# Quantum Information and Quantum Noise

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The term quantum information is really a synonym of the term quantum state, only viewed at a different angle. If a qubit has state

$|\psi\rangle =\alpha|0\rangle + \beta|1\rangle$

then the complex numbers $\alpha$ and $\beta$ are (up to a global phase) the quantum information stored in the qubit; instead of saying “qubit has state $|\psi\rangle$“, we can say “qubit store information $|\psi\rangle$

If we have a single qubit, we can’t pull down quantum information from the qubit into our classical world. We need many qubits storing identical information to measure $\alpha$ and $\beta$ with some precision; the more precision we want, the more qubits we need. We can’t also obtain $\alpha$ and $\beta$ by measuring in a single basis only, we need to measure in two different bases at least.

Pure states

$|\psi\rangle =\alpha|0\rangle + \beta|1\rangle$

are not the most general qubit’s states. The most general states are called mixed states and are described by density matrices. Density matrix $\rho$ of a pure state $|\psi\rangle$ is

$\rho =|\psi\rangle\langle\psi|=\begin{pmatrix}\alpha \\ \beta\end{pmatrix}(\alpha^* \beta^*)=\begin{pmatrix} |\alpha|^2 & \alpha\beta^* \\ \alpha^*\beta &|\beta|^2 \end{pmatrix}$

A valid density matrix must be Hermitian, positive semidefinite and have trace 1; vice versa, any Hermitian and positive semidefinite matrix with trace 1 is a valid density matrix.

An example of a density matrix of a non-pure state:

$\rho =p_0|0\rangle\langle 0|+p_1|1\rangle\langle 1|=\begin{pmatrix} p_0 & 0 \\ 0 &p_1 \end{pmatrix}$

where $p_0$ and $p_1$ are real, $p_0\geqslant 0$, $p_1\geqslant 0$, and $p_0 + p_1 = 1$

Non-pure states are also called noisy states. In the classical data processing noise is always bad and we should always get rid of the noise to obtain clean data. As we will see soon, the quantum noise is more interesting.

What does it mean that a qubit has mixed state

$\rho =\begin{pmatrix} p_0 & 0 \\ 0 &p_1 \end{pmatrix}$

Does it mean that a qubit really has a pure state $|0\rangle$ or $|1\rangle$, it just happened that we don’t know it exactly and model our incomplete knowledge by probabilities $p_0$ and $p_1$ ?

Well, this is subtle. It is possible that a qubit has a pure state that we don’t know exactly, but it is also possible that a qubit has no pure state.

What is important to understand, the above said is not some philosophy. The difference between the two cases has mathematical consequences in quantum mechanics, and in the end of the day the difference can be (statistically) measured.

Let us consider two-qubit EPR state

$|\psi_{1}\rangle =\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$

The density matrix of the state is

$\rho_{1} =\frac{1}{2}(|00\rangle + |11\rangle)(\langle 00| + \langle 11|)=\frac{1}{2}\begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{pmatrix}$

Each qubit in the pair the has probability $1/2$ of being in state $|0\rangle$ or state $|1\rangle$.

We can construct mixed state with the same property:

$\rho_{2} =\frac{1}{2}(|00\rangle\langle 00| + |11\rangle\langle 11|)=\frac{1}{2}\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$

In both cases the individual qubits have identical noisy states (only the two-qubit states are different). It looks like the EPR state and the second state are statistically identical, but John Bell using clever argument has shown that they are not: EPR state violates so-called Bell’s inequalities while the second state does not.

It is funny that the Bell’s discovery happened about 30 years after the related questions were raised in the famous EPR paper by Einstein himself, and all prominent physicists of the time were aware of the EPR paper; the discovery has waited 30 years for John Bell.

It is common knowledge today that the density matrix formalism mathematically captures physical difference of the states: two states with the same density matrix are physically indistinguishable, and two states with different density matrices are physically distinguishable; it seems like nobody understood this before the Bell’s discovery.

Another term to discuss quantum noise is coherence (the term coherence may have different meanings in physics, be aware). If an initially pure qubit’s state evolves into a noisy state, we say that the qubit has lost coherence. But there are different ways to loose coherence. The coherence of an individual qubit in a multiqubit system may leak into other qubits of the system so that the whole multiqubit system preserves coherence. This is controllable and reversible loss of coherence. If the multiqubit system is quantum computer, this process is an important part of quantum computation. In the quantum algorithms the individual qubits loose coherence at intermediate step and restore coherence (with high probability at least) in the end, before the final measurement.

The main problem with building quantum computers is that coherence uncontrollably leaks into environment, and the whole multiqubit system looses coherence; since we can’t control environment on the quantum level, the loss of coherence is irreversible. This process introduces really bad kind of quantum noise which destroys quantum computation.