ,k}}{{|}^{2}},\end{eqnarray} \tag{ 94 }$$

  so that the Liouville operators can be written as

  $$\begin{eqnarray}&&{{\mathcal{L}}_{k}}(f)={{g}_{k}}\left( d_{k}^{\dagger }f{{d}_{k}}-\frac{1}{2}\{d_{k}^{\dagger }{{d}_{k}},f\} \right)+{{{\rm e}}^{-\beta {{\nu }_{k}}}}{{g}_{k}}\left( {{d}_{k}}fd_{k}^{\dagger }-\frac{1}{2}\{{{d}_{k}}d_{k}^{\dagger },f\} \right).\end{eqnarray} \tag{ 95 }$$
• b.  
  Degenerate and zero-mode frequencies: The situation is slightly more complicated, when we allow for degenerate frequencies ν_k that can also be zero. Let us first investigate what happens, when some frequencies ${{\nu }_{k}}=\cdots ={{\nu }_{l}}=\xi$ are degenerate. We then have that the Lindblad operator has to be of the form

  $$\begin{eqnarray}&&{{S}_{\alpha }}(\xi )=\mathop{\sum }\limits_{{{\nu }_{l}}=\xi }{{s}_{\alpha ,l}}{{d}_{l}}.\end{eqnarray} \tag{ 96 }$$

  The term associated with the Bohr frequency ξ is then of the form

  $$\begin{eqnarray}&&{{\mathcal{L}}_{\xi }}=\mathop{\sum }\limits_{{{\nu }_{k}},{{\nu }_{l}}=\xi }{{\chi }_{k,l}}\left( d_{k}^{\dagger }f{{d}_{l}}-\frac{1}{2}\{d_{k}^{\dagger }{{d}_{l}},f\} \right)+{{{\rm e}}^{-\xi \beta }}{{\chi }_{l,k}}\left( {{d}_{k}}fd_{l}^{\dagger }-\frac{1}{2}\{{{d}_{k}}d_{l}^{\dagger },f\} \right),\end{eqnarray} \tag{ 97 }$$

  where we have defined the Hermitian matrix χ through the entries

  $$\begin{eqnarray}&&{{\chi }_{k,l}}=\mathop{\sum }\limits_{\alpha }{{G}^{\alpha }}(\xi )s_{\alpha ,k}^{*}{{s}_{\alpha ,l}}.\end{eqnarray} \tag{ 98 }$$

  Let U be a unitary transformation diagonalizing χ so that

  $$\begin{eqnarray}&&{{\chi }_{k,l}}=\mathop{\sum }\limits_{a}{{\lambda }_{a}}{{[U]}_{k,a}}{{[{{U}^{\dagger }}]}_{a,l}},\end{eqnarray} \tag{ 99 }$$

  and define $\tilde{d}_{a}^{\dagger }={{\sum }_{k}}d_{k}^{\dagger }{{[U]}_{k,a}}$ as well as ${{\tilde{d}}_{a}}={{\sum }_{k}}{{d}_{k}}{{[{{U}^{\dagger }}]}_{a,k}}$, so that the anti-commutation relations are preserved. With this transformation we may write

  $$\begin{eqnarray}&&{{\mathcal{L}}_{\xi }}=\mathop{\sum }\limits_{a\in \xi }{{\tilde{\mathcal{L}}}_{a}}(f),\end{eqnarray} \tag{ 100 }$$

  where now

  $$\begin{eqnarray}&&{{\tilde{\mathcal{L}}}_{a}}(f)={{\lambda }_{a}}\left( \tilde{d}_{a}^{\dagger }f{{\tilde{d}}_{a}}-\frac{1}{2}\{\tilde{d}_{a}^{\dagger }{{\tilde{d}}_{a}},f\} \right)+{{{\rm e}}^{-\xi \beta }}{{\lambda }_{a}}\left( {{\tilde{d}}_{a}}f\tilde{d}_{a}^{\dagger }-\frac{1}{2}\{{{\tilde{d}}_{a}}\tilde{d}_{a}^{\dagger },f\} \right).\end{eqnarray} \tag{ 101 }$$

  Let us now turn to the case where we have zero modes ${{\nu }_{k}}=0$. In this case we have a Lindblad operator of the following form

  $$\begin{eqnarray}&&{{S}_{\alpha }}(0)=\mathop{\sum }\limits_{{{\nu }_{k}}=0}{{s}_{\alpha ,k}}{{d}_{k}}+s_{\alpha ,k}^{*}d_{k}^{\dagger }=\mathop{\sum }\limits_{{{\nu }_{k}}=0}\frac{s_{\alpha ,k}^{*}+{{s}_{\alpha ,k}}}{2}{{w}_{2k-1}}-\frac{s_{\alpha ,k}^{*}-{{s}_{\alpha ,k}}}{2i}{{w}_{2k}}.\end{eqnarray} \tag{ 102 }$$

