Quantum States and Liouville Space

This page introduces the quantum-mechanical objects that appear throughout ProcessTensors.jl: state vectors, observables, density matrices, reduced states, Liouville-space vectorisation, and open-system dynamics.

The goal is not to replace a textbook on quantum mechanics or open quantum systems. Instead, this page gives the minimum theory needed to understand why the package moves between Hilbert-space and Liouville-space tensor-network representations.

For tensor-network vocabulary such as MPS, MPO, contraction, and bond dimension, see Tensor Networks in Physics.

Hilbert space and density matrices

Before going to the Liouville space, we shall have a quick revision of quantum mechanics in the Hilbert space. In ordinary closed-system quantum mechanics, the state of a pure quantum system is represented by a vector $|\psi\rangle$ in a Hilbert space. For a lattice of $N$ sites with local dimension $d$, a general state can be written as

\[|\psi\rangle = \sum_{s_1,\ldots,s_N} c_{s_1\cdots s_N} |s_1,\ldots,s_N\rangle.\]

The time evolution of a closed system is generated by a Hamiltonian $H$. In units where $\hbar=1$, the Schrödinger equation is

\[\frac{d}{dt}|\psi(t)\rangle = -iH|\psi(t)\rangle.\]

For a time-independent Hamiltonian,

\[|\psi(t)\rangle = U(t)|\psi(0)\rangle, \qquad U(t)=e^{-iHt}.\]

In tensor-network simulations, $|\psi\rangle$ is often represented as a matrix product state (MPS), while $H$ and other many-body operators are represented as matrix product operators (MPOs).

sites = siteinds("S=1/2", N)
ψ = MPS(sites, fill("Up", N))
H_mpo = MPO(H, sites)

Observables and expectation values

A physical observable is represented by an operator $O$. For a pure state, its expectation value is

\[\langle O\rangle = \langle\psi|O|\psi\rangle.\]

The variance of $O$ is

\[\operatorname{Var}(O) = \langle O^2\rangle - \langle O\rangle^2.\]

Two-point correlation functions compare observables at different sites. For example,

\[C_{AB}(i,j) = \langle A_iB_j\rangle - \langle A_i\rangle\langle B_j\rangle.\]

These are ordinary Hilbert-space quantities. They are the baseline against which density-matrix and Liouville-space formulas should be compared.

mz = expect(ψ, "Sz")                    # local expectations
expect_O = real(inner(ψ', O_mpo, ψ))    # operator expectation

ρ = to_dm(ψ)                            # pure state → density MPO
ρ_mix = to_dm([ψ1, ψ2]; coeffs=[p, 1-p])  # statistical mixture

Pure state vectors are not enough for open quantum systems. A subsystem interacting with an environment is generally not described by a single state vector, even if the combined system-environment state is pure.

The density matrix formalism gives a more general description of quantum states that contains these classical probabilities associated with each pure wavefunction.

For a pure state,

\[\rho = |\psi\rangle\langle\psi|.\]

For a statistical mixture of pure states,

\[\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|,\]

where

\[p_i \ge 0, \qquad \sum_i p_i = 1.\]

A valid density matrix satisfies three important properties:

\[\operatorname{Tr}(\rho)=1, \qquad \rho^\dagger=\rho, \qquad \rho \ge 0.\]

These mean that the state is normalised, Hermitian, and positive semidefinite.

Expectation values are computed as

\[\langle O\rangle = \operatorname{Tr}(O\rho).\]

For a pure state $\rho=|\psi\rangle\langle\psi|$, this reduces to the familiar expression $\langle\psi|O|\psi\rangle$.

ρ = to_dm(ψ)
ρ_mix = to_dm([ψ_a, ψ_b]; coeffs=[0.7, 0.3])

Reduced density matrices and partial traces

Open-system physics usually begins with a larger closed system made of a system $S$ and a bath or environment $B$. The joint density matrix lives on the composite Hilbert space

\[\mathcal{H}_{SB} = \mathcal{H}_S \otimes \mathcal{H}_B.\]

If the full state is $\rho_{SB}$, the reduced state of the system is obtained by tracing out the bath:

\[\rho_S = \operatorname{Tr}_B(\rho_{SB}).\]

Similarly, the reduced bath state is

\[\rho_B = \operatorname{Tr}_S(\rho_{SB}).\]

The partial trace is the mathematical operation behind the phrase “ignore the bath degrees of freedom.” It keeps all system observables correct:

\[\langle O_S\rangle = \operatorname{Tr}_{SB} \left[ (O_S\otimes I_B)\rho_{SB} \right] = \operatorname{Tr}_S \left[ O_S\rho_S \right].\]

A useful quantity derived from a reduced state is the von Neumann entropy,

\[S(\rho_S) = -\operatorname{Tr} \left( \rho_S\log\rho_S \right).\]

For a bipartite pure state, this entropy measures entanglement between $S$ and $B$. For a genuinely mixed state, it also contains classical and environmental uncertainty.

