Math fixes

docs/background.md

@@ -9,7 +9,7 @@ The dynamics is determined by: (1) a Hamiltonian, which corresponds to the unita
 These two types of terms enter the so-called *Lindblad equation* that determines the evolution of the *density matrix* of the system.
 Since we have $N$ qubits the Hilbert space has dimension $2^N$ and the density matrix is $2^N$ by $2^N$ in size. In practice this is a huge dimension unless $N$ is very small, and it therefore prevents a direct brute-force numerical solution of the Lindblad equation.
 
-The present code offers an approximate solution of the problem that can be very accurate for large systems (typically up to N~100 or more) if the geometry of the couplings between the qubits is one-dimensional. This approach can also be more efficient than a brute force approach in other geometries.
+The present code offers an approximate solution of the problem that can be very accurate for large systems (typically up to $N\sim 100$ or more) if the geometry of the couplings between the qubits is one-dimensional. This approach can also be more efficient than a brute force approach in other geometries.
 
 Two publications in which some simulation results obtained with this code have been described:
 
@@ -21,47 +21,47 @@ https://doi.org/10.1103/PhysRevB.102.064301
 
 An MPS is a particular way to encode
 a many-body wave-function using a set of matrices. Consider a system made of
-<img src="https://render.githubusercontent.com/render/math?math=N"> qubits, in a pure state
-<img src="https://render.githubusercontent.com/render/math?math=\left|\psi\right\rangle=\sum_{s_1,s_2,\cdots,s_N}\psi\left(s_1,s_2,\cdots,s_N\right)\left|s_1\right\rangle\left|s_2\right\rangle\cdots\left|s_N\right\rangle">. 
-In this expression the sum runs over the <img src="https://render.githubusercontent.com/render/math?math=2^N"> basis states (<img src="https://render.githubusercontent.com/render/math?math=s_i\in\left\{0,1\right\}">) and the wave-function is encoded
-into the function <img src="https://render.githubusercontent.com/render/math?math=\psi:\left\{s_i\right\}\to\psi\left(s_1,s_2,\cdots,s_N\right)">. An MPS is a state where the wave function is written
-<img src="https://render.githubusercontent.com/render/math?math=\psi\left(s_1,s_2,\cdots,s_N\right)={\rm Tr}\left[A^{(s_1)}_1A^{(s_2)}_2\cdots A^{(s_N)}_N\right]">
-where, for each qubit <img src="https://render.githubusercontent.com/render/math?math=i"> we have introduced two matrices
-<img src="https://render.githubusercontent.com/render/math?math=A^{(0)}_i"> and <img src="https://render.githubusercontent.com/render/math?math=A^{(1)}_i"> (for a local Hilbert space of dimension
-<img src="https://render.githubusercontent.com/render/math?math=D"> one needs
-<img src="https://render.githubusercontent.com/render/math?math=D"> matrices
-<img src="https://render.githubusercontent.com/render/math?math=A^{(0)}_i\cdots A^{(D)}_i">for each qubit). These matrices are in general rectangular
-(<img src="https://render.githubusercontent.com/render/math?math=d_i\times d_{i%2b1}">)
-and the wave-function is obtained by multiplying them. What determines the dimensions of the matrices ?
-If  the matrices are one-dimensional (scalar) one has  a trivial product state (and all product states can be written this way). On the other hand, if one allows for very large matrices, of size <img src="https://render.githubusercontent.com/render/math?math=2^N">, any arbitrary state can be written as an MPS. 
-In fact the MPS representation is really useful
-when the system has a moderate amount of bipartite entanglement.
+$N$ qubits, in a pure state
+$\left|\psi\right\rangle=\sum_{s_1,s_2,\cdots,s_N}\psi\left(s_1,s_2,\cdots,s_N\right)\left|s_1\right\rangle\left|s_2\right\rangle\cdots\left|s_N\right\rangle$. 
+In this expression the sum runs over the $2^N$ basis states ($s_i\in\left\{0,1\right\}$) and the wave-function is encoded
+into the function $\psi:\left\{s_i\right\}\to\psi\left(s_1,s_2,\cdots,s_N\right)$. An MPS is a state where the wave function is written
+$\psi\left(s_1,s_2,\cdots,s_N\right)={\rm Tr}\left[A^{(s_1)}_1A^{(s_2)}_2\cdots A^{(s_N)}_N\right]$
+where, for each qubit $i$ we have introduced two matrices
+$A^{(0)}_i$ and $A^{(1)}_i$ (for a local Hilbert space of dimension
+$D$ one needs
+$D$ matrices
+$A^{(0)}_i\cdots A^{(D)}_i$ for each qubit). These matrices are in general rectangular
+($d_i\times d_{i+1}$)
+and the wave-function is obtained by multiplying them. What determines the dimensions of the matrices?
+If  the matrices are one-dimensional (scalar) one has  a trivial product state (and all product states can be written this way).
+On the other hand, if one allows for very large matrices, of size $2^N$, any arbitrary state can be written as an MPS. 
+In fact the MPS representation is really useful when the system has a moderate amount of bipartite entanglement.
 As a "rule of thumb", to get a good MPS approximation of a given state, each
-matrix <img src="https://render.githubusercontent.com/render/math?math=A^{(s_i)}_i">, of size <img src="https://render.githubusercontent.com/render/math?math=d_{i-1}\times d_{i}">, should have a dimension
-<img src="https://render.githubusercontent.com/render/math?math=d_i"> of the order of
-<img src="https://render.githubusercontent.com/render/math?math=e^{S_{\rm vN}(i)}">, where
-<img src="https://render.githubusercontent.com/render/math?math=S_{\rm vN}(i)"> the von Neumann entropy of the subsystem
-<img src="https://render.githubusercontent.com/render/math?math=[i%2b1,\cdots,N]">.
-
-
-What about mixed states ? They can be represented using so-called matrix-product operators (MPO):
-<img src="https://render.githubusercontent.com/render/math?math=\rho=\sum_{a_1,a_2,\cdots,a_N}{\rm Tr}\left[M^{(a_1)}_1 M^{(a_2)}_2\cdots M^{(a_N)}_N\right]\sigma^a_1 \otimes \sigma^a_2 \otimes \cdots \sigma^a_N">
-where each <img src="https://render.githubusercontent.com/render/math?math=a_i"> can take four values
-<img src="https://render.githubusercontent.com/render/math?math=\in\{1,x,y,z\}">,
-<img src="https://render.githubusercontent.com/render/math?math=\sigma^{a_i}"> is a Pauli matrix or the identity acting on qubit
-<img src="https://render.githubusercontent.com/render/math?math=i">,
+matrix $A^{(s_i)}_i$, of size $d_{i-1}\times d_{i}$, should have a dimension
+$d_i$ of the order of
+$e^{S_{\rm vN}(i)}$, where
+$S_{\rm vN}(i)$ is the von Neumann entropy of the subsystem
+$[i+1,\cdots,N]$.
+
+
+What about mixed states? They can be represented using so-called matrix-product operators (MPO):
+$\rho=\sum_{a_1,a_2,\cdots,a_N}{\rm Tr}\left[M^{(a_1)}_1 M^{(a_2)}_2\cdots M^{(a_N)}_N\right]\sigma^a_1 \otimes \sigma^a_2 \otimes \cdots \sigma^a_N$
+where each $a_i$ can take four values
+$\in\{1,x,y,z\}$,
+$\sigma^{a_i}$ is a Pauli matrix or the identity acting on qubit
+$i$,
 and we have associated  four matrices
-<img src="https://render.githubusercontent.com/render/math?math=M_i^{(1)}">,
-<img src="https://render.githubusercontent.com/render/math?math=M_i^{(x)}">,
-<img src="https://render.githubusercontent.com/render/math?math=M_i^{(y)}"> and
-<img src="https://render.githubusercontent.com/render/math?math=M_i^{(z)}"> to each qubit.
+$M_i^{(1)},
+$M_i^{(x)}$,
+$M_i^{(y)}$ and
+$M_i^{(z)}$ to each qubit.
 
