Paper VII: Measurements and quantum channels

I.7 Measurements and quantum channels

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I.7.1 Packet measurements

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Let \(\rho\) be a density matrix and let \(\{E_k\}\) be a POVM:

\[ E_k\ge0, \qquad \sum_kE_k=I. \]

The outcome probabilities are

\[ p_k={\operatorname{Tr}}(E_k\rho). \]

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Theorem I.7.1: Block-diagonal measurement invariance

Let \(\{P_\alpha\}\) be a complete packet family, and suppose every measurement effect is packet diagonal:

\[ E_k=\sum_\alpha P_\alpha E_kP_\alpha. \]

Define the packet-dephased state

\[ {\mathcal D}_P(\rho)=\sum_\alpha P_\alpha\rho P_\alpha. \]

Then

\[ \boxed{ {\operatorname{Tr}}(E_k\rho) = {\operatorname{Tr}}(E_k{\mathcal D}_P(\rho)) } \]

for every outcome \(k\).

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Proof of Theorem I.7.1

Insert the packet decomposition of \(E_k\) and use cyclicity of the trace:

\[ {\operatorname{Tr}}(E_k\rho) = \sum_\alpha{\operatorname{Tr}}(P_\alpha E_kP_\alpha\rho) = \sum_\alpha{\operatorname{Tr}}(E_kP_\alpha\rho P_\alpha). \]

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The theorem identifies which coherences a declared measurement can detect. Observables that are diagonal in the packet decomposition cannot detect coherence between distinct packets. Coherence within a packet remains available to such observables.

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I.7 Measurements and quantum channels

I.7.1 Packet measurements

Theorem I.7.1: Block-diagonal measurement invariance

Proof of Theorem I.7.1