= Solution
Let a <projective measurement> have mutually <orthogonal projection>[orthogonal projections] $P_i$ satisfying $\sum_iP_i=I$. On a <density operator> $\rho$, the <Born rule> and the <Lüders rule> give
$$
\Pr(i)=\operatorname{Tr}(P_i\rho),
\qquad
\rho\longmapsto
\frac{P_i\rho P_i}{\operatorname{Tr}(P_i\rho)}
$$
conditional on outcome $i$. On the first part of a <tensor product of quantum systems>[bipartite quantum system], replace $P_i$ by $P_i\otimes I_2$.
The subsystems are isolated when there is no interaction term coupling them. Their <Hamiltonian operator> has the form
$$
H=H_1\otimes I_2+I_1\otimes H_2,
$$
so their subsequent <unitary time evolution> factorizes as $U_1\otimes U_2$.
The <reduced density matrix> of subsystem 1 is
$$
\boxed{\rho_1=\operatorname{Tr}_2\rho_{12}},
$$
where the <partial trace> is characterized by
$$
\operatorname{Tr}_1(M_1\rho_1)
=\operatorname{Tr}_{12}[(M_1\otimes I_2)\rho_{12}]
$$
for every local <observable> $M_1$. Thus $\rho_1$ contains exactly the statistics accessible by measurements on subsystem 1.
Suppose a projective measurement $\{Q_j\}$ is performed on subsystem 2 and its outcome is not communicated. The resulting nonselective state is
$$
\rho'_{12}=\sum_j(I_1\otimes Q_j)\rho_{12}(I_1\otimes Q_j).
$$
For every $M_1$, the <cyclic property of the trace> and $\sum_jQ_j^2=I_2$ give
$$
\begin{aligned}
\operatorname{Tr}[(M_1\otimes I_2)\rho'_{12}]
&=\sum_j\operatorname{Tr}[(M_1\otimes Q_j^2)\rho_{12}]\\
&=\operatorname{Tr}[(M_1\otimes I_2)\rho_{12}].
\end{aligned}
$$
Hence $\rho'_1=\rho_1$. No local projective measurement on subsystem 1 can reveal whether the remote unreported measurement occurred. This is <quantum no-signalling>, which prevents a choice made at a <spacelike separation> from transmitting information faster than light and makes the measurement formalism compatible with <relativistic causality>.
The proposed nondisturbing device would violate <no information without disturbance>. In an ordinary quantum instrument, let $M_{i\alpha}$ be the <Kraus operators> associated with classical output $i$. If every pure state $|\psi\rangle$ remains unchanged even conditional on the displayed outcome, every nonzero $M_{i\alpha}|\psi\rangle$ must be parallel to $|\psi\rangle$. A linear operator for which every vector is an <eigenvector> is a scalar multiple of the identity, so $M_{i\alpha}=c_{i\alpha}I$. Its output probability
$$
\sum_\alpha\langle\psi|M_{i\alpha}^\dagger M_{i\alpha}|\psi\rangle
=\sum_\alpha|c_{i\alpha}|^2
$$
is independent of the state. It cannot equal $\langle\psi|P_i|\psi\rangle$ for arbitrary projectors. Equivalently, repeated nondisturbing samples would permit <quantum state tomography> of one specimen and then <quantum cloning>, contradicting ordinary quantum theory.
Such devices would not *necessarily* enable <superluminal signalling>. One consistent operational extension could make every sequence of outputs depend only on the local <reduced density matrix> and local settings. Since an unreported remote measurement leaves that matrix unchanged, all local device statistics would remain unchanged too. Other extensions could add nonlocal outcome-dependent rules and permit signalling, but that behavior is additional to the device specification rather than forced by it.
Back to article page
