Measurement in Circuits

A measurement symbol in a quantum circuit is an interface from a quantum input to a classical record and, only when the circuit declares one, a retained quantum output. Its meaning is not fixed by an outcome probability alone: the circuit must state which register is measured, in which basis, how outcomes are encoded, what state-update rule is used, and whether an output wire survives.

At terminal readout, an outcome distribution may be the only required output. If the measured register is reused, the record is forgotten, or several fine outcomes are merged, the instrument matters. In particular, classically relabeling fine measurement results need not implement the same quantum process as measuring a coarse projector directly.

This page develops that circuit-facing audit for finite-dimensional projective measurements, using only the instrument notation needed to track records and later states. General probability rules, PVM and instrument theory, adaptive feedforward, detector physics, and advanced inference remain with their canonical owners.

Required background. Circuit Model supplies quantum and classical wires, register order, process composition, and circuit resource language. Measurement in a Chosen Basis supplies basis projectors, rotated readout, and the distinction between apparatus-frame and rotated-back output states.

Measurement Operations in a Circuit

Let $Q$ denote the measured quantum register, let $R$ collect every other retained quantum register, and let $M$ denote the classical outcome register. Declare the input tensor order as $Q\otimes R$. For computational reference-basis outcome $m$,

$$P_m = |m\rangle\langle m|_Q\otimes I_R, \qquad p_m = \operatorname{Tr}(P_m\rho).$$

An ideal projective branch is the subnormalized map

$$\mathcal I_m(\rho) = P_m\rho P_m, \qquad \rho_m = \frac{\mathcal I_m(\rho)}{p_m} \quad (p_m>0).$$

Here $\mathcal I_m$ retains $Q\otimes R$, and $\operatorname{Tr}_{QR}\mathcal I_m(\rho)=p_m$. Keeping that branch subnormalized until the outcome probability has been restored prevents a common record-building error. A general quantum instrument replaces the projectors by completely positive, trace-nonincreasing maps into a declared quantum output space whose sum is trace preserving. The associated effects fix the probabilities, but they do not by themselves fix the disturbance or the surviving output system. Projective Measurement owns the PVM and Lüders derivations, while Quantum Instruments owns the general structure.

If the outcome is ignored but the quantum output is retained, the result is

$$\Phi_{QR}(\rho) = \sum_m\mathcal I_m(\rho).$$

If the classical record and the measured quantum register are both retained, their normalized classical-quantum state is

$$\omega_{MQR} = \sum_m |m\rangle\langle m|_M \otimes \mathcal I_m(\rho), \qquad \operatorname{Tr}\omega_{MQR}=1.$$

Each quantum block inside this sum is still subnormalized. If $Q$ is destroyed or discarded while $R$ survives, the corresponding cq state is instead

$$\omega_{MR} = \sum_m |m\rangle\langle m|_M \otimes \operatorname{Tr}_Q\!\left[\mathcal I_m(\rho)\right], \qquad \operatorname{Tr}\omega_{MR}=1.$$

A different destructive instrument may map into a replacement or flag space; that output space must be declared rather than inferred from a meter symbol. A terminal outcome distribution alone licenses no claim about a post-measurement register that is unused or absent.

The Ten-Field Circuit-Measurement Record

Before calculating, complete the following record. A field may be marked $N/A$ only when the reason is stated.

1. Measurement task and licensed claim
   State whether the task predicts a distribution, a selected output, an unread channel, a classical-quantum record, or an experimental observable, and stop the claim at the available evidence.

2. Quantum registers, classical records, order, and basis
   Name every input, surviving output, and classical register. Declare tensor order, displayed bitstring order, bit significance, and the reference and desired bases.

3. Premeasurement state, preparation, and promise
   Give the normalized input or density operator, its preparation assumption, and any subspace or input promise. Separate an exact state used for analysis from an experimentally certified preparation.

4. Effects, projectors, and outcome encoding
   Specify the complete measurement, map each mathematical outcome to its classical code, and identify unused or invalid codes.

5. Instrument, state update, and destructiveness contract
   State the subnormalized outcome maps, their output space, the conditional and unread states when relevant, and whether the measurement is nondemolition, destructive, or record-only.

6. Basis-change circuit and time-order convention
   Give the active pre-rotation, distinguish written operator order from chronological gate order, and state whether a rotate-back is required.

7. Record retention, conditioning, coarse graining, and postprocessing
   Say which labels are kept, hidden, merged, or selected, and distinguish a classical map of recorded outcomes from a different quantum measurement.

8. Shots, resources, and uncertainty model
   Declare trials, settings, preparations, logical operations, readouts, and the assumptions behind any standard error or interval. Keep logical and physical resources separate.

9. Verification metric and evidence
   Check normalization, trace, positivity, forbidden outcomes, state or channel residuals, and the observable that distinguishes plausible alternatives.

10. Conclusion, stopping point, and canonical handoff
    Report exactly what has been established, name what remains unmodeled, and route specialist questions to their canonical owners.