  Consider the Majorana modes w_j with ${{d}_{k}}=\frac{1}{2}({{w}_{2k-1}}+{\rm i}{{w}_{2k}})$, so that the anti-commutation relations $\{{{w}_{k}},{{w}_{j}}\}=2{{\delta }_{kj}}$ hold. These operators are Hermitian ${{w}_{k}}=w_{k}^{\dagger }$. In these modes, the Liouvillian can be expressed as

  $$\begin{eqnarray}&&{{\mathcal{L}}_{0}}(f)=\mathop{\sum }\limits_{{{\nu }_{k}},{{\nu }_{l}}=0}{{\chi }_{kl}}\left( {{w}_{k}}f{{w}_{l}}-\frac{1}{2}\{{{w}_{k}}{{w}_{l}},f\} \right),\end{eqnarray} \tag{ 103 }$$

  where the χ_kl are the entries of a real, symmetric matrix χ that can be diagonalized by an orthogonal transformation O. In addition, there must be an even number of Majorana fermions, so they may be arbitrarily paired to define zero energy fermionic modes. By a similar argument as above we can introduce new Majorana modes ${{\tilde{w}}_{2k}},{{\tilde{w}}_{2k-1}}$ that are associated with the pair of eigenvalues $\lambda {{^{\prime} }_{k}}/2\geqslant {{\lambda }_{k}}/2\geqslant 0$ of the matrix χ and we can write

  $$\begin{eqnarray}&&{{\mathcal{L}}_{0}}(f)=\mathop{\sum }\limits_{k:{{\nu }_{k}}=0}\frac{{{\lambda }_{k}}}{2}\left( {{\tilde{w}}_{k}}f{{\tilde{w}}_{k}}-f \right)+\frac{\lambda {{^{\prime} }_{k}}}{2}\left( {{\tilde{w}}_{2k-1}}f{{\tilde{w}}_{2k-1}}-f \right).\end{eqnarray} \tag{ 104 }$$

Note that these transformations do not modify the form of the Hamiltonian H. We summarize the preceding discussion in the following Proposition, which allows us to only consider a canonical form of fermionic Davies generators that couple linearly to the bath. Moreover, since every Davies generator satisfies the KMS-condition, and is reversible with respect to the Gibbs state of the system Hamiltonian, we can immediately infer the form of the fixed point of the generator.

Proposition 11. Let $H=\sum _{k=1}^{N}{{\nu }_{k}}d_{k}^{\dagger }{{d}_{k}}$ be a free fermionic Hamiltonian that couples linearly via ${{S}_{\alpha }}=\sum _{k=1}^{N}{{s}_{\alpha ,k}}{{d}_{k}}+s_{\alpha ,k}^{*}d_{k}^{\dagger }$ to a thermal bath at inverse temperature β. Then the modes d_k may be chosen such that the dissipative Davies generator is given by

$$\begin{eqnarray}&&{{\mathcal{L}}_{\beta }}(f)=\mathop{\sum }\limits_{k=1}^{N}{{\mathcal{L}}_{k}}(f),\end{eqnarray} \tag{ 105 }$$

where we have defined the Majorana mode operators so that ${{d}_{k}}=\frac{1}{2}({{w}_{2k-1}}+{\rm i}{{w}_{2k}})$, and

$$\begin{eqnarray}&&{{\mathcal{L}}_{k}}(f)\mathop{:=}\limits_{{{\nu }_{k}}=0}\frac{{{\lambda }_{k}}}{2}\left( {{w}_{2k}}f{{w}_{2k}}-f \right)+\frac{\lambda {{^{\prime} }_{k}}}{2}\left( {{w}_{2k-1}}f{{w}_{2k-1}}-f \right),\end{eqnarray} \tag{ 106 }$$

$$\begin{eqnarray}&&{{\mathcal{L}}_{k}}(f)\mathop{:=}\limits_{{{\nu }_{k}}\ne 0}{{\lambda }_{k}}\left( d_{k}^{\dagger }f{{d}_{k}}-\frac{1}{2}\{d_{k}^{\dagger }{{d}_{k}},f\} \right)+{{\lambda }_{k}}{{{\rm e}}^{-\beta {{\nu }_{k}}}}\left( {{d}_{k}}fd_{k}^{\dagger }-\frac{1}{2}\{{{d}_{k}}d_{k}^{\dagger },f\} \right),\end{eqnarray} \tag{ 107 }$$

with $\lambda {{^{\prime} }_{k}}\geqslant {{\lambda }_{k}}\geqslant 0$. Furthermore, the thermal state

$$\begin{eqnarray}&&\sigma =\mathop{\prod }\limits_{k=1}^{N}\frac{1}{2{\rm cosh} (\frac{1}{2}\beta {{\nu }_{k}})}{{{\rm e}}^{-{\rm i}\frac{\beta }{2}{{\nu }_{k}}{{w}_{2k-1}}{{w}_{2k}}}}=\frac{{\rm exp} (-\beta H)}{{\rm tr}\;\left[ {\rm exp} (-\beta H) \right]}.\end{eqnarray} \tag{ 108 }$$

is the unique steady state of the generator in equation (105).

