Decoherence and mixed states

Learn the causes and consequences of quantum decoherence.

Objectives

Understand how a quantum system evolves when it gets entangled with its environment.
Experimentally verify the relation between decoherence and mixedness of quantum systems (like the qubits in a quantum register).

Background

Everyone has at least once in their life heard about the Schroedinger cat experiment (if not, this is the perfect opportunity!), which made scientists wonder about the reasons why certain nonclassical states are basically never observed for mesoscopic or macroscopic objects. This quantum-to-classical transition problem started the study of the phenomenon that we will describe below, known as decoherence, or even better, environment-induced decoherence. In few words, since realistic quantum systems are never perfectly isolated from the environment, they will interact and quickly become entangled with it. These correlations affect the information that we can get from measuring the system, and in particular counteract fragile properties as superposition, leading to a dynamical selection of more robust and stable states, known as pointer states. Decoherence owes its name to the fact that it is essentially an environment-induced destruction of quantum coherence. The theory that describes the behavior of non isolated systems is known as the theory of open quantum systems; it provides a different equation of motion, known as master equation, since their dynamics cannot be described with unitary time evolution.

Let's continue with discussing the formalism of density matrices for composite systems and then clarify why we should care about decoherence when it comes to quantum computers.

Theory

Consider a bipartite system made of the system $S$ of interest (the principal system) and the environment $E$ with the Hilbert spaces $\mathcal{H}_S$ and $\mathcal{H}_E$, respectively. The Hamiltonian of the total system is written as: $$H=H_S+H_E+H_{SE},$$where $H_S$ and $H_E$ are the Hamiltonians of the principal system and the environment, respectively, while $H_{SE}$ is the Hamiltonian of the system-environment interaction. These Hamiltonians are assumed to be time-independent in the following to simplify our discussion. Let $|\Psi(0)\rangle=|\psi_S(0)\rangle|\psi_E(0)\rangle$ be the initial state at $t=0$. The total state at $t>0$ is given by solving the Schrodinger equation as:$$|\Psi(t)\rangle = e^{-i H t}|\psi_S(0)\rangle |\psi_E(0)\rangle.$$ Let us consider the case $H_{SE}=0$ first. Then it is clear that:$$|\Psi(t) \rangle=|\psi_S(t) \rangle|\psi_E(t) \rangle,\quad \mathrm{where} \quad |\psi_S(t)\rangle = e^{-i H_S t}|\psi_S(0)\rangle, |\psi_E(t)\rangle = e^{-i H_E t}|\psi_E(0)\rangle $$If, on the other hand, $H_{SE}\neq 0$, such a separation is impossible. Even though the total state of system+environment remains a pure state $|\Psi(t)\rangle$, the state is entangled due to $H_{SE}$ and individual states $|\psi_S(t)\rangle$ and $|\psi_E(t)\rangle$ are not well-defined.

When dealing with composite systems, the formalism of density matrices is necessary for studying subsystems. Quantum states can be pure, i.e. states that cannot be written as a probabilistic mixture of other quantum states, or mixed, i.e. those that are instead a convex combination of pure states. If we want to study exclusively the principal system and forget the environment, we need to trace it away, i.e. we need to take the partial trace over the environment of the total system state:$$\rho_S=\sum_k \langle e_k|\Psi(t)\rangle \langle \Psi(t)|e_k \rangle=\sum_k (I_S\otimes \langle e_k|)|\Psi(t)\rangle \langle \Psi(t)|(I_S\otimes |e_k \rangle),$$where $I_S$ is the identity matrix of the system and ${| e_k \rangle}$ is a set of the complete orthonormal basis of the environment.

Let us now show that the state of the principal system is mixed due to entanglement with the environment. Consider an observable $A_S$ associated with the principal system $S$. The expectation value of $A_S$ with respect to $|\Psi(t)\rangle$ is:$$\mathrm{tr}(|\Psi(t)\rangle\langle \Psi(t)|(A_S\otimes I_E)) = $$$$\sum_{i,j} \langle s_i e_j|\Psi(t)\rangle\langle \Psi(t)|(A_S\otimes I_E)|s_i e_j\rangle=$$$$\sum_i \langle s_i | \left(\sum_j \langle e_j|\Psi(t)\rangle\langle \Psi(t)|e_j\rangle\right) A_S|s_i\rangle = \mathrm{tr} (\rho_S A_S),$$from which we conclude that the state of the principal system is a mixed state $\rho_S$ because the expectation value obtained is a convex combination of expectation values. By tracing out the degree of freedom of the environment, we lose the information of the environment and as a result, the state of the principal system is mixed.