Basis Choice and Pre-Measurement Rotations

Let $\{|e_a\rangle\}$ be the fixed apparatus basis and $\{|f_a\rangle\}$ the desired orthonormal basis. Define the column matrix $V$ by

$$V|e_a\rangle=|f_a\rangle.$$

The desired-basis measurement is implemented by applying $V^\dagger$ before fixed-reference readout, since

$$\langle e_a|V^\dagger|\psi\rangle = \langle f_a|\psi\rangle.$$

This is an active physical rotation, not merely a passive rewrite of coordinates. If $V^\dagger=AB$, then $B$ acts first in time. After a selected reference-basis outcome, a reusable output in the original frame requires applying $V$; a terminal probability-only measurement does not. Multiplying any $|f_a\rangle$ by an individual phase leaves its projector unchanged, but reversing noncommuting gates can change the implemented PVM.

For sign-sensitive Y-basis readout, this page fixes

$$Y= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \qquad |+y\rangle = \frac{|0\rangle+i|1\rangle}{\sqrt2}, \qquad |-y\rangle = \frac{|0\rangle-i|1\rangle}{\sqrt2}.$$

With $S=\operatorname{diag}(1,i)$,

$$V_Y=SH, \qquad V_Y^\dagger=HS^\dagger,$$

so the physical pre-readout order is $S^\dagger$ first and $H$ second. For

$$\rho = \frac{I+\mathbf r\cdot\boldsymbol{\sigma}}{2},$$

the same convention gives

$$p(\pm y) = \frac{1\pm r_y}{2}, \qquad r_y = \operatorname{Tr}(\rho Y) = -2\operatorname{Im}\rho_{01}.$$

The full basis-measurement workflow, including basis phases and output-frame choices, remains at Measurement in a Chosen Basis.

Outcomes, Classical Registers, and Bitstring Order

A drawn meter does not determine a software record convention. For an $n$-qubit register, state whether a displayed string is $q_{n-1}\cdots q_0$ or $q_0\cdots q_{n-1}$, which bit is most significant, and whether the register list uses the same order as the string. Changing a displayed label is classical re-encoding; permuting quantum wires is a physical operation.

Outcome bits also need an interpretation. For computational measurement of $Z$, a common convention is

$$b=0\longleftrightarrow z=+1, \qquad b=1\longleftrightarrow z=-1, \qquad z=(-1)^b.$$

This map must be stated rather than assumed. A qutrit stored in two classical bits likewise needs a code and an explicit invalid word. Observing an invalid code is evidence that the ideal record model is incomplete, not permission to renormalize it away silently.

If only part of a record is reported, its probability is a marginal. For displayed order $q_1q_0$,

$$p(q_1=0) = p(00)+p(01).$$

Probabilities of exclusive recorded outcomes are added; amplitudes are not. One shot produces one outcome. The full distribution is an ensemble statement estimated from repeated, comparably prepared trials.

Conditional States, Unread Measurements, and Discard

For $p_m>0$, the selected state is

$$\rho_m = \frac{\mathcal I_m(\rho)}{p_m}.$$

When $p_m=0$, the subnormalized branch is the zero operator and no normalized posterior state exists. Postselection therefore requires both the selected state and its success probability; it is not a deterministic transformation.

Three different objects answer three different questions:

• the family $\{\rho_m,p_m\}$ predicts a state after a known result;
• $\Phi_{QR}(\rho)=\sum_m\mathcal I_m(\rho)$ predicts the retained state after the result is forgotten;
• $\omega_{MQR}$ retains the classical result together with $Q\otimes R$, while $\omega_{MR}$ uses $\operatorname{Tr}_Q$ when $Q$ is discarded.

Tracing the classical record out of $\omega_{MQR}$ gives $\Phi_{QR}(\rho)$, not the unmeasured input in general. For example, unread computational Lüders measurement removes computational-basis coherences. This is a physical dephasing channel induced by the measurement interaction and discarded record, not a unitary on the measured qubit alone.

The State Update Rule owns the general selective/nonselective hierarchy. The compact Born Rule formula card is useful for probability lookup, but it does not determine the instrument or the output space.

Classical Postprocessing and Quantum Coarse Graining

Suppose a fine outcome $m$ is converted to a reported label $y$ by a classical stochastic map satisfying

$$q(y|m)\ge 0, \qquad \sum_y q(y|m)=1.$$

The resulting outcome operation is

$$\mathcal J_y = \sum_m q(y|m)\mathcal I_m.$$

For deterministic relabeling $y=f(m)$, this sums the subnormalized branches whose recorded labels share the same reported value. It describes measuring the fine alternatives first and then hiding their labels.

A direct coarse Lüders measurement is generally different. If

$$Q_y = \sum_{m\in y}P_m,$$

then

$$Q_y\rho Q_y = \sum_{m,n\in y}P_m\rho P_n$$

retains cross terms within the coarse subspace, whereas

$$\sum_{m\in y}P_m\rho P_m$$

removes them. The two procedures can agree on the distribution of $y$ while making different predictions for later interference. The canonical degenerate-outcome treatment is Degenerate Measurements and Lüders Rule.

