Rank-2 Tensor · Quantum Mechanics
Tensors in Quantum Mechanics
Application to quantum computers, MRI scanners, and GPS
Advanced TheoryEverything you have seen in the demo —> eigenvalues, eigenvectors, the quadratic form, trace and determinant invariance, passive rotation —> reappears in quantum mechanics, but now the tensor describes quantum states and energy levels instead of a car's resistance. Let us walk through the most important applications, starting with the density matrix and a real quantum computing problem.
The Density Matrix —> Tensor in Quantum Form
The density matrix \(\rho\) is a rank-2 tensor that completely describes the quantum state of a system. For a two-level quantum system —> called a qubit, the basic unit of a quantum computer —> it is a \(2 \times 2\) matrix that looks exactly like your tensor \(T\):
\(T = \begin{pmatrix} a & c \\ c & b \end{pmatrix}\)
// The quantum density matrix:
\(\rho = \begin{pmatrix} \rho_{11} & \rho_{12} \\ \rho_{21} & \rho_{22} \end{pmatrix}\)
// What each entry means:
\(\rho_{11}\) = probability of finding qubit in state \(|0\rangle\)
\(\rho_{22}\) = probability of finding qubit in state \(|1\rangle\)
\(\rho_{12}, \rho_{21}\) = quantum coherences —> interference between states
// The trace invariant becomes a probability law:
\(\text{Tr}(\rho) = \rho_{11} + \rho_{22} = \)1 --> total probability always = 1
In our demo, trace invariance means \(a + b\) stays constant under rotation. In quantum mechanics, trace invariance of \(\rho\) means total probability is conserved —> the qubit must be found somewhere, so probabilities always sum to 1. Same mathematics, deeper physical meaning.
A Qubit on Quantum Computer
Consider a single qubit on quantum computer, initially prepared in the state \(|+\rangle\) —> an equal superposition of \(|0\rangle\) and \(|1\rangle\). After interacting with its environment for a short time \(t\), the qubit loses some coherence. Its density matrix becomes:
\(\rho_0 = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix}\)
// After decoherence time t —> coherence decays:
\(\rho(t) = \begin{pmatrix} 0.5 & 0.5\,e^{-t/T_2} \\ 0.5\,e^{-t/T_2} & 0.5 \end{pmatrix}\)
// Where T2 is the decoherence time (~100 microseconds)
// At t = T2 (one decoherence time):
\(e^{-1} \approx 0.368\)
\(\rho(T_2) = \begin{pmatrix} 0.5 & 0.184 \\ 0.184 & 0.5 \end{pmatrix}\)
// At t >> T2 (fully decohered —> classical mixture):
\(\rho(\infty) = \begin{pmatrix} 0.5 & 0 \\ 0 & 0.5 \end{pmatrix}\) --> diagonal only, no coherence left
Now find the eigenvalues of \(\rho(T_2)\) —> using exactly formula from the demo with \(a = 0.5\), \(b = 0.5\), \(c = 0.184\):
\(\lambda^2 - \text{Tr}(\rho)\,\lambda + \det(\rho) = 0\)
// Trace and determinant:
\(\text{Tr}(\rho) = 0.5 + 0.5 = \)1.0
\(\det(\rho) = 0.5 \times 0.5 - 0.184^2 = 0.25 - 0.034 = \)0.216
// Discriminant:
\(\Delta = (a-b)^2 + 4c^2 = 0 + 4(0.184)^2 = \)0.135
\(\sqrt{\Delta} = \)0.368
// Eigenvalues:
\(\lambda_1 = \frac{1.0 + 0.368}{2} = \)0.684 --> dominant quantum state
\(\lambda_2 = \frac{1.0 - 0.368}{2} = \)0.316 --> secondary quantum state
// Purity of the qubit:
\(\text{Purity} = \text{Tr}(\rho^2) = \lambda_1^2 + \lambda_2^2\)
\(= 0.684^2 + 0.316^2 = 0.468 + 0.100 = \)0.568
// Pure state purity = 1.0, fully mixed = 0.5
// 0.568 means qubit is partially decohered
The Car Losing Its Shape
In our demo, the ellipse has a fixed shape —> its eigenvalues \(\lambda_1\) and \(\lambda_2\) never change. This represents a pure quantum state —> the qubit is perfectly coherent, like a car with a well-defined easy and hard push direction.
