17. Calculate quantum measurement and dephasing
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Work in \(\mathbb C^2\) with its usual inner product. The normalized state
\[\psi=\begin{pmatrix}3/5\\4i/5\end{pmatrix}
\]
has squared norm \(9/25+16/25=1\). This is a finite example for reading Paper VII.
Form the density matrix
Take the conjugate transpose \(\psi^{\ast}=(3/5,-4i/5)\). Then
\[\rho=\psi\psi^{\ast}
=\begin{pmatrix}9/25&-12i/25\\12i/25&16/25\end{pmatrix}.
\]
It is Hermitian and has trace one. Also \(\rho^2=\psi(\psi^{\ast}\psi)\psi^{\ast}=\rho\), so its purity \(\mathrm{tr}(\rho^2)\) is one. Positivity follows from \(v^{\ast}\rho v=|\psi^{\ast}v|^2\ge0\).
Measure in the coordinate basis
Use \(P_0=\mathrm{diag}(1,0)\) and \(P_1=\mathrm{diag}(0,1)\). The outcome probabilities are
\[p_0=\mathrm{tr}(P_0\rho)=9/25,\qquad
p_1=\mathrm{tr}(P_1\rho)=16/25.
\]
If the outcome is recorded, the corresponding normalized density matrices are \(P_0\) and \(P_1\). If the outcome is forgotten, the state is
\[\Phi(\rho)=P_0\rho P_0+P_1\rho P_1
=\begin{pmatrix}9/25&0\\0&16/25\end{pmatrix}.
\]
The two projectors are Kraus operators satisfying \(P_0^{\ast}P_0+P_1^{\ast}P_1=I\). The channel preserves trace. Its output purity is
\[(9/25)^2+(16/25)^2=337/625<1.
\]
See what a relative phase changes
The unitary \(U=\mathrm{diag}(1,-i)\) sends \(\psi\) to \(\phi=(3/5,4/5)^{\mathsf T}\). These two states have the same coordinate-basis probabilities, but they need not agree in another basis. Apply the Hadamard matrix
\[H=\frac1{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}.
\]
For \(H\psi\), the probabilities are \(|3+4i|^2/50=1/2\) and \(|3-4i|^2/50=1/2\). For \(H\phi\), they are \(7^2/50=49/50\) and \((-1)^2/50=1/50\). Relative phase therefore matters even when the first measurement probabilities agree.
Try it
Read the illustrated measurement chapter · Read the Kraus-channel chapter and damping figure
Continue with Quantum phase, measurement, and information loss for three-setting tomography, partial dephasing, coherent records, packet refinement, and a standalone exact Python program.
Repeat the probability and dephasing-purity calculations for \(\chi=(4/5,3/5)^{\mathsf T}\).
Answer: Coordinate probabilities are \(16/25\) and \(9/25\). The dephased purity is again \(337/625\). Applying \(H\) gives probabilities \(49/50\) and \(1/50\). The calculation concerns this finite system. Paper VII’s continuum targets require their separately stated constructions.
Papers, proofs, Lean, and code
Terms in this example: Complex conjugate · Adjoint matrix · Unitary matrix · Quantum amplitude · Density matrix · Matrix trace · Born probability · Purity of a density matrix · Dephasing · Kraus operators · Hermitian matrix · Positive semidefinite matrix · Coherence in a chosen basis
This introductory calculation covers one two-dimensional state, a unitary phase change and a measurement channel. Paper VII develops the finite quantum setting and separate continuum research targets. Quantum and Circuits provides a related route through arithmetic realizations and symmetry; the extended demonstration also reproduces its modulus-12 phase example. These subject routes add no formal claim correspondence.
Paper VII: Quantum Seed Calculus
Read · Written proofs · Lean coverage · Code and setup · Evidence
Quantum and Circuits
Read · Written proofs · Lean coverage · Code and setup · Evidence
Source passages:
Related library guide and code examples. The guide covers related constructions and their software interfaces.
Python arithmetic check for this example: quantum_measurement(). This checks the displayed rational, modular or finite-set calculations exactly.
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