Single Qubit States and Measurements

1 Learning Goals

Describe qubits and quantum measurements using ket notation
Connection ket notation to physical intuition from cryptography
Analyze novel situations using kets

2 What is a Qubit

A qubit is a quantum version of a bit. (The polarization of a single photon can encode one qubit, so you have some experience with qubits, but there are many other types: electron energy level, nuclear magnetic spin, quantum dots…)

Some familiar qubit states:

Photon	ket notation	English
vertically polarized	\(\ket{0}\)	ket 0, 0 state
horizontally polarized	\(\ket{1}\)	ket 1, 1 state
right diag. polarized	\(\ket{+}=\frac{1}{\sqrt{2}}\ket{0}+\frac{1}{\sqrt{2}}\ket{1})\)	plus state
right diag. polarized	\(\ket{+}=\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})\)	minus state

We call

\(\{\ket{0}\ket{1}\}\) the “standard basis”
\(\{\ket{+}\ket{-}\}\) the “Hadamard basis”

Note

Why do we call it a “qubit”??
* Any state can be written with \(\ket{0}\) and \(\ket{1}\) (similar to how any bit can be represented with \(0\) or \(1\))
* Basic quantum measurement has two outcomes

3 Writing a Qubit with Kets

We can mathematically represent the generic state of a qubit as

\[\ket{\psi}=a_0\ket{0}+a_1\ket{1} \qquad \textrm{such that } a_0,a_1\in\mathbb{C}, \textrm{ and } |a_0|^2+|a_1|^2=1 \tag{1}\]

The left side of the equation is read as “ket psi”, “state psi” or just “psi.” This is like how we have a variable \(x\) in an algebra equation. It is just the name of some element of our system.
\(a_0\) and \(a_1\) are called amplitudes, and they are complex numbers.
\(|a_0|^2+|a_1|^2=1\) is called the normalization condition
Note this tells us that any qubit state can be written as a (linear) combination of the standard basis states \(\ket{0}\) and \(\ket{1}\)

Examples

\(\ket{0}=1\ket{0}+0\ket{1}\)
\(\ket{-}=\frac{1}{\sqrt{2}}\ket{0}+\frac{-1}{\sqrt{2}}\ket{1}\)
Photon polarized at an angle \(\theta\) from vertical: \(\cos(\theta)\ket{0}+\sin{\theta}\ket{1}\)

Definition 1 A qubit is in a superposition state if \(a_0\) and \(a_1\) are both non-zero. We say such a state is in a superposition of \(\ket{0}\) and \(\ket{1}\).

Vector Note

It is often convenient to think of a qubit state as a vector in \(\mathbb{C}^2\). To do this we write: \[\ket{\psi}=\left( \begin{array}{c} a_0\\ a_1 \end{array}\right) \] with the amplitudes as the elements of the vector.

While we will not use this representation in class, it is very common.

4 Bras

If \[ \ket{\psi}=a_0\ket{0}+a_1\ket{1}\] then we define the “bra of psi” as \[\bra{\psi}=a_0^*\bra{0}+a_1^*\bra{1} \]

where \(a_0^*\) and \(a_1^*\) are the complex conjugates of \(a_0\) and \(a_1\). And we say \(\bra{\psi}\) is “bra psi” or the “bra of psi” and \(\bra{0}\) is “bra 0” and \(\bra{1}\) is “bra 1.”

Taking the bra works the same way even if you don’t have a state written in terms of \(\ket{0}\) and \(\ket{1}\). For example the bra of \(a_0\ket{\psi}\) is \(a_0^*\bra{\psi}\).

Vector Note

In vector notation, bras are the conjugate transpose of the corresponding vector. So we can write \(\bra{\psi}\) in vector notation as e write bras as \[\bra{\psi}=\left( \begin{array}{cc} a_0^*, a_1^* \end{array}\right) \] with the complex conjugates of the amplitudes as the elements of the row vector.

5 Brakets/Inner Product

We can stick a bra and a ket together to make a braket, or inner product.

