Observer in Quantum Mechanics

An observer can mean a measuring apparatus, a physical system carrying a record, an agent assigning probabilities, or a reference system defining coordinates. These roles need not coincide. Specifying the role turns a verbal claim about observation into a question about interactions, records, or probabilities. The broader conceptual question is introduced in What Is an Observer?.

Helpful background. Quantum instruments describe both outcome probabilities and state updates.

Records and conditional states

Consider a two-state system and an initially ready memory. An ideal measurement implements

$$U\lvert j\rangle\lvert R\rangle=\lvert j\rangle\lvert M_j\rangle, \qquad \langle M_0|M_1\rangle=0.$$

Linearity sends $\alpha\lvert0\rangle+\beta\lvert1\rangle$ to an entangled system–memory state. A device has encoded a distinguishable record whether or not a human reads it. Orthogonality is an idealization: real memories require sufficiently distinguishable, stable, retrievable states over the duration of the protocol.

The memory can be included in a larger quantum description, or its recorded outcome can condition predictions. These are different calculations. If $\mathcal I_j$ is the instrument map for result $j$, then

$$p_j=\operatorname{tr}\mathcal I_j(\rho),\qquad \rho_j=\frac{\mathcal I_j(\rho)}{p_j},\qquad \bar\rho=\sum_j\mathcal I_j(\rho).$$

The conditioned state uses a particular result; the nonselective state describes a procedure in which the result is not used. Neither formula adds a consciousness variable to the dynamics.

Two agents with different records

Prepare $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$ and measure $Z$. Alice reads $j=0$; Bob knows only that the measurement occurred. For a subsequent $Z$ measurement, Alice assigns probability one to $0$, while Bob assigns one half. Bob uses $I/2$, and Alice uses $\lvert0\rangle\langle0\rvert$.

Their assignments answer different information-conditioned questions. If Alice transmits the result through a reliable channel, Bob can condition on the same record. This example requires neither conflicting empirical frequencies nor an interpretation in which facts are relative. It does not settle Wigner’s friend, where an external observer attempts a coherent measurement of the entire laboratory.

Agents and reference systems

An agent chooses a preparation, intervention, or inference procedure. The apparatus realizes it. A quantum reference frame is instead a physical system relative to which observables are specified; it need not choose anything.

Relational quantum mechanics and QBism attach different significance to observer-indexed descriptions. Rovelli treats interactions among physical systems as central; Fuchs, Mermin, and Schack formulate probabilities as an agent’s judgments. These are interpretive proposals, not consequences of the instrument equations. See Relational and Agent-Centered Readings.

Exercise

Alice sends her binary result through a channel that flips it with probability $\epsilon$, with equal priors. Bob receives $0$. What probability should he assign to a repeated ideal $Z$ measurement returning $0$?

Solution

Bayes’ rule gives $p(j=0\mid\text{message }0)=1-\epsilon$. Bob’s state is $(1-\epsilon)\lvert0\rangle\langle0\rvert+\epsilon\lvert1\rangle\langle1\rvert$. At $\epsilon=1/2$ the message carries no information; at $\epsilon=0$ his prediction agrees with Alice’s.

References

• Fuchs, C. A., Mermin, N. D., and Schack, R., “An introduction to QBism with an application to the locality of quantum mechanics,” American Journal of Physics 82, 749–754 (2014), arXiv:1311.5253 — agent-centered interpretation.
• Rovelli, C., “Relational Quantum Mechanics,” International Journal of Theoretical Physics 35, 1637–1678 (1996), arXiv:quant-ph/9609002 — physical systems as relative observers.