Quantum Physics Foundations: Measurement, Outcomes, and State Update

1) Observables and outcomes: what the apparatus records

In the predictive framework of quantum physics, a measurement is not treated as passive “looking.” It is an interaction with a system implemented by an apparatus, and it is defined by the set of possible outcomes the apparatus can record and how those outcomes are associated with an observable.

An observable is the quantity a measurement device is designed to report (for example: a spin component along a chosen axis, or whether a photon passes a polarizer). Operationally, the observable is specified by:

• Which knob settings define it (e.g., polarizer angle, magnet orientation).
• Which discrete or continuous values can appear on the readout (the outcomes).
• Which outcome corresponds to which physical channel (e.g., “transmitted” vs “absorbed,” “up” vs “down”).

In many foundational examples, outcomes are discrete. For a two-outcome device, it is common to label outcomes as + and -, or 0 and 1. The key point is: the outcome is the value recorded by the apparatus, not a pre-existing value assumed to have been sitting there waiting to be revealed.

Measurement vs passive observation

In classical thinking, “observing” can be idealized as reading a property without changing it. In quantum prediction rules, a measurement is an operation that (i) produces an outcome with certain probabilities and (ii) changes what you should predict next. The “change” is not an optional philosophical add-on; it is part of the operational rule that matches experiments.

2) The Born rule (qualitative): from state to outcome probabilities

Before measuring, you describe the system by a state (whatever mathematical object your course uses to encode predictions). The measurement is described by a set of possible outcomes. The Born rule is the rule that maps:

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(state) + (chosen measurement/observable) → probabilities for each outcome

Qualitatively, it says: each outcome corresponds to a “component” of the state relative to that measurement, and the probability of that outcome is given by the squared size (magnitude) of that component.

For a two-outcome measurement with outcomes + and -, you can think in this minimal way:

• The state assigns a probability P(+) that the apparatus records +.
• The state assigns a probability P(-) that the apparatus records -.
• These probabilities add to 1: P(+)+P(-)=1.

What matters operationally is that the probabilities depend on both the state and the measurement setting. Changing the apparatus setting (e.g., rotating a polarizer) changes the mapping from state to probabilities.

A compact “prediction recipe”

1. Choose the observable (define the apparatus setting and the list of outcomes).
2. Use the Born rule to compute the probability of each outcome from the current state.
3. Sample once: the experiment produces one outcome in a single run.
4. Repeat many runs with the same preparation to estimate the probabilities empirically (frequencies approach the Born-rule probabilities).

3) State update (“collapse”) as an operational update rule

After an outcome is recorded, the predictive state you should use for subsequent measurements generally changes. This is called state update (often “collapse”), but in an operational course you can treat it as a rule for updating predictions conditioned on the observed outcome.

Operationally:

(prior state) + (measurement performed) + (outcome obtained) → (new state for future predictions)

This is analogous in spirit to conditioning in probability theory (“update your probabilities after learning new information”), but with a crucial difference: the update is tied to a physical measurement interaction and is constrained by the structure of quantum measurements.

Ideal projective measurement (minimal form)

For an ideal measurement of an observable with discrete outcomes, the update rule is:

• If outcome a is obtained, the post-measurement state becomes the state associated with outcome a (more precisely: the state is projected onto the subspace corresponding to a and then renormalized).

You do not need the full linear-algebra machinery to use the logic: after you get an outcome, you should predict as if the system is now in the state that would make that outcome repeat with certainty in an immediate re-measurement of the same observable (this is the repeatability property discussed next).

Step-by-step: prediction before and after a measurement

1. Before measuring, compute outcome probabilities using the Born rule from the current state.
2. Perform the measurement and record one outcome.
3. Update the state according to the outcome (collapse/update rule).
4. Use the updated state to compute probabilities for any subsequent measurement.

4) Repeatability and disturbance: compatible vs incompatible observables

Repeatability in ideal measurements

An ideal (projective) measurement of an observable has a key operational feature:

• If you measure the same observable twice in a row, with no intervening dynamics, you get the same outcome the second time with probability 1.

This is not a metaphysical claim that the value was “revealed.” It is a statement about the measurement rule: the first measurement prepares (updates to) a state that is an eigenstate of that observable corresponding to the recorded outcome, so the second measurement is certain.

Disturbance when measuring incompatible observables

Now consider two different observables (two different measurement settings). If they are incompatible, then:

• Measuring the first generally changes the probabilities for outcomes of the second.
• Measuring the second in between can destroy the repeatability of the first.

Operationally, incompatibility shows up as follows: there is no single state description that assigns definite (probability-1) outcomes to both observables simultaneously, except in special cases. So inserting a measurement of an incompatible observable typically “randomizes” the outcomes of the original observable when you measure it again.

In practice, you detect incompatibility by running sequences and comparing statistics:

• Sequence A: measure A then A again → perfect repeatability (ideal case).
• Sequence B: measure A, then B, then A → the final A outcomes are generally not perfectly correlated with the first A.