Decoherence is what happens when the environment randomly jostles the car from all directions. Over time the off-diagonal entry \(c\) decays to zero —> the coupling between the two directions vanishes. The ellipse slowly becomes a circle —> \(\lambda_1 = \lambda_2\) —> meaning the qubit has lost all directional preference. It no longer knows which quantum state it should be in.
A quantum computer must perform all its calculations before decoherence turns the ellipse into a circle. \(T_2 \approx 100\) microseconds is the time the ellipse stays elliptical. This is why quantum computing is so difficult —> you are racing against the clock before the tensor becomes isotropic.
Hamiltonian Eigenvalues Are Energy Levels
The Hamiltonian \(H\) is the rank-2 tensor whose eigenvalues are the allowed energy levels of a quantum system. The eigenvalue equation is:
\(H\,|\psi\rangle = E\,|\psi\rangle\)
// Exactly your tensor equation T.e = lambda.e
// but now lambda = E = energy level
// Example: two-level atom in a laser field
\(H = \begin{pmatrix} E_0 & \Omega/2 \\ \Omega/2 & E_1 \end{pmatrix}\)
// E0, E1 = bare energy levels (like a and b in your demo)
// Omega/2 = coupling from laser (like c in your demo)
// Energy eigenvalues using your formula:
\(E_{\pm} = \frac{(E_0 + E_1)}{2} \pm \frac{1}{2}\sqrt{(E_0 - E_1)^2 + \Omega^2}\)
// The splitting sqrt((E0-E1)^2 + Omega^2)
// is called the Rabi splitting —> directly measurable
The Rabi splitting \(\sqrt{(E_0-E_1)^2 + \Omega^2}\) is identical to your discriminant \(\sqrt{(a-b)^2 + 4c^2}\) — with \(E_0 \leftrightarrow a\), \(E_1 \leftrightarrow b\), and \(\Omega/2 \leftrightarrow c\). The laser coupling \(\Omega\) plays exactly the role of our slider \(c\) —> it tilts the principal axes and splits the energy levels apart.
Pauli Matrices —> Three Tensors for Spin
Quantum mechanical spin is described by three rank-2 tensors —> the Pauli matrices. Each one is a \(2 \times 2\) symmetric or Hermitian matrix, exactly like your tensor \(T\):
\(\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) \(\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\) \(\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)
// Eigenvalues of each Pauli matrix:
\(\lambda_1 = +1\) \(\lambda_2 = -1\) --> for ALL three matrices
// Physical meaning:
Measuring spin along ANY axis gives only +1/2 or -1/2
// Trace of each Pauli matrix:
\(\text{Tr}(\sigma_x) = \text{Tr}(\sigma_y) = \text{Tr}(\sigma_z) = \)0
// Determinant of each Pauli matrix:
\(\det(\sigma_x) = \det(\sigma_y) = \det(\sigma_z) = \)-1
MRI Scanners
Medical MRI flips hydrogen nuclear spins using radio waves. The Pauli matrix \(\sigma_x\) describes the spin-flip operation. Eigenvalues \(\pm 1\) correspond to spin-up and spin-down states —> the two states whose energy difference produces the MRI signal used to image tissue.
Quantum Gates
Every quantum logic gate in quantum computers is a \(2\times 2\) unitary tensor. The NOT gate is \(\sigma_x\), the phase gate uses \(\sigma_z\). Running a quantum algorithm means multiplying sequences of these rank-2 tensors —> matrix multiplication is the computation.