 
 
 ### Bond dimension and entanglement
 MPS can allow to store reliably a quantum state with a huge memory gain (compression) when the state is not too entangled. As explained above, 
 we should expect that, for a given level of accuracy, the size of the matrices in an MPS will scale as the exponential of the bipartite entanglement entropy (associated to the bi-partition on that 'bond'). In the context of MPS the matrix sizes
-<img src="https://render.githubusercontent.com/render/math?math=d_i"> are called 'bond dimensions'.
+$d_i$ are called 'bond dimensions'.
 
 The representation of the pure state in terms of an MPS is therefore all the more efficient as the corresponding pure-state is weakly entangled. By 'more efficient' we mean here that, for a given level of accuracy, the size of the matrices in the MPS will be smaller. Alternatively, if the matrix sizes are fixed, a weakly entangled state will be more precisely represented by an MPS than a highly entangled one.
 
@@ -79,7 +79,9 @@ We again refer here to the review by U. Schollwöck: https://doi.org/10.1016/j.a
 
 ### Vectorization
 
-A mixed state can be viewed as a pure state in some enlarged Hibert space with dimension squared. This is the so-called *vectorization*, and it is heavily used in this code. For a single qubit a density matrix can be viewed as one vector (= a pure state of some fictitious system) in a space of dimension 4. In this code a many-body density matrix is considered as a pure state of a (fictitious) system with (2^N)^2 = 4^N states.  In turn, such a pure state is encoded as an MPS (of a system with 4 states per site). The Lindblad super-operator acts linearly on density matrices. Since the present implementation encodes the density matrix as an MPS, the Lindblad super-operator is naturally encoded as an MPO.
+A mixed state can be viewed as a pure state in some enlarged Hibert space with dimension squared.
+This is the so-called *vectorization*, and it is heavily used in this code. For a single qubit a density matrix can be viewed as one vector (= a pure state of some fictitious system) in a space of dimension 4.
+In this code a many-body density matrix is considered as a pure state of a (fictitious) system with $(2^N)^2 = 4^N$ states.  In turn, such a pure state is encoded as an MPS (of a system with 4 states per site). The Lindblad super-operator acts linearly on density matrices. Since the present implementation encodes the density matrix as an MPS, the Lindblad super-operator is naturally encoded as an MPO.
 
 This can sometimes be source of confusion: the density matrix is of course an operator acting on the physical Hilbert space of the qubits, but, after vectorization, we interpret it as a pure state, and thus as an MPS.