The degree of mixture of a quantum state is measured by the von Neumann entropy defined as:$$S=-\mathrm{tr} \rho_S \log_2 \rho_S, $$ which takes the minimum value 0 for a pure state while maximum value $n$ for the maximally mixed state, where $n$ is the number of qubits in the principal system.

Motivation for the experiment

We have seen how decoherence means that a quantum system prepared in a pure state will evolve into a mixed state under interaction with its environment.

This phenomenon is one of the worse enemy for quantum computing; transition from a pure state to a mixed state is a degradation of quantum information and the output of quantum computation becomes less reliable in the presence of decoherence, which has thus to be suppressed. Qubits must be engineered in a way that minimizes environmental interactions, but still allows for enough control. Moreover, longevity of superposition states necessary for quantum information processing is crucial and very challenging, since those states are the most prone to decoherence. Decoherence is also called "noise" or "error" in some contexts, like for quantum error correcting code that provides the most promising road toward fault-tolerant quantum computing. Superconducting qubit systems were one of the first candidates and most prominent platforms used in explorative decoherence experiments. For this reason, it is interesting to understand how decoherence acts in such a setting like the Spark quantum computer.

Methods

The most well known experimental procedure to reconstruct the density matrix of a quantum system is quantum tomography. In this protocol several identical copies are prepared in the same state in order to perform a different measurement on each copy. To recover all the information about the state of the quantum systam, a tomographically complete basis of measurement operators is needed.

Only in the case of an infinite number of measurements the state would be reconstructed perfectly; in practice, one always needs to take into account the statistical error coming from the finite number of measurements.

The biggest limitation of quantum tomography is that the number of measurements scales exponentially with the number of qubits in the system, and thus it is achievable only for modest system sizes. For this reason, a lot of research has been conducted to find alternative methods to reduce the cost of this experimental procedure. The simplest reconstruction method for state tomography is linear inversion; it corresponds to retrieving the density matrix by inverting the relation $A\rho=p$, where $A$ is the measurement basis and $p$ is the vector of probabilities obtained from measuring all the operators in $A$ on many copies of $\rho$. But that's another story for a later time 😉.

Assignment

Consider a $3$-qubit system where one is the principal system $S$ and the rest acts as the environment $E$. The total system starts in the state $|\psi(0)\rangle = |+\rangle|00\rangle$. Calculate the reduced density matrix of the system $S$ and its von Neumann entropy.
The state above is gradually entangled by repeatedly applying the controlled-$R_x(\pi/8)$ gate (i.e. by applying the controlled-$R_x(k\pi/8)$ for different values of $k$). Write the corresponding circuit acting on $|\psi(0)\rangle$.
Run the different circuits on the quantum computer and reconstruct the state of the system via quantum tomography.
Calculate the entropy of the reduced state of the system $S$ for increasing $k$. At which value of $k$ all the information is lost?
Plot the entropy calculated above and the off-diagonal entry of the reduced state of the system $S$ against $k$. What behavior do you observe?

Questions for Preparation

Why is decoherence bad for quantum computing?
Take a Bell state and calculate the reduced density matrix of the first qubit. What state do you expect as result? Can you place it in the Bloch sphere?
Can you think of an example where entanglement is a resource we can use contrary to what happens in the decoherence process?

Literature

On-premises superconducting quantum computer for education and research (EPJ Quantum Technol. 11, 32, 2024).

The Theory of Open Quantum Systems (Breuer and Petruccione).

Quantum decoherence, M. Schlosshauer (Physics Reports, 2019).

On the interpretation of measurement in quantum theory, H. D. Zeh (Foundations of Physics, 1970).

Quantum State Tomography of a Single Qubit: Comparison of Methods, Roman Schmied (Journal of Modern Optics, 2016).