Subsystems and Local Observables

A local observable asks a question about one specified subsystem. Identity factors show which parts it does not read. Local and global observables develops joint measurements, correlation examples, and observables that cannot be implemented by local measurements alone.

Helpful background. Tensor products gives the composite-space notation; reduced states gives the local state used for expectation values. Here the operators are bounded, as they always are in finite dimensions.

Embedding a Local Observable

If $M_A$ acts on $\mathcal H_A$, its observable on $\mathcal H_A\otimes\mathcal H_B$ is

$$M_A\otimes I_B.$$

Likewise an observable on $B$ becomes $I_A\otimes N_B$. On a product vector,

$$(M_A\otimes I_B)(|a\rangle_A|b\rangle_B) =(M_A|a\rangle_A)|b\rangle_B.$$

Extend this action by linearity to entangled vectors. Operators on different factors commute, since multiplying them in either order gives $M_A\otimes N_B$.

The product observable $M_A\otimes N_B$ concerns joint correlations. Measuring the two local observables and multiplying their numerical outcomes obtains its statistics. By contrast, $M_A\otimes I+I\otimes N_B$ corresponds to the sum of those outcomes. Neither should be confused with an observable on only one subsystem.

Local Probabilities and Remote Records

The reduced state gives

$$\langle M_A\otimes I_B\rangle=\operatorname{Tr}(\rho_A M_A).$$

For local POVMs $\{E_a\}$ and $\{F_b\}$, without intervening interactions or adaptive settings,

$$p(a,b)=\operatorname{Tr}[\rho_{AB}(E_a\otimes F_b)].$$

Summing over $b$ replaces $\sum_bF_b$ by $I_B$, leaving $p(a)=\operatorname{Tr}(\rho_AE_a)$. Bob’s choice of a complete local measurement does not change Alice’s unread marginal. Selected subensembles can differ, but identifying them requires the remote record. The general instrument proof appears in conditional states and no signaling.

Quick Check

On two qubits, compare $Z\otimes I$, $I\otimes Z$, $Z\otimes Z$, and $Z\otimes I+I\otimes Z$ acting on $|01\rangle$, where $Z|0\rangle=|0\rangle$ and $Z|1\rangle=-|1\rangle$.

Solution

The state is an eigenvector of all four operators, with respective eigenvalues $+1,-1,-1,0$. The first two read separate subsystem values, the third their product, and the fourth their sum. These are distinct observables even though they can be evaluated from the same pair of local records.

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010, ch. 2.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018, ch. 2. Author’s manuscript.