Pure State Integration

Integration over the measure of random pure states $|\psi\rangle$ in $\mathbb{C}^d$, also known as the Fubini-Study measure.

Theory

A random pure state $|\psi\rangle$ can be theoretically generated by applying a random Haar-unitary matrix $U$ to a fixed reference state, e.g., $|0\rangle = (1, 0, \dots, 0)^T$.

\[|\psi\rangle = U |0\rangle\]

Consequently, the components of $\psi$ are simply the elements of the first column of the random unitary matrix $U$: $\psi_i = U_{i,1}$.

IntU.jl leverages this connection. The function dPsi creates a symbolic vector that is internally linked to the first column of a symbolic unitary matrix. Integration is then performed using the standard unitary Weingarten calculus.

Use dPsi(d) where d is the dimension of the state space.

Usage

  1. Basic Integration using @integrate

The @integrate macro identifies psi as the random state vector.

using IntU, Symbolics
@variables d
# E[|psi_1|^2] = 1/d
@integrate abs(psi[1, 1])^2 dPsi(d)
# Output: 1/d
  1. Manual Integration
using IntU, Symbolics

@variables d
# Use SymbolicMatrix for the state vector
psi = SymbolicMatrix(:psi, :psi)

# 1. Average of |psi_1|^2
res = integrate(abs(psi[1, 1])^2, dPsi(d))
println(res)
# Output: 1/d

# In element-wise mode, we can use matrix entries
integrate(abs(psi[1, 1] * conj(psi[2, 1]))^2, dPsi(d))
# Output: 1/(d*(d+1))

Pitfalls

  • Indexing: psi behaves like a vector, but since it is a SymbolicMatrix, it requires two indices: psi[i, 1]. Internally psi[i, 1] corresponds to U[i, 1].
  • Normalization: The standard measure assumes $\langle \psi | \psi \rangle = 1$. The integral volume is normalized to 1. $\int d\psi = 1$.

References

  1. Życzkowski, K., & Sommers, H. J. (2001). Induced measures in the space of mixed quantum states. Journal of Physics A: Mathematical and General, 34(35), 7111.
  2. Bengtsson, I., & Życzkowski, K. (2017). Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press.