Qubits and Measurement
Qubit is the quantum system that we want to study and is in some quantum state. Observable is the physical quantity 物理量 that we want to query about and is represented by Hermitian matrices. Measurement is the actual action that we perform a query and read out a classic outcome.
Table of Contents
1. Qubits
The unit of computation is qubit (short for quantum bits). A qubit in a quantum computer consists of two perfectly distinguishable statuses of a (usually microscopic) quantum system. We usually use a unit vector to refer to the quantum state.
Notations in Quantum Computing
In quantum computing, vectors are denoted in bracket:
- \( \ket{a} \) means a column vector \(\vec{a} = \begin{pmatrix}x\\y\end{pmatrix}\)
- \( \bra{a} \) means the adjoint of row vector \( \vec{a} \), i.e., take conjugate1 of each element \( a^{\dag} = \begin{pmatrix}x^{\star} & y^{\star}\end{pmatrix} \)
- \( \braket{a|b} \) means the dot product of these 2 vectors.
- \( \ket{a}\bra{b} \) means a 2-by-2 matrix.
There exist 2 basis vectors: ground state \( \ket{0}=\begin{pmatrix}1\\0\end{pmatrix} \) and excited state \( \ket{1}=\begin{pmatrix}0\\1\end{pmatrix} \).
A qubit can be in other states than \( \ket{0} \) and \( \ket{1} \). In general, an arbitrary qubit state is
\[ \ket{\psi} = \alpha\ket{0}+\beta\ket{1} = \begin{pmatrix}\alpha\\\beta\end{pmatrix}, \text{ where } \alpha,\beta \in \mathbb{C}, |\alpha|^{2}+|\beta|^{2}=1 \]
The coefficients \(\alpha,\beta\) are called amplitudes of \( \ket{\psi} \).
Some Important Conclusions
- \( \braket{\psi|\phi}^{\star} = \braket{\phi|\psi} \). This implies \( |\braket{\psi|\phi}|^{2} = \braket{\psi|\phi}\braket{\phi|\psi} \)
2. Measurement of Qubits
The outcome of a qubit cannot be read directly. We must perform a measurement.
- Measurement
- Each measurement can only produce one outcome \( 0 \) or \( 1 \). Although a single measurement can only have one outcome, different measurement may have different outcomes. But, we can statistically figure out the state of the qubit after a large amount of measurements, that is the Born’s Rule.
Born’s Rule (Basic)
The probability of getting the outcome \(x\) when measuring any quantum state \( \ket{\psi} \) in the computational basis is \[ \mathbb{P}(x) = | \braket{x|\psi} |^{2}, \quad x=0,1 \]
\(\ket{0}\) means that we can only get \(0\) when we measure it; while \(\ket{1}\) means we can only get \(1\) when measuring it.
2.1. Bloch Sphere, Global Phase, and Relative Phase
Any qubit can be visualized as a point \( (\theta, \phi) \) on a 2D sphere (named Bloch sphere).
Induction of Bloch Sphere
2.2. Measurement in Any Arbitrary Basis
- Orthonormal qubit basis
- Is a collection \( \{ \ket{\phi_{0}}, \ket{\phi_{1}} \} \) of 2 orthogonal states
Measuring in an arbitrary basis means the action to check whether the state is in \(\ket{\phi_{0}}\) or \(\ket{\phi_{1}}\).
Some important basis:
- Hadamard basis (x-basis). \(\ket{\plusmn} = \frac{1}{\sqrt{2}}(\ket{0} \plusmn \ket{1})\)
- y-basis. \(\ket{\plusmn i} = \frac{1}{\sqrt{2}}(\ket{0} \plusmn i\ket{1})\)
Born’s Rule (Generalized)
The probability of getting outcome \(x\) when measuring any quantum state \( \ket{\psi} \) in basis \( \{ \ket{\phi_{0}}, \ket{\phi_{1}} \} \) is
\[ P(x) = |\braket{\phi_{x}|\psi}|^{2} \]
for \(x=0,1\)
2.3. Measurement of an Observable
Suppose we have already know all the possible states of a qubit it can collapse into, say \(\{ \ket{\phi_{i}} \}\). If we measure the qubit, the qubit must collapse into one of these states and returns a real value (readout outcomes) \(h_{i}\).
Note these possible states must be orthonormal. Therefore, we can use a single matrix to unify readout outcomes and corresponding states (orthonormal basis), i.e., the Hermitian2 matrix:
\[ H = \sum_{i} h_{i}\ket{\phi_{i}}\bra{\phi_{i}} \]
Moreover, given a certain initial state \(\psi\), we can compute the expectation of outcomes \( \sum_{i} h_{i} p_{i} \), where \( p_{i} = |\braket{\psi|\phi_{i}}|^{2} \). Indeed,
\[ \sum_{i} h_{i} p_{i} = \braket{\psi|H|\psi} \]
Take Schodinger’s Cat as an example, all possible states are \(\ket{\text{alive}}, \ket{\text{dead}}\).
Note: the observable only collapses into one state, here, no measurement is performed. The Hermitian matrix describes a mixture of states inside this observable.