# Anthony Opipari

PhD Candidate, University of Michigan

# Summary and Interpretation of Quantum Decoherence

Final course project for EECS 598: Quantum Information, Probability and Computation

Winter 2020. Electrical & Computer Engineering Department, University of Michigan.

Authors: Anthony Opipari & Donato Mastropietro

This post is based on work done by myself and my project partner, Donato Mastropietro, as our final project for EECS 598: Quantum Information, Probability and Computation at the University of Michigan. We were very fortunate to be taught by Professor S. Sandeep Pradhan. Throughout the course, I was continually fascinated by the question of how the classical properties of physical systems we experience on a daily basis can be compatible with the underlying quantum state of our universe. Based on my questions, Professor Pradhan suggested I look into the theory of Quantum Decoherence and I’m grateful I was able to make this the topic of my final project. I continue to find Quantum Decoherence the most interesting theory I’m aware of. This course was completed as part of my Master’s degree.

## Abstract

In this report, we summarize a selection of works on the topic of quantum decoherence. Quantum decoherence provides a theory for how classical properties emerge from the governing principles of quantum mechanics. This theory has many implications for the field of quantum computing, which in many cases requires the stability of quantum coherence for solving computational problems. In this report we will summarize the measurement and preferred basis problems followed by a discussion of environment-induced decoherence with worked examples.

## I. Introduction

Quantum mechanics introduced entirely new properties to physical science. In particular, the quantum entanglement and quantum superposition properties predicted by the axioms of quantum mechanics represent a substantial departure from the classical properties of classical mechanics. In addition to their novelty, these properties seem to be completely at odds with our every day classical experience as observers. However experimental support for the existence of quantum phenomena has been repeatedly shown (e.g. the double slit experiment). Thus, the axioms of quantum mechanics are experimentally validated but appear incompatible with our experiences. The theory of quantum decoherence claims to reconcile this situation by providing a model for how the classical properties we are familiar with emerge from quantum systems.

In addition to its relevance for physical science, quantum decoherence is an important topic for the experimental realization of quantum computers. The efficiency of many quantum algorithms are contingent on quantum properties which, according to quantum decoherence, can be influenced or even destroyed as we shall discuss in section Section V.

This paper is meant to serve as a discussion of this topic by summarizing the work in $[1]$ $[2]$ $[3]$ $[4]$.

## II. Von Neumann Measurement Scheme

In 1932, John von Neumann introduced a new mathematical framework for measuring quantum systems. Exploiting properties of quantum entanglement, he developed a way to correlate a quantum dynamical system and its measurement apparatus. The method he devised, named eponymously for von Neumann, considered the measurement device to itself be a quantum system.

Before von Neumann, measurement devices were seen as purely classical systems - a view deemed the Copenhagen interpretation. This had the restriction that anytime the measurement device was used, a collapse of the quantum state occurred.

With von Neumann’s scheme for measurement, we can define a Hilbert space for the detector, $\mathscr{H}_{\mathcal{D}}$ . Further we can describe the detector according to some basis given by $\{\ket{d_\uparrow}, \ket{d_\downarrow}\}$ . This is a powerful scheme because now we can entangle the system and the detector in what’s known as a ‘pre-measurement’ thus capturing information about the system without causing the state to collapse. The scheme works as follows. Say the system we want to measure is in its own Hilbert space, $\mathscr{H}_\mathcal{S}$, with some initial state $\ket{\psi_\mathcal{S}} = (\alpha\ket{\uparrow}+\beta\ket{\downarrow})$, where spin up and spin down states are represented by the up and down arrows, respectively; these form a basis for our system. During pre-measurement, which takes place in the composite system’s Hilbert space, $\mathscr{H}_\mathcal{S}\mathbin{\mathop{\otimes}}\mathscr{H}_\mathcal{D}$, we capitalize on the continuous time evolution and linearity of the Schrödinger Equation to cause the state of the detector to transition into states associated with the basis states of the system. Take the example below; let the detector initially be spin down. Then if the system is spin up, the detector will transition to be spin up whereas if the system is spin down, the detector will transition to be spin down.

