Quantum State Tomography

State tomography reconstructs an unknown density matrix from measurements on many identically prepared copies.

Reconstructing a state

A single measurement reveals almost nothing about an unknown quantum state, and measurement destroys it. Quantum state tomography works around this by using many identical copies. By measuring different observables on different copies and collecting statistics, one estimates enough expectation values to reconstruct the full density matrix rho. It is the experimental route from a physical system to its complete mathematical description.

The qubit case

A single-qubit density matrix is fixed by three real numbers, the Bloch vector components , , . Measuring each Pauli on a fraction of the copies estimates these, and rho = (I + X + Y + Z)/2. For n qubits one measures all 4^n Pauli expectation values, so the number of settings and the total measurements grow exponentially with system size.

import numpy as np
I=np.eye(2);X=np.array([[0,1],[1,0]])
Y=np.array([[0,-1j],[1j,0]]);Z=np.array([[1,0],[0,-1]])
def rebuild(ex,ey,ez):
    return 0.5*(I + ex*X + ey*Y + ez*Z)
print(np.round(rebuild(0,0,1),3))  # |0><0|

Making it physical

Raw linear inversion of noisy data can yield a matrix with negative eigenvalues, which is not a valid state. Maximum-likelihood and Bayesian estimation instead fit the most probable valid density matrix to the data, guaranteeing positivity and unit trace. These methods also provide error bars, essential for trusting the reconstruction.

Scaling and modern methods

Full tomography is infeasible for large systems because of the exponential cost in both measurements and classical post-processing. Scalable alternatives exploit structure: compressed sensing reconstructs low-rank states from few measurements, matrix-product-state tomography handles weakly entangled states efficiently, and classical shadows estimate many properties from randomized measurements without building the whole matrix. State tomography and its efficient descendants are the standard tools for verifying that hardware prepared the state it was supposed to.