In this article, we first recall the basic postulates of quantum computing. Then we mainly focus on the mathematical structures behind it.

Postulates of Quantum Mechanics

Postulate 1: Associated to any isolated physical system is a complex Hilbert space (a.k.a. state space of the system). The system is completely described by its state vector, which is a unit vector in the system’s state space.

Quantum mechanics does not tell us what the state space is for a given physical system, nor does the state vector. Figuring that out for a specific system is a difficult problem for which physicists have developed many intricate and beautiful rules, for example quantum electrodynamics.

The simplest quantum mechanical system is the qubit. A qubit has a two-dimensional state space. Suppose $\ket{0}$ and $\ket{1}$ form an orthonormal basis for that state space

Quantum mechanics does not tell us which unitary operators $U$ describe real-world quantum dynamics. In the case of single qubits, it turns out that any unitary operator at all can be realized in realistic systems.

A more refined version of this postulate can be given which describes the evolution of a quantum system in continuous time. From this more refined postulate we will recover Postulate 2.

Postulate 2’: The time evolution of the state of a closed quantum system is described by the Schrödinger equation,

\[i\hbar \frac{\mathrm{d}\ket{\psi}}{\mathrm{d}t}=H\ket{\psi},\]

where $H$ is a fixed Hermitian operator known as the Hamiltonian of the closed system.

When experimentists observe the system, it becomes no longer closed.

Posulate 3: Quantum measurements are described by a collection ${M_m}$ of measurement operators. These are operators acting on the state space of the system being measured. The index m refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is $\ket{\psi}$ immediately before the measurement then the probability that result $m$ occurs is given by

\[p(m)=\bra{\psi}M_m^\dagger M_m\ket{\psi},\]

and the state of the system after the measurement is

\[\frac{M_m\ket{\psi}}{\sqrt{\bra{\psi}M_m^\dagger M_m\ket{\psi}}}.\]

The measurement operators satisfy the completeness equation,

\[\sum_m M_m^\dagger M_m =I.\]

Postulate 4: The state space of a composite physical system is the tensor product of the state spaces of the component physical systems.

Density Matrix

The density matrix formulation is mathematically equivalent to the state vector appraoch, but it provides a convenient tool for describing quantum systems whose state is not completely known.

Suppose a quantum system is in one of a number of states $\ket{\psi_i}$, where $i$ is an index, with respective probabilities $p_i$. We shall call ${ p_i, \ket{\psi_i} }$ an ensemble of pure states. The density operator for the system is defined by the equation:

\[\rho \equiv \sum_i p_i\ket{\psi_i}\bra{\psi_i}.\]

Reformulation of Posulate 2: $\rho \mapsto U\rho U^\dagger$.

Reformulation of Posulate 3: $p(m)=\tr(M_m\rho M_m^\dagger)$, and the density matrix after measurement is

\[\frac{M_m\rho M_m^\dagger}{\tr(M_m\rho M_m^\dagger)}\]

Characterization of density matrices

An operator $\rho$ is the density operator associated to some ensemble ${ p_i, \ket{\psi_i} }$ if and only if

(i) (Trace condition) $\rho$ has trace equal to one.

(ii) (Positive condition) $\rho$ is a positive operator.

When do two sets of ensemble generate the same density matrix?

We say the set $\ket{\tilde{\psi}_i}$ generates the operator $\rho\equiv \sum_i\ket{\tilde{\psi}_i}\bra{\tilde{\psi}_i}$.

The sets $\ket{\tilde{\psi}_i}$ and $\ket{\tilde{\varphi}_i}$ generate the same density matrix if and only if

\[\ket{\tilde{\varphi}_i}=\sum_j u_{ij}\ket{\tilde{\psi}_i},\]

where $u_{ij}$ is a unitary matrix of complex numbers.

Composite Systems

(Schmidt decomposition) For every vector $\ket{\psi}\in \cH_A\otimes\cH_B$, there exist orthonormal basis $\{e_j\in \cH_A\}$ and $\{f_j\}\in\cH_B$ such that

\[\ket{\psi}=\sum_{j=1}^d\lambda_j\ket{e_j}\otimes\ket{f_j}\]

This is easily proved by an application of singular value decomposition.

(Tricks with maximal entanglement) If all Schmidt coefficients of a pure state $\phi$ are $\lambda_j=\frac{1}{d}$, then $\phi$ is called maximally entangled of dimension $d$. Every maximally entanged state is of the form $\phi=(\mathbb 1\otimes U)\ket{\Omega}$ where $U$ is any unitary and

\[\ket{\Omega}=\frac{1}{\sqrt{d}}\sum_{j=1}^d \ket{j}\otimes\ket{j}.\]

Questions

Understand Choi’s matrix paper and Quantum Process Tomography. Lindblad’s paper also.

Learn something from Moyang Cheng, Yuchen Guo, Hanyu Xue, Yuxiang Wang.