\begin{align} \ket{\uparrow}\ket{d_\downarrow}&\rightarrow\ket{\uparrow}\ket{d_\uparrow}\nonumber\newline \ket{\downarrow}\ket{d_\downarrow}&\rightarrow\ket{\downarrow}\ket{d_\downarrow}\nonumber \end{align}

A consequence of this correlation is that we can now measure the state of the detector to discover information about the state of the system - in essence, measuring the system (by establishing a system-detector correlation) without causing a collapse. Our composite system is now in a state $\ket{\Phi^c}$, given below, and is ready to be measured.

\begin{align} \ket{\Phi^c} = \alpha\ket{\uparrow}\ket{d_\uparrow}+\beta\ket{\downarrow}\ket{d_\downarrow} \nonumber \end{align}

## III. The Measurement Problem

To speak further on measurement, we will first consider the problems which arise when taking measurements in the quantum world. According to the axioms of quantum mechanics, quantum systems like our composite state, $\ket{\Phi^c}$ above, may exist in a superposition of states preceding a measurement. Upon measuring, the state of the system collapses to a definite configuration non-deterministically. This presents a challenge for our natural intuition as observers since we only ever observe a single outcome and cannot observe the ‘full’ superposition of possible outcomes.

For our classically trained minds, it is nonsensical that systems can be in two states at a single time. Take a simple coin for example. In our classical world both the head’s and tail’s side cannot face upwards at once. Nonetheless, quantum superposition allows these two states to exist simultaneously. This is a great example of how the Copenhagen interpretation introduces a division between classical and quantum dynamics by defining the observer (us in this case) as a part of the classical world, which distinctly separates the observer from the quantum system under observation.

Evidence now exists showing even macroscopic objects exhibit non-classical behavior (e.g. the Weber bar). But even with this evidence and recognition of quantum theory, our predilections rooted in classicality cause us to resist accepting quantum phenomenon and thus defines the measurement problem.

## IV. The Preferred Basis Problem

The study of quantum decoherence finds its origins in the difficulty to reconcile standard unitary quantum dynamics and standard interpretations of quantum mechanics with how observers end up with determinate results. Let us continue to use the basis for our system and detector as defined in Section II to investigate the problem further. Let us say that the previously defined states are vectors such that they form the eigenbasis for the Pauli-$z$ matrix and thus define the spin along the z-axis for our system. Let us next define another basis for our system and detector which will form the eigenbasis for the Pauli-$x$ matrix and thus define the spin of the system along the x-axis; as defined below, we shall call them dot and cross of our system and detector, respectively.

\begin{align} \ket{\odot}=\frac{\ket{\uparrow}+\ket{\downarrow}}{\sqrt{2}} ,\quad \ket{\otimes}=\frac{\ket{\uparrow}-\ket{\downarrow}}{\sqrt{2}}\nonumber \end{align}
\begin{align} \ket{d_\otimes}=\frac{\ket{d_\uparrow}+\ket{d_\downarrow}}{\sqrt{2}} ,\quad \ket{d_\odot}=\frac{\ket{d_\downarrow}-\ket{d_\uparrow}}{\sqrt{2}}\nonumber \end{align}
\begin{align} \sigma_z&=\ket{\uparrow}\bra{\uparrow}-\ket{\downarrow}\bra{\downarrow}=\ket{d_\uparrow}\bra{d_\uparrow}-\ket{d_\downarrow}\bra{d_\downarrow}\nonumber\newline \sigma_x &= \ket{\odot}\ket{\odot}-\ket{\otimes}\ket{\otimes} = \ket{d_\odot}\ket{d_\odot}-\ket{d_\otimes}\ket{d_\otimes}\nonumber \end{align}
\begin{align} \alpha=-\beta=\frac{1}{\sqrt{2}}\nonumber \end{align}

Further, let us remember the definition of our composite system after the pre-measurement and now define the constants alpha and beta to be equal and opposite. With the system set up, we can perform a change of basis from that of the eigenstates of the Pauli-Z to those of the Pauli-X matrix.