3.2.1. Block decomposition of the canonical fermionic Davies generator

Any Liouvillian with Lindblad operators that are linear in the fermionic creation and annihilation operators can be brought to Jordan normal form, with the Jordan blocks corresponding to dynamical excitations on an extended (doubled) Fock space [40]. Given that we consider fermionic Davies generators, reversibility guarantees that the Liouvillian can be diagonalized in the doubled Fock space. Hence, these operators have a very natural block decomposition resulting from the diagonalization of the Liouvillian. In order to prove hypercontractivity for linear fermionic Davies Generators, we can work with a coarser decomposition as long as it meets the requirements of lemma 7. Before we proceed to the block decomposition we will state some necessary definitions.

Definition 12. Let $k\in \{1,\ldots ,N\}$, then we define two sets of operators, the even mode operators $\{{{f}_{k}}\}$ and the odd mode operators $\{{{\tilde{f}}_{k}}\}$.

• Let us by abuse of notation, define the operators ${{f}_{k}}(b)={{f}_{k}}({{b}_{2k-1}},{{b}_{2k}})$, so that
  $$\begin{eqnarray}&&{{f}_{k}}(0,0)=\mathbb{1},\;\;\;\;\;\;{{f}_{k}}(1,1)={{w}_{2k-1}}{{w}_{2k}}{\rm exp} \left( \frac{{\rm i}{{\nu }_{k}}\beta }{2}{{w}_{2k-1}}{{w}_{2k}} \right),\end{eqnarray} \tag{ 109 }$$
  $$\begin{eqnarray}&&{{f}_{k}}(1,0)=\sqrt{{\rm cosh} \left( \frac{\beta {{\nu }_{k}}}{2} \right)}{{w}_{2k-1}},\;\;\;\;\;\;{{f}_{k}}(0,1)=\sqrt{{\rm cosh} \left( \frac{\beta {{\nu }_{k}}}{2} \right)}{{w}_{2k}}.\end{eqnarray} \tag{ 110 }$$
  Note that for ${{\nu }_{k}}=0$ this reduces to ${{f}_{k}}({{b}_{2k-1}},{{b}_{2k}})=w_{2k-1}^{{{b}_{2k-1}}}w_{2k}^{{{b}_{2k}}}$.
• The operators $\{{{\tilde{f}}_{k}}\}$ are related to $\{{{f}_{k}}\}$ through the identification
  $$\begin{eqnarray}&&{{\tilde{f}}_{k}}(b)={{\tilde{f}}_{k}}({{b}_{2k-1}},{{b}_{2k}})={{w}_{2k-1}}{{w}_{2k}}{{f}_{k}}({{b}_{2k-1}},{{b}_{2k}}).\end{eqnarray} \tag{ 111 }$$

Moreover, given a string $b=({{b}_{1}},\ldots ,{{b}_{2N}})\in {{\{0,1\}}^{2N}}$, we define

$$\begin{eqnarray}&&f(b)=\left\{ \begin{array}{ccccccccccccccc} \mathop{\prod }\limits_{k=1}^{N}{{f}_{k}}(b)\ \ \ {\rm if}\ \ \ |b|\ \ {\rm is}\ {\rm even}, \\ \mathop{\prod }\limits_{k=1}^{N}{{\tilde{f}}_{k}}(b)\ \ \ {\rm if}\ \ \ |b|\ \ {\rm is}\ {\rm odd}, \\ \end{array} \right.\end{eqnarray} \tag{ 112 }$$

with $|b|={{\sum }_{k}}{{b}_{k}}$. Furthermore we define the following sets of eigen-operators, for $n=0,\ldots ,2N$

$$\begin{eqnarray}&&{{\mathcal{E}}_{n}}=\left\{ f(b)|\ \ b\in {{\{0,1\}}^{2N}}\ \ {\rm and}\ \ |b|=n \right\}.\end{eqnarray} \tag{ 113 }$$

These definitions form the basis of the block decomposition for the fermionic Davies generators. The blocks will be defined in terms of the span of the product of these mode operators. In fact, we will see that the f(b) are eigenvectors of the Liouvillian. Moreover, note the distinction between an even and an odd number of excitations $|b|$ . These eigen-operators induce a decomposition that will meet the requirements of lemma 7.

Lemma 13. Let ${{\mathcal{L}}_{\beta }}$ be a linear fermionic Davies generator in canonical form as in proposition 11, then the semigroup ${{T}_{t}}={\rm exp} (t{{\mathcal{L}}_{\beta }})$ preserves the following block decomposition.