5) Case studies with branching outcomes

Case study 1: Polarization measurement sequences

Consider single photons and ideal polarizers. Use the following operational labels:

• H / V: horizontal/vertical polarization measurement (outcomes: transmitted in the H channel vs V channel).
• D / A: diagonal/anti-diagonal measurement at 45° (outcomes: D vs A).

Assume ideal devices: each measurement produces one of two outcomes and updates the state to match that outcome.

Sequence 1: Measure the same basis twice (repeatability)

Setup: Prepare a photon, measure in the H/V basis, then immediately measure again in the H/V basis.

Step-by-step logic:

1. First H/V measurement yields either H or V with probabilities given by the Born rule from the initial state.
2. State update: if the outcome was H, the photon is now in state H; if V, it is now in state V.
3. Second H/V measurement: outcome repeats with certainty (ideal repeatability). So H→H always, V→V always.

Operational prediction: perfect correlation between the two readouts.

Sequence 2: Insert an incompatible measurement (disturbance)

Setup: Measure H/V, then measure D/A, then measure H/V again.

Branching tree view:

Start → measure H/V → (H branch) or (V branch)  [with Born-rule probabilities from the initial state]

Now focus on what happens after the first outcome:

• If the first outcome is H, the state is H. Measuring D/A on state H gives two possible outcomes (D or A) with nonzero probabilities (in the ideal 45° case, they are equal). The state updates to D or A.
• From either D or A, a final H/V measurement again has two possible outcomes with nonzero probabilities (again equal in the ideal 45° case).

Step-by-step operational prediction:

1. First H/V measurement produces H or V.
2. Second measurement in the rotated basis produces D or A, and updates the state accordingly.
3. Third measurement in H/V no longer perfectly matches the first outcome; the intermediate measurement has changed the state relevant to H/V predictions.

What you can check experimentally: Compare the correlation between the first and last H/V outcomes with and without the intermediate D/A measurement. With the intermediate incompatible measurement, the correlation is reduced (in the ideal symmetric case, it can be driven to zero).

Sequence 3: “Filtering” and conditional predictions

Setup: Prepare photons, measure H/V, and keep only the runs where the result was H (post-selection). Then measure D/A on the kept photons.

Operational meaning: The first measurement plus selection prepares a known input state (H) for the second stage. You can then apply the Born rule to predict the D/A outcome probabilities for that prepared state. This is a practical way to turn “measurement + conditioning” into a state-preparation procedure.

Case study 2: Stern–Gerlach-style logic with branching outcomes

Consider a two-outcome measurement of a spin component along a chosen axis, implemented conceptually like a Stern–Gerlach device. Label outcomes:

• +z and -z for measuring spin along the z-axis.
• +x and -x for measuring spin along the x-axis.

The essential operational features are the same as polarization: two outcomes, Born-rule probabilities, and state update to the corresponding outcome state.

Branching outcomes as a decision tree

Sequence: Measure along z, then along x, then along z again.

You can represent the logic as a tree of conditional predictions:

Step-by-step operational prediction:

1. First z measurement yields +z or -z.
2. Update: the system is now in the corresponding z state.
3. Measure x: outcomes are distributed according to the Born rule for an ±z input (in the symmetric ideal case, 50/50).
4. Update: the system is now in +x or -x.
5. Measure z again: outcomes are distributed according to the Born rule for an ±x input (again 50/50 in the symmetric ideal case).

Key operational lesson: The middle measurement changes the state in a way that matters for later predictions, even if you “only recorded” the outcome and did nothing else. In the formalism, recording the outcome is part of what defines which conditional state you should use next.

Optional refinement: ignoring outcomes vs conditioning on outcomes

Sometimes you perform a measurement but do not keep track of which outcome occurred (or you average over it). Operationally, this differs from conditioning on a specific recorded outcome:

• Conditioned update: you know the outcome, so you use the corresponding post-measurement state for that branch.
• Unconditioned update: you do not know (or do not use) the outcome, so your next predictions use a mixture of the possible post-measurement states weighted by their probabilities.

This distinction matters in multi-step experiments because “having the record” changes what you can predict for subsequent steps.

6) Separating formal prediction rules from interpretation questions

What the formal rule predicts (minimum operational package)

• Given a state and a chosen observable, the Born rule gives probabilities for each possible outcome.
• When an outcome is recorded, you update the state according to the measurement’s update rule (ideal case: project onto the outcome state/subspace and renormalize).
• Immediate repetition of the same ideal measurement yields the same outcome with certainty.
• Intervening incompatible measurements generally change later outcome probabilities and can destroy repeatability.
• Multi-step experiments are handled by branching: compute probabilities at each step using the current (possibly updated) state, and condition on recorded outcomes when you have them.

Interpretation questions (kept separate)

Questions like “Did the system have the value before measurement?”, “What physically causes collapse?”, or “Is collapse real or just information update?” are interpretation-level questions. The operational framework used in this chapter does not require answering them to make correct predictions. For the minimum needed foundation, you treat measurement as a well-defined procedure with (i) outcome probabilities and (ii) a state-update rule that tells you how to predict what happens next.