\begin{align} \ket{\Phi^c} &= \frac{1}{\sqrt{2}}(\ket{\uparrow}\ket{d_\uparrow}-\ket{\downarrow}\ket{d_\downarrow})\nonumber\newline &=\frac{1}{\sqrt{2}}((\frac{\ket{\uparrow}\ket{d_\uparrow}-\ket{\downarrow}\ket{d_\downarrow}}{2})-(\frac{-\ket{\uparrow}\ket{d_\uparrow}+\ket{\downarrow}\ket{d_\downarrow}}{2}))\nonumber\newline &=\frac{1}{\sqrt{2}}((\frac{\ket{\uparrow}\ket{d_\uparrow}+\ket{\uparrow}\ket{d_\downarrow}-\ket{\downarrow}\ket{d_\uparrow}-\ket{\downarrow}\ket{d_\downarrow}}{2})\nonumber\newline &\;\;\;\;-(\frac{\ket{\uparrow}\ket{d_\downarrow}-\ket{\uparrow}\ket{d_\uparrow}+\ket{\downarrow}\ket{d_\downarrow}-\ket{\downarrow}\ket{d_\uparrow}}{2}))\nonumber\newline &=\frac{1}{\sqrt{2}}((\frac{\ket{\uparrow}-\ket{\downarrow}}{\sqrt{2}}\frac{\ket{d_\uparrow}+\ket{d_\downarrow}}{\sqrt{2}})-(\frac{\ket{\uparrow}+\ket{\downarrow}}{\sqrt{2}}\frac{\ket{d_\downarrow}-\ket{d_\uparrow}}{\sqrt{2}}))\nonumber\newline &=\frac{1}{\sqrt{2}}(\ket{\otimes}\ket{d_\otimes}-\ket{\odot}\ket{d_\odot})\nonumber \end{align}

Thus we have shown that this state can be expressed equivalently in terms of more than one basis. This implies the state is not yet in a definite configuration and that the set of possible outcomes for a measurement have not yet been chosen. To understand this, first consider the system in its density operator form given by $\rho^c=\ket{\Phi^c}\bra{\Phi^c}$. Next we will perform two distinct measurements. The first measurement will be defined by the Pauli-Z observable with eigenbasis, $\{\ket{d_\uparrow}, \ket{d_\downarrow}\}$, and the second will be defined by the Pauli-X observable with eigenbasis, $\{\ket{d_\otimes}, \ket{d_\odot}\}$. Note these are projective measurements.

Measurement with $\sigma_z$:

\begin{align} &P_{+1,z}=I\otimes\ket{d_\uparrow}\bra{d_\uparrow}\nonumber\newline &P_{-1,z}=I\otimes\ket{d_\downarrow}\bra{d_\downarrow}\nonumber\newline &p(m=+1)=Tr[P_{+1,z}\rho^c]=\frac{1}{2}\nonumber\newline &\rho_{+1}^{c\prime}=\frac{P_{+1,z}\rho^cP_{+1,z}^\dagger}{Tr[P_{+1,z}\rho^c]}=\ket{\uparrow}\bra{\uparrow}\ket{d_\uparrow}\bra{d_\uparrow}\nonumber\newline &p(m=-1)=Tr[P_{-1,z}\rho^c]=\frac{1}{2}\nonumber\newline &\rho_{-1}^{c\prime}=\frac{P_{-1,z}\rho^cP_{-1,z}^\dagger}{Tr[P_{-1,z}\rho^c]}=\ket{\downarrow}\bra{\downarrow}\ket{d_\downarrow}\bra{d_\downarrow}\nonumber \end{align}