• 1.  
  We have a block decomposition of the form

  $$\begin{eqnarray}&&\mathcal{B}=\mathop{\oplus }\limits_{n=0}^{2N}{{\mathcal{B}}_{n}}.\end{eqnarray} \tag{ 114 }$$

  The blocks are defined as ${{\mathcal{B}}_{n}}={\rm span}\ \ {{\mathcal{E}}_{n}}$, where $\mathcal{E}$_n was introduced in definition 12.
• 2.  
  The spectrum of ${{\mathcal{L}}_{\beta }}=\oplus _{n=0}^{2N}{{\mathcal{L}}_{\beta }}{{|}_{{{\mathcal{B}}_{n}}}}$ is contained in the interval

  $$\begin{eqnarray}&&{\rm spec}\left( {{\mathcal{L}}_{\beta }}{{|}_{{{\mathcal{B}}_{n}}}} \right)\in \left( -\infty ,-\Lambda n \right],\end{eqnarray} \tag{ 115 }$$

  where $\Lambda ={\rm min} _{j=1}^{N}{{\Lambda }_{j}}$, and we define ${{\Lambda }_{j}}={{\lambda }_{j}}(\frac{1+{{{\rm e}}^{-\beta {{\nu }_{j}}}}}{2}).$

Proof. Recall that the Davies generator is of the form ${{\mathcal{L}}_{\beta }}=\sum _{j=1}^{N}{{\mathcal{L}}_{j}}$. Before we proceed we present a decomposition of $\mathcal{L}$_j into w-Majorana fermions. Together with ${{d}_{j}}=\frac{1}{2}\left( {{w}_{2j-1}}+i{{w}_{2j}} \right)$ one can verify that for ${{\nu }_{j}}\ne 0$ we have

$$\begin{eqnarray}\begin{array}{rcl} {{\mathcal{L}}_{j}}(f) & = & {{\lambda }_{j}}\frac{1+{{{\rm e}}^{-\beta {{\nu }_{k}}}}}{4}\left( {{w}_{2j-1}}f{{w}_{2j-1}}+{{w}_{2j}}f{{w}_{2j}}-2f \right) \\ {} & {} & +i{{\lambda }_{j}}\frac{1-{{{\rm e}}^{-\beta {{\nu }_{k}}}}}{4}\left( {{w}_{2j-1}}f{{w}_{2j}}-{{w}_{2j}}f{{w}_{2j-1}}-\{{{w}_{2j-1}}{{w}_{2j}},f\} \right). \\ \end{array}\end{eqnarray} \tag{ 116 }$$

Whereas for ${{\nu }_{k}}=0$ we already had that by definition

$$\begin{eqnarray}&&{{\mathcal{L}}_{k}}(f)=\frac{{{\lambda }_{k}}}{2}\left( {{w}_{2k}}f{{w}_{2k}}-f \right)+\frac{\lambda {{^{\prime} }_{k}}}{2}\left( {{w}_{2k-1}}f{{w}_{2k-1}}-f \right).\end{eqnarray} \tag{ 117 }$$

By making use of the anti-commutation relations of the Majorana fermions, $\{{{w}_{k}},{{w}_{j}}\}=2{{\delta }_{kj}}$, we have for any $f(b)\in {{\mathcal{E}}_{n}}$, that

$$\begin{eqnarray}&&{{\mathcal{L}}_{\beta }}(f(b))=-\left( \mathop{\sum }\limits_{k:{{\nu }_{k}}=0}{{\lambda }_{k}}{{b}_{2k-1}}+\lambda {{^{\prime} }_{k}}{{b}_{2k}}+\mathop{\sum }\limits_{k:{{\nu }_{k}}\ne 0}{{\lambda }_{k}}\frac{1+{{{\rm e}}^{-{{\nu }_{k}}\beta }}}{2}({{b}_{2k-1}}+{{b}_{2k}}) \right)\;f(b).\end{eqnarray} \tag{ 118 }$$

Hence, all f(b) are in fact eigenvectors of ${{\mathcal{L}}_{\beta }}$. For a bit string b with $|b|=n$, choosing $\Lambda ={{{\rm min} }_{j}}{{\Lambda }_{j}}$, we can always find the upper bound

$$\begin{eqnarray}&&-\;\left( \mathop{\sum }\limits_{{{\nu }_{k}}=0}{{\lambda }_{k}}{{b}_{2k}}+\lambda {{^{\prime} }_{k}}{{b}_{2k-1}}+\mathop{\sum }\limits_{{{\nu }_{k}}\ne 0}{{\lambda }_{k}}\frac{1+{{{\rm e}}^{-{{\nu }_{k}}\beta }}}{2}({{b}_{2k-1}}+{{b}_{2k}}) \right)\leqslant -\Lambda |b|=-\Lambda n.\end{eqnarray} \tag{ 119 }$$

Identifying n with the number of set bits in b, we see that subspaces $\mathcal{B}$_n are naturally preserved. A simple counting argument shows that the full operator space $\mathcal{B}(\mathcal{H})$ is spanned by the ${{2}^{2N}}$ eigenvectors. Moreover we can directly estimate the bound on the spectrum as given in the Lemma. □

3.2.2. The log-Sobolev constant for canonical fermionic Davies generators

Two criteria that are required by lemma 7 to ensure hypercontractivity are already met by the lemma 13 for linear fermionic Davies generators. To obtain an estimate for the log-Sobolev constant α, we need to prove a norm bound for all ${{f}_{n}}\in {{\mathcal{B}}_{n}}$ for the $2\to 4$ norm. That is we need a bound of the form $\parallel {{f}_{n}}{{\parallel }_{2\to 4,\sigma }}\leqslant {{C}^{n}}$ for some constant C and any $n\in \mathbb{N}$. Before we proceed to prove such a bound, we state a proposition which will facilitate the derivation.