Full state:

      system S              bath B
    ┌──────────┐        ┌──────────┐
    │          │        │          │
    │  ρ_SB    │  --->  │ trace B  │
    │          │        │          │
    └──────────┘        └──────────┘

Reduced state:

      ρ_S = Tr_B(ρ_SB)

ρ = to_dm(ψ)                            # full-chain density MPO
# trace out other sites via delta contractions on bra/ket legs (tutorial)
ρ1 = one_site_reduced_density_matrix(ρ, sites, 1)

Liouville space and vectorisation

A density matrix is an operator on Hilbert space,

\[\rho \in \mathcal{B}(\mathcal{H}),\]

where $\mathcal{B}(\mathcal{H})$ denotes the space of linear operators on $\mathcal{H}$.

The key observation is that $\mathcal{B}(\mathcal{H})$ is itself a Hilbert space when equipped with the Hilbert–Schmidt inner product,

\[\langle\langle A | B \rangle\rangle = \operatorname{Tr}(A^\dagger B).\]

This operator Hilbert space is called Liouville space.

If $\dim(\mathcal{H})=d$, then

\[\dim\mathcal{B}(\mathcal{H}) = d^2.\]

This is why Liouville-space simulations are more expensive than pure-state Hilbert-space simulations, but also why density matrices can be treated with state-vector tensor-network tools.

sites_L = liouv_sites(sites)            # create once, reuse everywhere
ρL = to_liouville(ρ; sites=sites_L)
ρ_back = to_hilbert(ρL; sites=sites)    # round-trip check

First-level vectorisation

Choose an orthonormal basis $\{|j\rangle\}$ of $\mathcal{H}$. The basic vectorisation rule is

\[|j\rangle\langle k| \longmapsto |j\rangle\otimes|k\rangle.\]

Equivalently, for an operator

\[A = \sum_{j,k} A_{jk} |j\rangle\langle k|,\]

we define

\[|A\rangle\rangle = \sum_{j,k} A_{jk} |j\rangle\otimes|k\rangle.\]

This is the simple linear-algebra move behind Liouville space: an operator becomes a vector in a larger space.

A more invariant way to say this is

\[\mathcal{B}(\mathcal{H}) \simeq \mathcal{H}\otimes\mathcal{H}^*.\]

The second factor is the dual space. After a basis is chosen, it is often identified computationally with another copy of $\mathcal{H}$.

Column-major vectorisation convention

Different communities use different vectorisation conventions. The convention matters because it determines the Kronecker-product formulas for left and right multiplication.

ProcessTensors.jl uses column-major vectorisation, matching Julia's native array ordering.

For a one-site density matrix

\[\rho = \begin{pmatrix} \rho_{11} & \rho_{12} \\ \rho_{21} & \rho_{22} \end{pmatrix},\]

the vectorised state is

\[|\rho\rangle\rangle = \operatorname{vec}(\rho) = \begin{pmatrix} \rho_{11} \\ \rho_{21} \\ \rho_{12} \\ \rho_{22} \end{pmatrix}.\]

Equivalently, the first matrix index changes fastest.

Matrix:

       column 1        column 2

        ρ₁₁              ρ₁₂
        ρ₂₁              ρ₂₂

Column-major vector:

       [ρ₁₁, ρ₂₁, ρ₁₂, ρ₂₂]ᵀ

Operators as superoperators

A superoperator is a linear map acting on operators. After vectorisation, it becomes an ordinary operator acting on Liouville-space vectors.

The central identity is

\[\operatorname{vec}(A\rho B) = (B^{\mathsf{T}}\otimes A) \operatorname{vec}(\rho).\]

Two special cases are especially important:

\[\operatorname{vec}(A\rho) = (I\otimes A) |\rho\rangle\rangle,\]

and

\[\operatorname{vec}(\rho B) = (B^{\mathsf{T}}\otimes I) |\rho\rangle\rangle.\]

So left multiplication and right multiplication are represented by different Liouville-space operators.

In package notation:

# OpSum terms use _L / _R suffixes before vectorisation
os = OpSum()
os += 1.0im, "Sx_L", 1              # Sx ρ
os += -1.0im, "Sx_R", 1             # ρ Sx

Channels as Liouville-space operators

A deterministic physical evolution of density matrices is described by a completely positive trace-preserving map,

\[\Phi: \mathcal{B}(\mathcal{H}_{\mathrm{in}}) \rightarrow \mathcal{B}(\mathcal{H}_{\mathrm{out}}).\]

Such maps are quantum channels.

In Hilbert-space notation, a channel is written as a map,

\[\rho_{\mathrm{in}} \longmapsto \Phi(\rho_{\mathrm{in}}).\]

After vectorisation, it becomes a matrix-like object,

\[|\rho_{\mathrm{out}}\rangle\rangle = \Phi |\rho_{\mathrm{in}}\rangle\rangle.\]

This is the computational reason for using Liouville space in this package: density matrices become MPS-like objects, while channels and generators become MPO-like objects.