Measurement with $\sigma_x$:

\begin{align} &P_{+1,x}=I\otimes\ket{d_\otimes}\bra{d_\otimes}\nonumber\newline &P_{-1,x}=I\otimes\ket{d_\odot}\bra{d_\odot}\nonumber\newline &p(m=+1)=Tr[P_{+1,x}\rho^c]=\frac{1}{2}\nonumber\newline &\rho_{+1}^{c\prime}=\frac{P_{+1,x}\rho^cP_{+1,x}^\dagger}{Tr[P_{+1,x}\rho^c]}=\ket{\otimes}\bra{\otimes}\ket{d_\otimes}\bra{d_\otimes}\nonumber\newline &p(m=-1)=Tr[P_{-1,x}\rho^c]=\frac{1}{2}\nonumber\newline &\rho_{-1}^{c\prime}=\frac{P_{-1,x}\rho^cP_{-1,x}^\dagger}{Tr[P_{-1,x}\rho^c]}=\ket{\odot}\bra{\odot}\ket{d_\odot}\bra{d_\odot}\nonumber \end{align}

The above results are key to understanding the preferred basis problem. For the same state, $\rho^c$, applying different observables leads to different resulting collapsed states. In the case above, measuring with the Pauli-Z and Pauli-X observables, resulted in the same outcome distribution yet different final states. This implies that before measurement, our state was not in a definite configuration and that the set of possible resulting states had not yet been chosen. These results are in direct contradiction with our understanding of classical measurements since in the classical world, before measurement our state is in one of a known set of possible outcomes.

The correlation of this system further implies that we have measured both observables, $\sigma_z$ and $\sigma_x$, and thus the spin along the X- and the Z-axes, at the same time. But this leads to another contradiction: how can we have measured both at the same time if these observables do not commute;

\begin{align} [\sigma_z, \sigma_x]\neq 0\nonumber \end{align}

Non-commutation indicates that the application of one observable before the next should alter the outcomes yet we know this is untrue since we found the outcomes to be correlated. These contradictions motivate and define the preferred basis problem.

## V. Environment-Induced Decoherence

In this section, we will summarize the central results of environment-induced decoherence and discuss how they resolve the challenges presented by the measurement and preferred basis problems. For this summary we will extend the discussion from Section II of a von Neumann system-detector scheme by introducing a quantum environment $\mathcal{E}$.

To begin, consider again our quantum system, $\mathcal{S}$, and our quantum detector $\mathcal{D}$ each in their own respective Hilbert space, $\mathscr{H}_\mathcal{S}, \& \mathscr{H}_\mathcal{D}$. We now introduce the quantum environment, $\mathcal{E}$, which exists in $\mathscr{H}_\mathcal{E}$. We assume the quantum environment is a large system representing all remaining particles in the environment.

Upon interaction, our composite system $\Psi^c$ is given by,

\begin{align} \ket{\Psi^i} &= \ket{\psi_\mathcal{S}}\ket{d_\downarrow}\ket{\mathcal{E}}\nonumber\newline &= (\alpha\ket{\uparrow}+\beta\ket{\downarrow})\ket{d_\downarrow}\ket{\mathcal{E}}\nonumber\newline &\rightarrow \alpha\ket{\uparrow}\ket{d_\uparrow}\ket{\mathcal{E}_\uparrow}+\beta\ket{\downarrow}\ket{d_\downarrow}\ket{\mathcal{E}_\downarrow} = \ket{\Psi^c}\nonumber \end{align}

Thus, we can observe that just as before, the superposition in $\mathscr{H}_\mathcal{S}$ is extended to the full composite system in $\mathscr{H}_\mathcal{S}\otimes\mathscr{H}_\mathcal{D}\otimes\mathscr{H}_\mathcal{E}$. Note the corresponding density operator representation $\rho^c=\ket{\Psi^c}\bra{\Psi^c}$.

At this point decoherence can provide a resolution to the preferred basis problem using the tridecompositional uniqueness theorem from $[5]$. This theorem states that if a state $\psi\in\mathscr{H}_1\otimes\mathscr{H}_2\otimes\mathscr{H}_3$ can be expressed as a decomposition $\psi=\sum_i c_i\ket{a_i}\ket{b_i}\ket{c_i}$, then such a decomposition must be unique. Thus by including the third system, $\mathcal{E}$, we are guaranteed that the resulting state cannot be represented in terms of a different basis, in contrast to the example in Section IV which used a composite state of only two subsystems. The basis forming such a decomposition is referred to as the preferred basis.