Proposition 14. Let

$$\begin{eqnarray}&&Q({{v}^{(1)}},{{v}^{(2)}},{{v}^{(3)}},{{v}^{(4)}})={\rm tr}\;\left[ {{\sigma }^{1/4}}\ {{v}^{(1)\dagger }}\ {{\sigma }^{1/4}}\ {{v}^{(2)}}\ {{\sigma }^{1/4}}\ {{v}^{(3)\dagger }}\ {{\sigma }^{1/4}}\ {{v}^{(4)}} \right],\end{eqnarray} \tag{ 120 }$$

where the ${{v}^{(i)}}\in {{\mathcal{E}}_{n}}$ are taken as a shorthand notation for ${{v}^{(i)}}:=f({{b}^{(i)}})$, for some $|{{b}^{(i)}}|=n$.

• 1.  
  The following bound on the supremum of ${{Q}_{\beta }}$ holds

  $$\begin{eqnarray}&&\mathop{{\rm max} }\limits_{\{{{v}^{(i)}}\}}|Q({{v}^{(1)}},{{v}^{(2)}},{{v}^{(3)}},{{v}^{(4)}})|\leqslant {{{\rm e}}^{n\beta \nu }},\end{eqnarray} \tag{ 121 }$$

  where $\nu ={{{\rm max} }_{j}}|{{\nu }_{j}}|$ is the largest single particle frequency of H.
• 2.  
  Furthermore $|Q({{v}^{(1)}},{{v}^{(2)}},{{v}^{(3)}},{{v}^{(4)}})|\gt 0$ only when, ${{b}^{(1)}}\oplus {{b}^{(2)}}\oplus {{b}^{(3)}}\oplus {{b}^{(4)}}\in {{\{(00),(11)\}}^{N}}$, where $\oplus$ denotes addition modulo 2.

Proof. For ease of notation, we define

$$\begin{eqnarray}&&{{\eta }_{k}}={{\left[ 2{\rm cosh} \left( \frac{\beta {{\nu }_{k}}}{2} \right) \right]}^{-1}}{\rm exp} \left( -\frac{{\rm i}\beta {{\nu }_{k}}}{2}{{w}_{2k-1}}{{w}_{2k}} \right),\end{eqnarray} \tag{ 122 }$$

so that $\sigma =\Pi _{k=1}^{N}{{\eta }_{k}}$. We have that for any power of $\alpha \in \mathbb{R}$, $[{{\tilde{f}}_{k}}(a,b),\eta _{j}^{\alpha }]\;=[{{f}_{k}}(a,b),\eta _{j}^{\alpha }]=0,$ whenever $k\ne j$. This follows because η_k is quadratic in the Majorana modes. Furthermore, we naturally have that all Majorana fermions anti-commute at different sites. Therefore, the absolute value of $Q({{v}^{(1)}},{{v}^{(2)}},{{v}^{(3)}},{{v}^{(4)}})$ can be written as a product of partial traces each pertaining to mode k. We consider the even mode case (i.e. $|b|$ even) first

$$\begin{eqnarray}&&\begin{array}{l} \left| Q({{v}^{(1)}},{{v}^{(2)}},{{v}^{(3)}},{{v}^{(4)}}) \right| \\ \quad =\left| {\rm tr}\;\left[ {{\sigma }^{1/4}}f({{b}^{(1)}}){{\sigma }^{1/4}}f({{b}^{(2)}}){{\sigma }^{1/4}}f({{b}^{(3)}}){{\sigma }^{1/4}}f({{b}^{(4)}}) \right] \right| \\ \quad =\mathop{\prod }\limits_{k=1}^{N}\left| {\rm t}{{{\rm r}}_{k}}\;\left[ \eta _{k}^{1/4}f_{k}^{\dagger }{{({{b}^{(1)}})}^{\dagger }}\eta _{k}^{1/4}{{f}_{k}}({{b}^{(2)}})\eta _{k}^{1/4}f_{k}^{\dagger }({{b}^{(3)}})\eta _{k}^{1/4}{{f}_{k}}({{b}^{(4)}}) \right] \right|. \\ \end{array}\end{eqnarray} \tag{ 123 }$$

