Quantum Information Helpers

A collection of utility functions to assist with calculations common in Quantum Information.

Functions

`purity(rho)`

Calculates the purity $\gamma = \mathrm{tr}(\rho^2)$.

For pure states, $\gamma = 1$.
For maximally mixed states, $\gamma = 1/d$.

`fidelity(rho, sigma)`

Calculates the fidelity between two states.

Note: This implements the "Uhlmann fidelity" squared form for states commonly used in some contexts, or simply overlap $\mathrm{tr}(\rho \sigma)$. Check implementation: Currently defined as tr(rho * sigma). For pure states $|\psi\rangle, |\phi\rangle$, this equals $|\langle \psi | \phi \rangle|^2$.

`partial_trace(M, dims, subsystem)`

Symbolically computes the partial trace of a composite system.

M: Matrix to trace.
dims: Tuple of dimensions of subsystems, e.g., (2, 2) for two qubits.
subsystem: Index of the subsystem to trace out (1 or 2, etc.).

Example: Average Purity

Calculating the average purity of a subsystem when the global system is in a random pure state.

using IntU, Symbolics

# System Dimensions: d_A = 2, d_B = 2
d_A = 2
d_B = 2
d = d_A * d_B

# Random Unitary on full system
U = SymbolicMatrix(:U, :U, d)
measure = dU(d)

# Pure state |psi> = U |00> (first column of U)
# We form the density matrix rho = |psi><psi|
# rho_{ij} = U_{i,1} * conj(U_{j,1})

# We can conceptually construct the partial trace.
# But IntU provides helpers if you work with explicit indices.
# Or we can compute the integral of Purity(rho_A).

# < Purity(rho_A) > = (d_A + d_B) / (d + 1)
# For d_A=d_B=2, d=4 -> (2+2)/5 = 0.8

val = average_purity((d_A, d_B), 1) 
# Note: average_purity is a helper that returns the analytical formula or value
println(val)
# Output: 4//5 (for d_A=2, d_B=2)

References

Nielsen, M. A., & Chuang, I. L. (2010). Quantum computation and quantum information. Cambridge university press.
Watrous, J. (2018). The theory of quantum information. Cambridge University Press.