With our composite system $\rho^c$, we now consider the consequences of:

1. Loss of access to $\mathcal{E}$. Effects of decoherence emerge when the degrees of freedom present in $\mathcal{E}$ are lost from the observer. Such effects can be modeled from the perspective of an observer by tracing over $\mathcal{E}$ in the composite state $\rho^c$.
2. Orthonormal environment states. Decoherence requires the environment states $\{\mathcal{E}_i\}$ associated with detector states $\{d_i\}$ be orthonormal: $\ket{\mathcal{E}_i}\bra{\mathcal{E}_j}=\delta_{ij}$.

Under these conditions the system reduces according to,

\begin{align} \rho_{\mathcal{SD}}&=Tr_\mathcal{E}\ket{\Psi}\bra{\Psi}=\sum_i\braket{\mathcal{E}_i|\Psi}\braket{\Psi|\mathcal{E}_i} \nonumber \newline &=\sum_i\bra{\mathcal{E}_i}(\alpha\alpha^*\ket{\uparrow}\bra{\uparrow}\ket{d_\uparrow}\bra{d_\uparrow}\ket{\mathcal{E}_\uparrow}\bra{\mathcal{E}_\uparrow}\nonumber \newline &\quad\quad+\alpha\beta^*\ket{\uparrow}\bra{\downarrow}\ket{d_\uparrow}\bra{d_\downarrow}\ket{\mathcal{E}_\uparrow}\bra{\mathcal{E}_\downarrow}\nonumber\newline &\quad\quad+\beta\alpha^*\ket{\downarrow}\bra{\uparrow}\ket{d_\downarrow}\bra{d_\uparrow}\ket{\mathcal{E}_\downarrow}\bra{\mathcal{E}_\uparrow}\nonumber\newline &\quad\quad+\beta\beta^*\ket{\downarrow}\bra{\downarrow}\ket{d_\downarrow}\bra{d_\downarrow}\ket{\mathcal{E}_\downarrow}\bra{\mathcal{E}_\downarrow})\ket{\mathcal{E}_i} \nonumber\newline &=(\alpha\alpha^*\ket{\uparrow}\bra{\uparrow}\ket{d_\uparrow}\bra{d_\uparrow}\braket{\mathcal{E}_\uparrow|\mathcal{E}_\uparrow}\braket{\mathcal{E}_\uparrow|\mathcal{E}_\uparrow}\nonumber\newline &\quad\quad+\alpha\beta^*\ket{\uparrow}\bra{\downarrow}\ket{d_\uparrow}\bra{d_\downarrow}\braket{\mathcal{E}_\uparrow|\mathcal{E}_\uparrow}\braket{\mathcal{E}_\downarrow|\mathcal{E}_\uparrow}\nonumber\newline &\quad\quad+\beta\alpha^*\ket{\downarrow}\bra{\uparrow}\ket{d_\downarrow}\bra{d_\uparrow}\braket{\mathcal{E}_\uparrow|\mathcal{E}_\downarrow}\braket{\mathcal{E}_\uparrow|\mathcal{E}_\uparrow}\nonumber\newline &\quad\quad+\beta\beta^*\braket{\downarrow|\downarrow}\braket{d_\downarrow|d_\downarrow}\braket{\mathcal{E}_\uparrow|\mathcal{E}_\downarrow}\braket{\mathcal{E}_\downarrow|\mathcal{E}_\uparrow})\nonumber\newline &\quad\quad+(\alpha\alpha^*\ket{\uparrow}\bra{\uparrow}\ket{d_\uparrow}\bra{d_\uparrow}\braket{\mathcal{E}_\downarrow|\mathcal{E}_\uparrow}\braket{\mathcal{E}_\uparrow|\mathcal{E}_\downarrow}\nonumber\newline &\quad\quad+\alpha\beta^*\ket{\uparrow}\bra{\downarrow}\ket{d_\uparrow}\bra{d_\downarrow}\braket{\mathcal{E}_\downarrow|\mathcal{E}_\uparrow}\braket{\mathcal{E}_\downarrow|\mathcal{E}_\downarrow}\nonumber\newline &\quad\quad+\beta\alpha^*\ket{\downarrow}\bra{\uparrow}\ket{d_\downarrow}\bra{d_\uparrow}\braket{\mathcal{E}_\downarrow|\mathcal{E}_\downarrow}\braket{\mathcal{E}_\uparrow|\mathcal{E}_\downarrow}\nonumber\newline &\quad\quad+\beta\beta^*\ket{\downarrow}\bra{\downarrow}\ket{d_\downarrow}\bra{d_\downarrow}\braket{\mathcal{E}_\downarrow|\mathcal{E}_\downarrow}\braket{\mathcal{E}_\downarrow|\mathcal{E}_\downarrow}) \nonumber\newline &=(\alpha\alpha^*\ket{\uparrow}\bra{\uparrow}\ket{d_\uparrow}\bra{d_\uparrow}11+\alpha\beta^*\ket{\uparrow}\bra{\downarrow}\ket{d_\uparrow}\bra{d_\downarrow}10\nonumber\newline &\quad\quad+\beta\alpha^*\ket{\downarrow}\bra{\uparrow}\ket{d_\downarrow}\bra{d_\uparrow}01+\beta\beta^*\braket{\downarrow|\downarrow}\braket{d_\downarrow|d_\downarrow}00)\nonumber\newline &\quad\quad+(\alpha\alpha^*\ket{\uparrow}\bra{\uparrow}\ket{d_\uparrow}\bra{d_\uparrow}00+\alpha\beta^*\ket{\uparrow}\bra{\downarrow}\ket{d_\uparrow}\bra{d_\downarrow}01\nonumber\newline &\quad\quad+\beta\alpha^*\ket{\downarrow}\bra{\uparrow}\ket{d_\downarrow}\bra{d_\uparrow}10+\beta\beta^*\ket{\downarrow}\bra{\downarrow}\ket{d_\downarrow}\bra{d_\downarrow}11) \nonumber \newline &=(\alpha\alpha^*\ket{\uparrow}\bra{\uparrow}\ket{d_\uparrow}\bra{d_\uparrow}+\beta\beta^*\ket{\downarrow}\bra{\downarrow}\ket{d_\downarrow}\bra{d_\downarrow}) \nonumber\newline &=\rho^\prime_{\mathcal{SD}} \nonumber \end{align}

