Local Tomography

Local tomography means that the state of a composite system is determined by joint statistics of local measurements. It allows classical comparison of outcomes and correlations; it does not mean that the separate reduced states alone determine the joint state. The property constrains a theory’s composition rule.

Product observables span complex quantum operators

For finite-dimensional complex Hilbert spaces, choose Hermitian operator bases $\{F_\mu\}_{\mu=0}^{d_A^2-1}$ and $\{G_\nu\}_{\nu=0}^{d_B^2-1}$. Their tensor products form a basis of Hermitian operators on $A\otimes B$. Thus the numbers

$$c_{\mu\nu}=\operatorname{Tr} [\rho_{AB}(F_\mu\otimes G_\nu)]$$

determine $\rho_{AB}$ after accounting for the basis normalization. Each expectation can be recovered from local spectral measurements and paired outcome records. No entangling detector is needed in principle.

The real dimension of the unnormalized state space is $K_A=d_A^2$, so $K_{AB}=K_AK_B$. Subtract one parameter for normalized states only after taking this product.

For two qubits,

$$\rho_{AB}=\frac14\biggl[I\otimes I +\sum_i r_i\sigma_i\otimes I +\sum_j s_jI\otimes\sigma_j +\sum_{ij}T_{ij}\sigma_i\otimes\sigma_j\biggr].$$

The two local Bloch vectors provide six parameters; the nine correlations supply the rest. Two Bell states can have identical local Bloch vectors and different $T_{ij}$.

A real-quantum counterexample

In real Hilbert-space quantum theory, allowed density operators and effects are real symmetric matrices. A single rebit has $K=3$, while two rebits have a four-dimensional real Hilbert space and hence $K_{AB}=4(5)/2=10$, exceeding $3^2$.

An explicit missing direction is $\sigma_y\otimes\sigma_y$, which is real symmetric even though $\sigma_y$ itself is not a real observable. The states

$$\rho_\pm=\tfrac14(I\otimes I\pm\sigma_y\otimes\sigma_y)$$

are positive and normalized. Every product of real local effects has the same expectation in both, because $\operatorname{Tr}(E\sigma_y)=0$ for real symmetric $E$. A global measurement can distinguish them. The failure is therefore a precise statement about accessible effects, not incomplete experimental skill.

Limits of the principle

Local tomography is about identifiability, not efficiency. The number of required settings and samples can grow rapidly, and noise makes inversion difficult. It also assumes known local measurement operators; a device-independent Bell analysis makes a different set of assumptions.

Nor does local tomography mean separability. Entangled states are fully included in the complex-quantum expansion. GPT reconstructions use the property to restrict alternatives without excluding quantum correlations.

Exercises

Do the reduced density operators determine whether two qubits are entangled?

Solution

No. A Bell state and $I\otimes I/4$ both have marginals $I/2$, while the former is entangled and the latter is a product state. The correlation coefficients distinguish them. This illustrates why tomography needs paired local records, not only each laboratory’s histogram.

References

• Hardy, Lucien, “Quantum Theory From Five Reasonable Axioms,” arXiv:quant-ph/0101012 (2001). Paper.
• Hardy, Lucien, and William K. Wootters, “Limited holism and real-vector-space quantum theory,” Foundations of Physics 42, 454–473 (2012). Paper.