Moreover, observe that at each side $[{{w}_{2k-1}}{{w}_{2k}},\eta _{k}^{\alpha }]=0$ and since ${{\tilde{f}}_{k}}(a,b)\;={{w}_{2k-1}}{{w}_{2k}}{{f}_{k}}(a,b)$, we get that the same factorization argument also holds for the odd mode case

$$\begin{eqnarray}&&\begin{array}{l} {\rm t}{{{\rm r}}_{k}}\;\left[ \eta _{k}^{1/4}\tilde{f}_{k}^{\dagger }{{({{b}^{(1)}})}^{\dagger }}\eta _{k}^{1/4}{{\tilde{f}}_{k}}({{b}^{(2)}})\eta _{k}^{1/4}\tilde{f}_{k}^{\dagger }({{b}^{(3)}})\eta _{k}^{1/4}{{\tilde{f}}_{k}}({{b}^{(4)}}) \right] \\ \quad ={\rm t}{{{\rm r}}_{k}}\;\left[ \eta _{k}^{1/4}f_{k}^{\dagger }{{({{b}^{(1)}})}^{\dagger }}\eta _{k}^{1/4}{{f}_{k}}({{b}^{(2)}})\eta _{k}^{1/4}f_{k}^{\dagger }({{b}^{(3)}})\eta _{k}^{1/4}{{f}_{k}}({{b}^{(4)}}) \right], \\ \end{array}\end{eqnarray} \tag{ 124 }$$

since the factors ${{w}_{2k-1}}{{w}_{2k}}$ cancel. It therefore suffices to discuss the even mode case only, since the Q functions will behave in the same way for the odd mode case.

• 1.  
  Each partial trace ${\rm t}{{{\rm r}}_{k}}\;\left[ \cdot \right]$ appearing as a factor has ${{4}^{4}}$ possible input values of ${{\{(b_{2k-1}^{(i)},b_{2k}^{(i)})\}}_{i=1\ldots 4}}$. By interpreting the trace as a Hilbert–Schmidt inner product of operator in different forms, we may conclude that all the tuples $(b_{2k-1}^{(i)},b_{2k}^{(i)})$ should be equal for all i in order to maximize the trace. Form the remaining four candidates, it can be verified directly that the largest value is always reached for $(b_{2k-1}^{(i)},b_{2k}^{(i)})=(1,1)$ and is given by

  $$\begin{eqnarray}&&\begin{array}{l} {\rm tr}\;\left[ \eta _{k}^{1/4}f_{k}^{\dagger }{{(1,1)}^{\dagger }}\eta _{k}^{1/4}{{f}_{k}}(1,1)\eta _{k}^{1/4}f_{k}^{\dagger }(1,1)\eta _{k}^{1/4}{{f}_{k}}(1,1) \right] \\ \quad =\;\frac{{\rm cosh} \left( \frac{3}{2}\beta {{\nu }_{k}} \right)}{{\rm cosh} \left( \frac{1}{2}\beta {{\nu }_{k}} \right)}\leqslant {\rm exp} \left( \beta |{{\nu }_{k}}| \right). \\ \end{array}\end{eqnarray} \tag{ 125 }$$

  Choosing $\nu ={{{\rm max} }_{k}}|{{\nu }_{k}}|$, note that each factor in equation (123) can contribute at most ${\rm exp} \left( \beta \nu \right)$. That is for $|{{b}^{(i)}}|=n$ excitations, we always find the bound $|Q({{v}^{(1)}},{{v}^{(2)}},{{v}^{(3)}},{{v}^{(4)}})|\leqslant {\rm exp} {{\left( \beta \nu \right)}^{n}}$ as stated above.
• 2.  
  The only contributions that can remain in equation (123), are those from factors which behave as ${\rm t}{{{\rm r}}_{k}}\;[{{w}_{2k-1}}{{w}_{2k}}\eta _{k}^{\alpha }]$ and ${\rm t}{{{\rm r}}_{k}}\;[\eta _{k}^{\alpha }]$ itself for some power $\alpha \in \mathbb{R}$, since the partial trace over any product of Majorana modes not proportional to the identity vanishes. Then, we have to require that locally at each mode k, an even number of Majorana modes get paired. This implies the condition in the proposition above.

□

We can now state the essential norm bound for each subspace of n excitations $\mathcal{B}$_n.

Lemma 15. Let ${{f}_{n}}\in {{\mathcal{B}}_{n}}$ denote an element with n-fermionic excitations of the canonical fermionic Davies generator, then the following norm bound holds

$$\begin{eqnarray}&&\parallel {{f}_{n}}\parallel _{4,\sigma }^{4}\leqslant {{2}^{8n}}{\rm exp} \left( n\beta \nu \right)\parallel {{f}_{n}}\parallel _{2,\sigma }^{4},\end{eqnarray} \tag{ 126 }$$

where $\nu ={{{\rm max} }_{k}}|{{\nu }_{k}}|$ denotes the maximal single particle frequency of H.