This reduction process, which is induced by the environment conditions outlined above, is the central result of decoherence. The reduced state $\rho_{\mathcal{SD}}$ is diagonal with respect to the basis $\{\ket{\uparrow}\ket{d_\uparrow}, \ket{\uparrow}\ket{d_\downarrow}, \ket{\downarrow}\ket{d_\uparrow}, \ket{\downarrow}\ket{d_\downarrow}\}$, which is referred to as the preferred basis. We will briefly discuss how nature chooses this basis in Section VII. This process can thus be summarized using the density matrix representation as,

\begin{align} \rho_{\mathcal{SD}}&=\begin{bmatrix} \alpha\alpha^* & 0 & 0 & \alpha\beta^* \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \beta\alpha^* & 0 & 0 & \beta\beta^* \end{bmatrix}\nonumber\\ &\xrightarrow{\text{decoherence}}\begin{bmatrix} \alpha\alpha^* & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \beta\beta^* \end{bmatrix}\nonumber \end{align}

The remaining terms along the diagonal represent a classical probabilistic mixture of classical correlations that would result from performing a measurement (using the preferred apparatus observable, see Section VII for details) on the now incoherent state. The off-diagonal terms which were destroyed in the decoherence process represent quantum correlations. Another important observation is that the resulting density operator $\rho_{\mathcal{SD}}$ cannot be decomposed into a superposition, in contrast to the state of the system before decoherence. This result then provides a resolution to the measurement problem since a state in superposition that undergoes decoherence has been shown to lose its superposition and instead exhibit only classical properties.