Electron Spin Resonance
ESR spectroscopy identifies free radicals in chemical and biological samples by measuring spin eigenvalue transitions. The resonance frequency directly gives the eigenvalue splitting \(\Delta E = 2\lambda\) of the spin tensor in a magnetic field.
LCD Screens
Every pixel in your phone screen manipulates the polarisation tensor of light —> a \(2\times 2\) Jones matrix with eigenvalues controlling which light passes. The display works by switching between different eigenvalue states of the polarisation tensor millions of times per second.
Hydrogen Spectral Lines are Tensor Eigenvalues
The hydrogen atom Hamiltonian is a rank-2 tensor in an infinite-dimensional function space. Its eigenvalues —> the energy levels —> are:
\(E_n = -\frac{13.6\,\text{eV}}{n^2}\) \(n = 1, 2, 3, \ldots\)
\(E_1 = \)-13.6 eV --> ground state (most stable)
\(E_2 = \)-3.4 eV --> first excited state
\(E_3 = \)-1.51 eV --> second excited state
// Photon emitted when electron drops from n=3 to n=2:
\(\Delta E = E_3 - E_2 = -1.51 - (-3.4) = \)1.89 eV
\(\lambda_{photon} = hc/\Delta E = \)656 nm --> red H-alpha line
// These eigenvalues are invariants:
// same whether atom is in Paris, on Mars, or near a black hole
When astronomers observe a galaxy billions of light years away, they identify hydrogen by its spectral lines at 656 nm, 486 nm, and 434 nm. These wavelengths are the eigenvalues of the hydrogen Hamiltonian tensor —> invariants that are identical everywhere in the universe. The same tensor invariance you verify in GeoGebra by dragging \(\theta\) operates across the entire cosmos.
Demo vs Quantum Mechanics —> Side by Side
| Demo | Quantum Mechanics | Real Life |
|---|---|---|
| Tensor T = [[a,c],[c,b]] | Density matrix \(\rho\) or Hamiltonian \(H\) | Qubit state in quantum computer |
| Eigenvalues \(\lambda_1, \lambda_2\) | Energy levels \(E_1, E_2\) | Hydrogen spectral lines —> 656 nm, 486 nm |
| Eigenvectors \(\hat{e}_1, \hat{e}_2\) | Quantum states \(|\psi_1\rangle, |\psi_2\rangle\) | Spin-up and spin-down in MRI |
| Trace = a + b (invariant) | \(\text{Tr}(\rho) = 1\) (probability conserved) | Qubit always found in some state |
| Determinant = ab - c² (invariant) | Purity \(\text{Tr}(\rho^2)\) conserved in closed system | Quantum information not lost |
| Slider c controls tilt | Laser coupling \(\Omega\) mixes states | Rabi oscillations in atomic clocks |
| Passive rotation —> ellipse unchanged | Change of basis —> physics unchanged | GPS works regardless of satellite orientation |
| c = 0 principal frame | Energy eigenbasis —> diagonal H | Stationary states of atoms |
| Ellipse becomes circle —> isotropic | Fully mixed state —> maximum decoherence | Qubit destroyed by environment |
The Ellipse Is Everywhere in Physics
In our demo, the ellipse \(ax^2 + 2cxy + by^2 = 1\) is the geometric face of the tensor. Its shape —> governed by \(\lambda_1\) and \(\lambda_2\) is invariant under rotation. This single geometric fact, dressed in different mathematical clothing, appears at every scale of physics.
The hydrogen atom has an ellipse in energy space —> its eigenvalues \(E_n\) are invariant across the universe. The qubit on quantum computer has an ellipse on the Bloch sphere —> decoherence shrinks it toward a circle. The MRI scanner measures which axis of the spin ellipse the nuclear spin points along. The GPS satellite carries a clock whose tick rate is governed by the stress-energy tensor —> a rank-2 tensor invariant.
Every time you drag \(\theta\) in demo and watch the components change while the ellipse stays still —> you are watching the most fundamental symmetry in all of physics: physical reality is independent of how we choose to describe it.