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A pseudo-pure state is a density matrix of the formwhere $|\psi ⟩$ is a normalized pure state, $I$ is the identity matrix on a $d$-dimensional space and here $0\le p\le 1$. The term $I/d$ is the maximally mixed state. Under a unitary transformation $U$, this term stays fixed and the selected pure state becomes $U|\psi ⟩$. This is why liquid-state NMR quantum computing can follow pure-state-like transformations in part of a highly mixed ensemble.^[ref]

For a traceless matrix representing an observable $O$, direct evaluation of the trace givesThe identity contribution vanishes because $\mathrm{tr}O=0$. Thus $p$ scales the detectable expectation. It is not itself the purity: expanding ${\rho }^2$ and taking its trace yieldsFor example, for two qubits with $p=1/10$, the purity is $103/400$, close to the maximally mixed value $1/4$, although the selected pure state still controls the traceless signal.

The preparation penalty can also be derived explicitly. Take $n$ independent equal spin-one-half systems with one-spin density matrix $(I+\eta Z)/2$, where $Z$ is a Pauli matrix and $0\le \eta \le 1$ is their individual polarization. Their largest population is $a=((1+\eta )/2{)}^n$. A preparation that preserves this population and averages all other populations to equality has $d=2^n$ andIndeed the selected population of $\rho$ is $[1+(d-1)p]/d=a$. In the weak-polarization regime $n\eta \ll 1$, the binomial theorem gives $p\simeq n\eta /(2^n-1)$. If the energy separation is $\hslash \omega$, set $ϵ=\hslash \omega /(k_{B}T)$; a thermal two-level system has $\eta =\tanh (ϵ/2)\simeq ϵ/2$. This gives $p\simeq nϵ/[2(2^n-1)]$, with every parameter defined. Here $\hslash$ is the reduced Planck constant, $k_B$ the Boltzmann constant and $T$ the temperature. The exponential signal loss concerns this specified averaging preparation; it is not a universal bound on every initialization method.

The reduced Planck constant is $\hslash =h/(2\pi )$, with the same action units as the Planck constant. Since angular frequency is $\omega =2\pi \nu$, a photon's energy can be written $E=\hslash \omega$ instead of $h\nu$.

The reduced Planck constant occurs naturally in the Schrödinger equation, the canonical commutation relation $[X,P]=i\hslash I$, and the Heisenberg uncertainty principle $\Delta X\Delta P\ge \hslash /2$. Comparing a characteristic action with $\hslash$ helps assess a semiclassical approximation, although a large ratio alone does not guarantee that all quantum interference becomes unobservable.

The spatial translation operator is the unitary operator generated by momentum. For displacement $\mathbf{y}$,where $\hslash$ is the reduced Planck constant. In a position representation in quantum mechanics, $\mathbf{P}=-i\hslash \nabla$, so its action on a wavefunction isA localized wave packet is therefore displaced by $+\mathbf{y}$. A momentum eigenstate of eigenvalue $\mathbf{p}$ instead acquires the phase $e^{-i\mathbf{p}\cdot \mathbf{y}/\hslash }$.

The canonical commutation relation givesThus the expectation value of the position operator in the translated quantum state increases by $\mathbf{y}$.

For an operator-valued scalar field, the compatible translation convention isInfinitesimally this means $[P_j,\varphi (\mathbf{x})]=i\hslash {\partial }_j\varphi (\mathbf{x})$. The field's coordinate argument labels where a local observable is evaluated; $\mathbf{X}$ is the position operator of a particle. These meanings explain their differently ordered conjugation formulas. ^[ref]

The spin projection is the component of a particle's spin angular momentum along a specified spatial direction. For the $z$ direction, an eigenstate of the angular momentum operator $S_z$ has measured value $m_s\hslash$, where $m_s=-s,-s+1,\dots ,s$ and $\hslash$ is the reduced Planck constant. For spin one-half, the two possible results are $+\hslash /2$ and $-\hslash /2$.

The direction belongs to the measurement arrangement: a state with definite $S_z$ generally has uncertain $S_x$. A Stern-Gerlach experiment separates the corresponding components through a spatially varying magnetic field. This is a component measurement, rather than a claim that the full spin angular momentum points along one of two classical directions.