## VI. Quantum Coin Flip

We now continue our analysis using an example of a quantum coin toss. The goal of this section is to illustrate how the incoherent state resulting from decoherence exhibits classical properties familiar to our experience as observers. For this discussion let us represent the top facing and bottom facing sides of the coin as two separate qubits each living in their own Hilbert space, $\mathscr{H}_{Top}$ and $\mathscr{H}_{Bottom}$ respectively. For simplicity, we will use the computational basis as the preferred basis,

\begin{align} \ket{H_{Top}}&=\ket{H_{Bottom}}=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \ket{T_{Top}}=\ket{T_{Bottom}}=\begin{bmatrix} 0 \\ 1 \end{bmatrix}\nonumber \end{align}

Therefore, the corresponding preferred apparatus observable is given by the Pauli-$z$ observable,

\begin{align} \sigma_z&= (1)\begin{bmatrix} 1 \\ 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \end{bmatrix} +(-1)\begin{bmatrix} 0 \\ 1 \end{bmatrix} \begin{bmatrix} 0 & 1 \end{bmatrix}\nonumber\\ &=(1)\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} +(-1)\begin{bmatrix} 0 & 0\\ 0 & 1 \end{bmatrix}\nonumber\\ &=\alpha_{+1}P_{+1}+\alpha_{-1}P_{-1}\nonumber \end{align}

We will construct our coin such that it begins in a superposition given by,

\begin{align} \ket{\Psi_{coin}}&\in\mathscr{H}_{Top}\otimes\mathscr{H}_{Bottom}\nonumber\newline \ket{\Psi_{coin}}&=\frac{1}{\sqrt{2}}(\ket{H_{Top}}\ket{T_{Bottom}}+\ket{T_{Top}}\ket{H_{Bottom}})\nonumber \end{align}

and in corresponding density operator form,

\begin{align} \rho_{coin}&=\ket{\Psi_{coin}}\bra{\Psi_{coin}}\nonumber\\ &=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0\\ \end{bmatrix}\nonumber\\ &\xrightarrow{\text{decoherence}}\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0\\ \end{bmatrix}\nonumber\\ &=\frac{1}{2}(\ket{H_{Top}}\bra{H_{Top}}\ket{T_{Bottom}}\bra{T_{Bottom}}\nonumber\\ &\;\;\;\;+\ket{T_{Top}}\bra{T_{Top}}\ket{H_{Bottom}}\bra{H_{Bottom}})\nonumber\\ &=\rho^\prime_{coin}\nonumber \end{align}

Now, let us consider a measurement of this system using the von Neumann scheme explained in Section II. Just as in the case of a real coin flip, the top facing side of the coin will be the detector for the state of the bottom facing side. In this example, $\mathscr{H}_{Top}$ is analogous to $\mathscr{H}_{\mathcal{D}}$ and $\mathscr{H}_{Bottom}$ is analogous to $\mathscr{H}_{\mathcal{S}}$. With the measurement scheme established, we can now define measurement operators using the preferred apparatus observable: $(P_{+1}\otimes I)$ and $(P_{-1}\otimes I)$. Now consider the two possible outcomes, either we observe heads on top (outcome $m=+1$) or tails on top (outcome $m=-1$):

Probability of outcome:

\begin{align} p(\alpha_{+1})&=Tr((P_{+1}\otimes I)^\dagger(P_{+1}\otimes I)\rho^\prime_{coin})\nonumber \\ &=Tr(\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & \frac{1}{2} & 0 & 0\\ 0 & 0 & \frac{1}{2} & 0\\ 0 & 0 & 0 & 0 \end{bmatrix})\nonumber\\ &=Tr(\begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & \frac{1}{2} & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix})\nonumber\\ &=\frac{1}{2}\nonumber \end{align}