Basic rules:

Bras, kets, and numbers should be thought of as “multiplying” each other if there is no other symbol between them. Factoring and distributing and associativity work like expected for multiplication.
Brakets result in numbers/scalars
Brakets of standard basis standard basis states are always \(0\) or \(1\): \[\braket{0}{0}=1, \quad \braket{1}{1}=1, \quad \braket{0}{1}=0, \quad \braket{1}{0}=0 \tag{2}\]
Numbers (“scalars”) commute (can switch order) with bras and kets. For example: \[\left(\frac{1}{\sqrt{2}}\bra{0}\right)\left(\frac{1}{\sqrt{3}}\ket{0}\right)=\frac{1}{\sqrt{6}}\braket{0}{0}=\frac{1}{\sqrt{6}}\times 1= \frac{1}{\sqrt{6}}\]
Numbers (“scalars”) distribute over bras and kets. For example: \[\frac{1}{\sqrt{2}}\left(\bra{0}+\bra{1}\right)=\frac{1}{\sqrt{2}}\bra{0}+\frac{1}{\sqrt{2}}\bra{1} \]
Brakets distribute. For example: \[\left(\frac{1}{\sqrt{2}}\bra{0}+\frac{i}{\sqrt{2}}\bra{1}\right)\left(\frac{1}{\sqrt{3}}\ket{0}-\frac{2i}{\sqrt{3}}\ket{1}\right)= \frac{1}{\sqrt{6}}\braket{0}{0}+\frac{i}{\sqrt{6}}\braket{1}{0}+ \frac{2}{\sqrt{6}}\braket{0}{1}-\frac{i}{\sqrt{6}}\braket{1}{1} \]
Bras, kets do not in general commute. For example, \[\braket{\psi}{\phi}\neq \braket{\phi}{\psi}\] But it is true is that \[\braket{\psi}{\phi}= \left(\braket{\phi}{\psi}\right)^*\]

Definition 2 We say that two states \(\ket{\psi}\) and \(\ket{\phi}\) are orthogonal if \(\braket{\psi}{\phi}=0\). (Note \(\braket{\psi}{\phi}=0\) if and only if \(\braket{\phi}{\psi}=0\), so it doesn’t matter which order you take the inner product with.)

Vector Note

In the vector formalism, braket corresponds to matrix multiplication between a row vector and a column vector (the bra is the row vector and the ket is the column vector). If \[\ket{\psi}=\left( \begin{array}{c} a_0\\ a_1 \end{array}\right) \qquad \ket{\phi}=\left( \begin{array}{c} b_0\\ b_1 \end{array}\right) \qquad \] then \[ \braket{\phi}{\psi}=\left( \begin{array}{cc} b_0^*, b_1^* \end{array}\right)\left( \begin{array}{c} a_0\\ a_1 \end{array}\right)=b_0^*a_0+ b_1^*a_1. \] We see the braket/inner product results in a scalar output, as expected.

6 Qubit Measurement

A quantum qubit measurement is represented mathematically by an orthonormal set of kets: \(M=\{\ket{\phi_1},\ket{\phi_2}\}\). We often call \(M\) the measurement basis. This measurement is asking whether the qubit is in the state \(\ket{\phi_1}\) or \(\ket{\phi_2}.\)

Definition 3 We say that two states \(\ket{\psi}\) and \(\ket{\phi}\) are orthonormal if they are orthogonal (See Definition 2) and are both normalized. A ket \(\ket{\psi}\) is normalized if it obeys the normalization condition from Equation 1 (or equivalently, \(\braket{\psi}{\psi}=1\)).

If you measure a state \(\ket{\psi}\) with the measurement \(M=\{\ket{\phi_1},\ket{\phi_2}\}\), then the laws of quantum mechanics tell us that:

With probability \(|\braket{\phi_1}{\psi}|^2\), we get outcome \(\ket{\phi_1}\) (the qubit is in state \(\ket{\phi_1}\)), and the state \(\ket{\psi}\) collapses to \(\ket{\phi_1}\).
With probability \(|\braket{\phi_2}{\psi}|^2\), we get outcome \(\ket{\phi_2}\) (the qubit is in state \(\ket{\phi_2}\)), and the state \(\ket{\psi}\) collapses to \(\ket{\phi_2}\).