Proof. The proof follows the ideas from [6]. Any ${{f}_{n}}\in {{\mathcal{B}}_{n}}$ can be written as

$$\begin{eqnarray}&&{{f}_{n}}=\mathop{\sum }\limits_{|b|=n}\alpha (b)f(b).\end{eqnarray} \tag{ 127 }$$

Note that the operators in definition 12 were chosen to be orthonormal with respect to the weighted inner product ${\rm tr}\;[{{\sigma }^{1/2}}{{f}_{n}}({{b}^{(1)}}){{\sigma }^{1/2}}{{f}_{n}}({{b}^{(2)}})]={{\delta }_{{{b}^{(1)}},{{b}^{(2)}}}}$. We therefore have that $\parallel {{f}_{n}}\parallel _{2,\sigma }^{2}={{\sum }_{b}}|\alpha (b){{|}^{2}}$. Direct evaluation of the four-norm of f yields

$$\begin{eqnarray}&&\parallel {{f}_{n}}\parallel _{4,\sigma }^{4}=\mathop{\sum }\limits_{{{b}^{(1)}},{{b}^{(2)}},{{b}^{(3)}},{{b}^{(4)}}}\alpha {{({{b}^{(1)}})}^{*}}\alpha ({{b}^{(2)}})\alpha {{({{b}^{(3)}})}^{*}}\alpha ({{b}^{(4)}})Q({{v}^{(1)}},{{v}^{(2)}},{{v}^{(3)}},{{v}^{(4)}}).\end{eqnarray} \tag{ 128 }$$

By proposition 14, we always have $|Q({{v}^{(1)}},{{v}^{(2)}},{{v}^{(3)}},{{v}^{(4)}})|\leqslant {\rm exp} (n\beta \nu )$. Moreover, recall that Q is only different from zero, when more than one mode is occupied at each site, or pair of sites. We therefore have a bound with the constrained sum ${{\sum }^{\prime }}$ taking into account the occupancy as stated in proposition 14:

$$\begin{eqnarray}&&\begin{array}{l} \parallel {{f}_{n}}\parallel _{4,\sigma }^{4}\leqslant {\rm exp} (n\beta \nu )\sum _{{{b}^{(1)}},{{b}^{(2)}},{{b}^{(3)}},{{b}^{(4)}}}^{\prime }|\alpha ({{b}^{(1)}})||\alpha ({{b}^{(2)}})||\alpha ({{b}^{(3)}})||\alpha ({{b}^{(4)}})| \\ \quad \equiv {\rm exp} (n\beta \nu ){{S}_{n}}. \\ \end{array}\end{eqnarray} \tag{ 129 }$$

Let us now express the rhs of equation (129) in terms of new discrete vectors ${{X}^{(i)}}\subseteq [1,N]$, ${{X}^{(ij)}}\subseteq [1,N]$ and ${{K}^{(i)}}\in {{\{(0,1),(1,0),(1,1)\}}^{|{{X}^{(i)}}|}}$, which given ${{X}^{(i)}}$ is isomorphic to the set of functions ${{X}^{(i)}}\to \{(0,1),(1,0),(1,1)\}$ defined as

$$\begin{eqnarray}&&{{X}^{(i)}}:=\{j\in [1,N]:b_{2j-1}^{(i)}+b_{2j}^{(i)}\gt 0\},\end{eqnarray} \tag{ 130 }$$

$$\begin{eqnarray}&&{{X}^{(ij)}}:={{X}^{(i)}}\cap {{X}^{(j)}},\end{eqnarray} \tag{ 131 }$$

$$\begin{eqnarray}&&{{K}^{(i)}}(j):=(b_{2j-1}^{(i)},b_{2j}^{(i)}))\ {\rm for}\ {\rm all}\ j\in {{X}^{(i)}}.\end{eqnarray} \tag{ 132 }$$

It is clear that the $\{{{b}^{(i)}}\}$ determine all these quantities. In the cases where $\alpha ({{b}^{(i)}})\ne 0$, the following converse is also true. The $\{{{K}^{(i)}}\}$ together with the $\{{{X}^{(ij)}}\}$ uniquely determine the $\{{{b}^{(i)}}\}$. Additional consistency relations such as ${{X}^{(12)}}\cap {{X}^{(34)}}={{X}^{(23)}}\cap {{X}^{(14)}}$ may be required to guarantee the existence of the $\{{{b}^{(i)}}\}$. In terms of the new variables

$$\begin{eqnarray}&&{{S}_{n}}=\mathop{\sum }\limits_{\{{{K}^{(i)}}\}}\mathop{\sum }\limits_{\{{{X}^{(ij)}}\}}|\alpha ({{b}^{(1)}})||\alpha ({{b}^{(2)}})||\alpha ({{b}^{(3)}})||\alpha ({{b}^{(4)}})|.\end{eqnarray} \tag{ 133 }$$