State after collapse:

\begin{align} \rho^\prime_{coin}&=\frac{(P_{+1}\otimes I)\rho_{coin}(P_{+1}\otimes I)^\dagger}{p(\alpha_{+1})}\nonumber\\ &=(\frac{1}{\frac{1}{2}})\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & \frac{1}{2} & 0 & 0\\ 0 & 0 & \frac{1}{2} & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}\nonumber\\ &=(2)\begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & \frac{1}{2} & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}\nonumber\\ &=\begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}\nonumber\\ &=\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \otimes \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\nonumber\\ &=\begin{bmatrix} 1\\ 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \end{bmatrix} \otimes \begin{bmatrix} 0 \\ 1 \end{bmatrix} \begin{bmatrix} 0 & 1 \end{bmatrix}\nonumber\\ &=\ket{H_{Top}}\bra{H_{Top}} \otimes \ket{T_{Bottom}}\bra{T_{Bottom}}\nonumber\\ &=(\ket{H_{Top}}\ket{T_{Bottom}})(\bra{H_{Top}}\bra{T_{Bottom}})\nonumber \end{align}
2. Case: Tails on top

Probability of outcome:

\begin{align} p(\alpha_{-1})&=Tr((P_{-1}\otimes I)^\dagger(P_{-1}\otimes I)\rho^\prime_{coin})\nonumber\\ &=Tr(\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix})\nonumber\\ &=Tr(\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix})\nonumber\\ &=\frac{1}{2}\nonumber \end{align}

State after collapse:

\begin{align} \rho^\prime_{coin}&=\frac{(P_{-1}\otimes I)\rho_{coin}(P_{-1}\otimes I)^\dagger}{p(\alpha_{-1})} & \nonumber\\ &=(\frac{1}{\frac{1}{2}})\begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & \frac{1}{2} & 0 & 0\\ 0 & 0 & \frac{1}{2} & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} & \nonumber\\ &=(2)\begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & \frac{1}{2} & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} & \nonumber\\ &=\begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} & \nonumber\\ &=\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \otimes \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} & \nonumber\\ &=\begin{bmatrix} 0\\ 1 \end{bmatrix} \begin{bmatrix} 0 & 1 \end{bmatrix} \otimes \begin{bmatrix} 1 \\ 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \end{bmatrix} & \nonumber\\ &=\ket{T_{Top}}\bra{T_{Top}} \otimes \ket{H_{Bottom}}\bra{H_{Bottom}} & \nonumber\\ &=(\ket{T_{Top}}\ket{H_{Bottom}})(\bra{T_{Top}}\bra{H_{Bottom}}) & \nonumber \end{align}

This quantum coin example illustrates the key result of environment-induced decoherence; a quantum state initially in a superposition undergoes decoherence and loses its quantum correlations resulting in a purely classical state. In this particular example, the resulting classical state has a 50% probability of being observed with a heads on top and a 50% probability of being observed with a tails on top. In addition, for each of these possible scenarios we know with certainty that the bottom side of the coin will show the opposite engraving (if top is heads then bottom is tails, if top is tails then bottom is heads). Thus, the classical properties our intuition expects have emerged from a quantum system following the decoherence process.

## VII. Environment-Induced Superselection

We now present a very brief summary of environment-induced superselection. This theory attempts to explain how the preferred basis is chosen for a particular apparatus-environment system. Observe that the system will remain stable in a preferred basis state through the decoherence process. With this observation, superselection defines a criterion for stability that the projection operators corresponding to system-apparatus correlations should commute with the apparatus-environment Hamiltonian, $[P_i,H_{\mathcal{AE}}]=0$. The projection operators satisfying this criterion then form what is referred to as the preferred apparatus observable, $\mathcal{O}_\mathcal{A}=\sum_i\alpha_i P_i$.

## VIII. Conclusion

In this report we have summarized the key elements of quantum decoherence. First, we introduced the von Neumann measurement scheme, a useful tool for considering effects of quantum entanglement. Next we explained how the measurement and preferred basis problems relate quantum properties like superposition to our classical experiences. We then summarized the conditions and consequences of environment-induced decoherence and interpreted the results through a quantum coin example. Finally, we briefly discussed the theory of environment-induced superselection and how a preferred basis is chosen. This concludes our analysis and summary of quantum decoherence as presented in $[1]$ $[2]$ $[3]$ $[4]$.

## References

$[1]$
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