You can not control the outcome of a quantum measurement. The outcomes are determined probabilistically.

Example

Suppose a clockwise polarized photon encounters a right diagonally polarized filter. What is the probability that no photon exits the polarizer? (Clockwise and counter-clockwise polarized photons are used for 3D movies - one type of polarization for each eye.)

The photon’s polarization is represented by \(\frac{1}{\sqrt{2}}\ket{0}+\frac{i}{\sqrt{2}}\ket{1}\) in ket notation. The vertically polarized filter is “asking” is the photon right diagonally polarized (\(\ket{+}\)) or left diagonally polarized \((\ket{-})\), so it corresponds to the measurement \(M=\{\ket{+},\ket{-}\}\). If the photon collapses to a the left diagonally polarized state, it will get destroyed by the polarizer, so for this problem, we want to calculate the probability of outcome \(\ket{-}\).

First we need to determine \(\bra{-}\) \[ \ket{-}=\frac{1}{\sqrt{2}}\ket{0}+\frac{1}{\sqrt{2}}\ket{1}\rightarrow \textrm{ (bra) }\rightarrow \left(\frac{1}{\sqrt{2}}\right)^*\bra{0}+\left(\frac{1}{\sqrt{2}}\right)^*\bra{1}= \frac{1}{\sqrt{2}}\bra{0}+\frac{1}{\sqrt{2}}\bra{1}=\bra{-} \] Now we take the braket/inner product between \(\bra{-}\) and \(\frac{1}{\sqrt{2}}\ket{0}+\frac{i}{\sqrt{2}}\ket{1}\): \[ \begin{align} &\left(\frac{1}{\sqrt{2}}\bra{0}+\frac{1}{\sqrt{2}}\bra{1}\right)\left(\frac{1}{\sqrt{2}}\ket{0}+\frac{i}{\sqrt{2}}\ket{1}\right)&\\ &=\frac{1}{2}\braket{0}{0}+\frac{1}{2}\braket{1}{0}+\frac{i}{2}\braket{0}{1}+\frac{i}{2}\braket{1}{1}& \textrm{ distributing bra/ket inner product}\\ &=\frac{1}{2}+\frac{i}{2}& \textrm{ Using rule for inner product of s.b. states} \end{align}. \] Now we just need to take the absolute value squared to determine the probability. To take the absolute value squared, we use the trick that \(|a|^2=aa^*\): \[ \left|\frac{1}{2}+\frac{i}{2}\right|^2=\left(\frac{1}{2}+\frac{i}{2}\right)\left(\frac{1}{2}-\frac{i}{2}\right)=\frac{1}{4}+\frac{i}{4}-\frac{i}{4}+\frac{1}{4}=\frac{1}{2} \]

Thus there is a \(1/2\) probability that the measurement outcome is \(\ket{-}\), in which case the photon collapses to the state \(\ket{-}\) (left diagonally polarized) and then is blocked by the filter).

Group Exercises

If a photon with polarization \(\frac{1}{\sqrt{3}}\ket{0}+\sqrt{\frac{2}{3}}\ket{1}\) encounters a clockwise polarized filter (corresponding the measurement \(M=\{\frac{1}{\sqrt{2}}(\ket{0}+i\ket{1}),\frac{1}{\sqrt{2}}(\ket{0}-i\ket{1})\}\), what is the probability that a photon emerges, and what is its polarization.
If a photon with polarization \(\cos(30^\circ)\ket{0}+i\sin(30^\circ)\ket{1}\) encounters a vertically polarized filter followed by a single photon detector, what is the probability of a detection or no detection?
If we have a qubit state \(\ket{\psi}=a_0\ket{0}+a_1\ket{1}\), what is \(\braket{\psi}{\psi}\)?