ρL = to_liouville(ρ; sites=sites_L)   # density matrix as Liouville MPS
L_mpo = liouvillian_mpo(H, sites_L)     # channel generator as Liouville MPO

Dynamics in Liouville space

Hamiltonian dynamics

Closed-system density matrices obey the von Neumann equation

\[\frac{d\rho}{dt} = -i[H,\rho].\]

Expanding the commutator,

\[\frac{d\rho}{dt} = -iH\rho + i\rho H.\]

After vectorisation, this becomes

\[\frac{d}{dt} |\rho\rangle\rangle = \mathcal{L}_H |\rho\rangle\rangle,\]

where

\[\mathcal{L}_H = -i(I\otimes H) + i(H^{\mathsf{T}}\otimes I).\]

Using the package suffix language, this is the same structure as

\[\mathcal{L}_H = -iH_L + iH_R.\]

Here $H_L$ means “left multiplication by $H$” and $H_R$ means “right multiplication by $H$” in the Liouville-space convention used by the package.

L_mpo = liouvillian_mpo(H, sites_L)
ρL = tdvp(ρL, L_mpo, -1im * dt; alg=TDVP(), maxdim=32)

Open Markovian dynamics

Many Markovian open quantum systems are described by a Master equation in the GKLS form. The most commonly used version is the Local Master Equations (LME), given by

\[\frac{d\rho}{dt} = -i[H,\rho] + \sum_\mu \gamma_\mu \left( L_\mu\rho L_\mu^\dagger - \frac{1}{2} \left\{ L_\mu^\dagger L_\mu, \rho \right\} \right).\]

Here $H$ generates coherent Hamiltonian dynamics, while the jump operators $L_\mu$ describe dissipative processes such as decay, dephasing, or particle loss.

The anticommutator is

\[\left\{ L_\mu^\dagger L_\mu, \rho \right\} = L_\mu^\dagger L_\mu\rho + \rho L_\mu^\dagger L_\mu.\]

After vectorisation, the whole equation becomes

\[\frac{d}{dt} |\rho\rangle\rangle = \mathcal{L} |\rho\rangle\rangle,\]

where $\mathcal{L}$ is the Liouvillian superoperator.

For one jump operator $L$ with rate $\gamma$, the dissipative part has the structure

\[\gamma \left( L^*\otimes L - \frac{1}{2} I\otimes L^\dagger L - \frac{1}{2} (L^\dagger L)^{\mathsf{T}}\otimes I \right).\]

In package language, these correspond to the _Jump, _LdagL_L, and _LdagL_R terms.

using ITensors.Ops: Trotter
jumps = [(γ, "Sm", 1)]                  # jump rate, operator, site
L_mpo = liouvillian_mpo(H, sites_L; jump_ops=jumps)
ρL = tebd(ρL, L_mpo, dt, T; alg=Trotter{2}(), maxdim=32)

Hilbert-to-Liouville dictionary

The following table summarises the main translations used throughout this documentation.

Id_L = to_liouville(Id_mpo; sites=sites_L)
Sz_L = to_liouville(Sz_mpo; sites=sites_L)
Tr_ρ = inner(Id_L, ρL)
expect_Sz = inner(Sz_L, ρL)           # same sites_L on both factors

Further reading

This page is meant to be a compact bridge into the rest of the package. For more background, the following resources are useful.

Quantum states and density matrices

1. John Preskill, Lecture Notes for Quantum Computation, Chapter 2 and Chapter 3

   A clear introduction to density matrices, measurements, quantum operations, and channels from a quantum-information perspective.

2. Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information

   A standard reference for density matrices, partial traces, quantum channels, and open-system language in finite-dimensional quantum mechanics.

3. Mark M. Wilde, Quantum Information Theory

   A freely available advanced reference with careful treatments of density operators, partial traces, purifications, and quantum channels.

Liouville space and vectorisation

1. J. A. Gyamfi, “Fundamentals of Quantum Mechanics in Liouville Space.”

   A pedagogical introduction to finite-dimensional Liouville space, vectorisation, Kronecker products, and open-system master equations.

2. QuTiP documentation

   Useful for practical examples of density matrices, Lindblad master equations, superoperators, and numerical open-system simulation.

3. QuTiP: An open-source Python framework for the dynamics of open quantum systems.

   A software-oriented reference showing how open quantum dynamics, master equations, and Monte Carlo methods are organised in a numerical package.

Open quantum systems

1. Heinz-Peter Breuer and Francesco Petruccione, The Theory of Open Quantum Systems

   A standard textbook reference for reduced density matrices, master equations, Markovian limits, and non-Markovian open-system dynamics.

2. Ángel Rivas and Susana F. Huelga, Open Quantum Systems: An Introduction

   A compact and readable introduction to the mathematical structure of open quantum dynamics.

3. G. Lindblad, “On the Generators of Quantum Dynamical Semigroups.”

   The original mathematical reference for the Lindblad generator of Markovian quantum dynamics.