Postulates of Quantum Mechanics

Alexei Gilchrist

Quantum mechanics can be encapsulated by a set of postulates. In this set we will take the Hilbert space \(\mathcal{H}\) as the primary object.

Quantum postulates

The core of quantum theory may be phrased as a series of postulates. There are different approaches that one could take, and in any approach there are aesthetic choices about what could be considered the ‘simplest’ or most ‘fundamental’ postulates. Below is a fairly standard set when taking the Hilbert space as the primary object.

1 Representing states

The following postulates capture how states are represented in quantum mechanics, how multiple systems are combined and the joint state is expressed, and also the converse of ignoring or discarding a system.

Pure States

The states of an isolated physical system are completely described by norm-1 vectors in a Hilbert space.

The following postulate generalises the previous one by allowing for a statistical description of the physical state. It turns out that this extended description is necessary once discarding of systems and measurements are taken into account.

Mixed States

The states of an isolated physical system are described by trace-1 positive semi-definite linear operators called density operators.

If we have two systems described by either vectors or density matrices, we can describe the joint state using the tensor product.

Tensor product

Given the states of component physical systems, the state of the composite system is described by the tensor products of the states of the component systems.

\[\begin{align*}|\psi_a\rangle_a\otimes|\psi_b\rangle_b &= |\psi\rangle_{ab} \\ \rho_a\otimes\rho_b &= \rho_{ab}\end{align*}\]

Finally, the inverse of combining systems is discarding some of them, which is effected by taking the partial trace over the discarded systems.

Partial trace

Discarding a subsystem corresponds to taking the partial-trace over the discarded systems of the density operator representing the joint state of the systems.

\[\begin{align*}\rho_a &= \mathrm{Tr}_b(\rho_{ab})\\ \rho_b &= \mathrm{Tr}_a(\rho_{ab})\end{align*}\]

In general, when taking the partial trace the resulting state described by the reduced density operator will be mixed.

2 Evolution of states

Unitary evolution

The evolution of an isolated quantum system is described by a unitary transformation. So that

\[|\psi(t_1)\rangle = U(t_1,t_0) |\psi(t_0)\rangle\]
or equivalently for density matrices
\[\rho(t_1) = U(t_1,t_0)\rho(t_0)U(t_1,t_0)^\dagger\]

Equivalently, we can say

Evolution Equations

The time evolution of the state vector obeys the Schrödinger equation:

\[i\hbar \frac{d}{dt}|\psi(t)\rangle = H |\psi(t)\rangle\]
Similarly the time evolution of the density matrix obeys the von Neumann equation
\[i\hbar \frac{d}{dt}\rho(t) = [H, \rho(t)]\]

CPTP maps

The evolution of a closed quantum system can be described by a completely positive trace preserving (CPTP) map of the form

\[\mathcal{E}(\rho) = \sum_j K_j\rho K_j^\dagger.\]

The map takes the system from state \(\rho\) to state \(\mathcal{E}(\rho)\). The operators \(K_j\) satisfy the condition \(\sum_j K_j^\dagger K_j=I\).

3 Measurements

In contrast to classical physics, the mechanism of measurements is explicitly stated as postulates in quantum mechanics.

Observables

Observable quantities are represented by Hermitian operators as they have real eigenvalues. The eigenvalues are the possible measurement results. Say the spectral decomposition a Hermitian operator grouped by distinct eigenvalues is

\[ A = \sum_j a_j \mathbb{P}_j\]
where \(\mathbb{P}_j\) is the projector onto the eigenspace of \(a_j\). Then the probability of getting result \(a_j\) is
\[ p(a_j) = \mathrm{Tr}( \mathbb{P}_j\, \rho\, \mathbb{P}_j^\dagger ) = \mathrm{Tr}( \mathbb{P}_j\, \rho) \]
and if the result \(a_j\) is obtained, the system is left in the state
\[ \rho_{j} = \frac{ \mathbb{P}_j\, \rho\, \mathbb{P}_j^\dagger }{ p(m_j) } = \frac{ \mathbb{P}_j\, \rho\, \mathbb{P}_j }{ p(m_j) } \]

The intermediate forms in the above equations are to emphasise the connection with generalised measurements introduced below. For the post-measurement state, the fact that projectors are Hermitian was used, and for the probability of measurement, the cyclic property of the trace was also used.

The following is a specialisation of the Observables postulate

Born Rule

If the state of some system is \(|\psi\rangle\), the probability of finding it in state \(|\phi\rangle\) is \(|\langle \phi| \psi\rangle|^2\).

When we combine tensor products and partial traces together with observables we arrive at the most general formulation of a quantum measurement

Generalised measurements

Measurement results \(m_j\) are associated with operators \(M_j\) such that \(\sum_j M_j^\dagger M_j = I\).

The probability of getting result \(m_j\) given the state \(\rho\) is

\[ p(m_j) = \mathrm{Tr}(M_j\, \rho\, M_j^\dagger)\]

After getting result \(m_j\) the post-measurement state is

\[ \rho_{j} = \frac{ M_j\, \rho\, M_j^\dagger }{ p(m_j) } \]

The condition on the measurement operators ensures the completeness of the probability:

\[\begin{align*} \sum_j p(m_j) &= \sum_j \mathrm{Tr}(M_j\, \rho\, M_j^\dagger) = \sum_j \mathrm{Tr}( M_j^\dagger M_j\, \rho) \\ &= \mathrm{Tr}\left(\sum_j M_j^\dagger M_j\, \rho\right) = \mathrm{Tr}(\rho) = 1\end{align*}\]

This last version is a specialisation of generalised measurements when we are not interested in the post-measurement state, only in the probabilities of results.

POVMs

Each measurement result \(m_j\) is associated with a positive semi-definite operator \(E_j\), such that \(\sum_j E_j = I\) and the probability of getting the result \(m_j\) is

\[ p(m_j) = \mathrm{Tr}(E_j\, \rho) \]
The operators \(E_j\) are called “Effects”.

POVM stands for positive operator valued measure.

Using the cyclic property of the trace, we can identify the POVM effects with the measurement operators:

\[ p(m_j) = \mathrm{Tr}(M_j\, \rho\, M_j^\dagger) = \mathrm{Tr}(M_j^\dagger M_j\, \rho) = \mathrm{Tr}(E_j\, \rho)\]
i.e. \(E_j = M_j^\dagger M_j\), and since \(A^\dagger A\ge 0\) for any operator \(A\), we have \(E_j \ge 0\).

© Copyright 2021 Alexei Gilchrist