Recall that ${{X}^{(ij)}}={{X}^{(ji)}}$, so that we can now repeatedly apply the Cauchy–Schwartz inequality. We obtain first for ${{X}^{(12)}}$ and ${{X}^{(34)}}$

$$\begin{eqnarray}&&\begin{array}{l} {{S}_{n}}\leqslant \mathop{\sum }\limits_{\{{{K}^{(i)}}\}}\mathop{\sum }\limits_{{{X}^{(13)}},{{X}^{(24)}},{{X}^{(14)}},{{X}^{(23)}}}\sqrt{\mathop{\sum }\limits_{{{X}^{(12)}}}|\alpha ({{b}^{(1)}}){{|}^{2}}}\sqrt{\mathop{\sum }\limits_{{{X}^{(12)}}}|\alpha ({{b}^{(2)}}){{|}^{2}}} \\ \qquad \qquad \qquad \qquad \sqrt{\mathop{\sum }\limits_{{{X}^{(34)}}}|\alpha ({{b}^{(3)}}){{|}^{2}}}\sqrt{\mathop{\sum }\limits_{{{X}^{(34)}}}|\alpha ({{b}^{(4)}}){{|}^{2}}}. \\ \end{array}\end{eqnarray} \tag{ 134 }$$

After four additional applications of Cauchy–Schwartz inequality to the remaining summation indices ${{X}^{(ij)}}$ one obtains the following bound

$$\begin{eqnarray}&&\begin{array}{l} {{S}_{n}}\leqslant \mathop{\sum }\limits_{\{{{K}^{(i)}}\}}\sqrt{\mathop{\sum }\limits_{{{X}^{(12)}},{{X}^{(13)}},{{X}^{(14)}}}|\alpha ({{b}^{(1)}}){{|}^{2}}}\sqrt{\mathop{\sum }\limits_{{{X}^{(21)}},{{X}^{(23)}},{{X}^{(24)}}}|\alpha ({{b}^{(2)}}){{|}^{2}}} \\ \qquad \qquad \sqrt{\mathop{\sum }\limits_{{{X}^{(31)}},{{X}^{(32)}},{{X}^{(34)}}}|\alpha ({{b}^{(3)}}){{|}^{2}}}\sqrt{\mathop{\sum }\limits_{{{X}^{(41)}},{{X}^{(42)}},{{X}^{(43)}}}|\alpha ({{b}^{(4)}}){{|}^{2}}}. \\ \end{array}\end{eqnarray} \tag{ 135 }$$

Observing that for a fixed ${{b}^{(i)}}$ with n non-zero bits, there are at most ${{2}^{2n}}$ subsets ${{X}^{(ij)}}$ such that ${{\bigcup }_{j}}{{X}^{(ij)}}={{X}^{(i)}}$ that are summed over, we have the final bound

$$\begin{eqnarray}&&{{S}_{n}}\leqslant {{2}^{4n}}\mathop{\prod }\limits_{i=1}^{4}\mathop{\sum }\limits_{{{K}^{(i)}}}\sqrt{\mathop{\sum }\limits_{{{b}^{(i)}}}|\alpha ({{b}^{(i)}}){{|}^{2}}}\leqslant {{2}^{8n}}\parallel {{f}_{n}}\parallel _{\sigma ,2}^{4}.\end{eqnarray} \tag{ 136 }$$

The last inequality follows from the application of Cauchy–Schwartz to the sum over ${{K}^{(i)}}$ and observing that ${{K}^{(i)}}$ may take no more than ${{4}^{n}}$ different values under the assumption that $|{{b}^{(i)}}|=n$. Together with (129) we arrive at the claim. □

We can now state the logarithmic Sobolev constant for the canonical class of linear fermionic Davies generators which only depends on two numbers, the spectral gap Λ, which is temperature dependent and the largest single mode frequency ν. These numbers are model dependent and have to be evaluated for each system individually.

Theorem 16. Let ${{\mathcal{L}}_{\beta }}$ denote a linear fermionic Davies generator in canonical form. Moreover let $\Lambda ={{{\rm min} }_{k}}\;{{\Lambda }_{k}}$ and $\nu ={{{\rm max} }_{k}}|{{\nu }_{k}}|$, denote the weakest mode coupling and the largest single particle frequency respectively. Then the log-Sobolev constant is bounded by

$$\begin{eqnarray}&&\frac{\Lambda }{\beta \nu +14}\leqslant \alpha \leqslant \Lambda .\end{eqnarray} \tag{ 137 }$$

Proof. The result follows from the previous discussion. Lemma 13 and lemma 129 give the necessary conditions for the corollary 8 with $\lambda =\Lambda$ and $C={{2}^{2}}{\rm exp} \left( \beta /4\nu \right)$. Hence

$$\begin{eqnarray}&&\alpha \geqslant \frac{\lambda }{{\rm log} \left( {{C}^{4}}{{2}^{8}}{{e}^{2}} \right)}=\frac{\Lambda }{{\rm log} \left( {{{\rm e}}^{\beta \nu }}{{2}^{16}}{{e}^{2}} \right)}\geqslant \frac{\Lambda }{\beta \nu +14}.\end{eqnarray} \tag{ 138 }$$

The upper bound on α follows from the spectral gap bound on the log-Sobolev constant [16]. □