For two-qubit computational parity,

$$Q_0 = \frac{I+Z\otimes Z}{2} = P_{00}+P_{11},$$ $$Q_1 = \frac{I-Z\otimes Z}{2} = P_{01}+P_{10}.$$

An even/odd result is not a full Bell-state label. Direct parity primitives and ancilla constructions belong to Multi-Qubit Gates; the named states belong to Bell States.

Terminal, Destructive, and Mid-Circuit Boundaries

Terminal record-only use. If no quantum output is consumed later, a declared probability distribution and record encoding may be sufficient. It does not imply that a measured system survives.

Destructive measurement. A destructive detector has an output space that omits the measured subsystem or replaces it by a declared flag state. Destructiveness is an instrument property, not a consequence of drawing the meter at the end of a diagram.

Nondemolition or reusable output. If later gates act on the measured register, the circuit must specify the selected or unread state, the frame in which it is returned, and any reset. Identical first-outcome probabilities do not guarantee identical later behavior.

Measured classical control. A recorded result may choose a later operation, but that process is not a coherent controlled unitary. Controlled Operations owns the coherent block map and its off-diagonal branches. This page owns the classical record on the measured side of the comparison.

Some terminal measurements can be deferred by replacing a measured classical-control step with coherent control and measuring later. Such an identity requires a specified final record and the absence of intervening uses of the removed coherence. Adaptive branching, reset, repeated measurement, abstract controller dependencies, and dynamic scheduling belong to Mid-Circuit Measurement and Feedforward; this page stops at the convention-complete measurement node and its record.

Logical equivalence also says nothing about detector latency, assignment fidelity, QND behavior, crosstalk, or physical cost. Those claims belong to Control, Readout, and Calibration.

Shots, Estimators, and Verification

For $N$ stable independent repetitions with $K$ outcome categories,

$$(n_1,\ldots,n_K) \sim \operatorname{Multinomial}(N;p_1,\ldots,p_K), \qquad \widehat p_m = \frac{n_m}{N}.$$

The estimator covariance is

$$\operatorname{Cov}(\widehat p_m,\widehat p_n) = \frac{p_m\delta_{mn}-p_mp_n}{N}.$$

The negative off-diagonal covariance expresses normalization: category frequencies are not independent. A model standard error substitutes the declared $p_m$; a plug-in standard error substitutes $\widehat p_m$. Neither is automatically a confidence interval. Hoeffding’s bounded-variable inequality can provide a finite-sample tail bound, but it still assumes a stable sampling model.

A circuit record should distinguish at least four uncertainty sources:

• multinomial variation from a finite number of trials;
• preparation and instrument mismatch;
• assignment, invalid-code, leakage, and calibration error;
• drift or correlation between nominally repeated trials.

The elementary ideal verification metrics used here include the probability sum, maximum category residual, total variation distance,

$$D_{\mathrm{TV}}(\widehat p,p) = \frac12\sum_m|\widehat p_m-p_m|,$$

trace and positivity checks for branches, and an interference-sensitive observable when two instruments share a coarse distribution. Negative probabilities, nonpositive branches, trace loss for a complete instrument, or a probability sum different from one are stopping failures.

Advanced likelihood design, confidence regions, nuisance parameters, and estimator risk belong to Quantum Measurement as Estimation. The IID formulas above exclude drift, correlated shots, assignment error, and an incorrect measurement model.

Ideal Semantics, Readout Models, and Canonical Handoffs

An evidence statement should stop at the lowest layer actually checked:

1. Exact circuit semantics proves algebraic probabilities, branches, and record maps under an ideal measurement contract.
2. Numerical simulation adds floating-point and truncation tolerances but still does not establish a physical detector.
3. Calibrated experiment needs a preparation model, detector response, uncertainty, held-out validation, drift controls, and hardware provenance.

A readout-assignment matrix $A_{y|m}=p(y|m)$ is a forward classical model. Its calibration, uncertainty, inversion, regularization, and mitigation are not derived here. Likewise, ideal counts of rotations and readout calls do not imply equal duration, detector complexity, QND character, or fidelity.

SPAM Errors owns the operational preparation–process–measurement record, assignment orientation and license, nonidentifiability, gauge, context transfer, and validation boundary. Measurement Error Mitigation owns finite-record inversion or forward fitting of a validated terminal classical response, including constraints, regularization, calibration uncertainty, covariance, scaling assumptions, overhead, and residual validation; this page retains ideal circuit measurement semantics, record maps, bit order, estimators, and finite-shot checks.

The Born Rule owns general probability assignments; Projective Measurement and Measurement in a Chosen Basis own the projective and rotated-basis derivations; and Quantum Instruments owns arbitrary outcome maps. This page stops after translating those objects into a convention-complete circuit record.

Phase Kickback owns the eigenstate or character-state mechanism that prepares a returned relative phase and chooses the task-specific $X/Y$ verification targets. This page retains the basis rotations, outcome records, estimators, finite-shot uncertainty, and detector-model boundary used to measure those targets.

For orientation, the chapter gateway places measurement beside gates and computation models. The Quantum Information and Computation overview, What Is Quantum Information?, Quantum Information Roadmap, and Math Needed for Quantum Information provide broader task and prerequisite paths.

The source basis is deliberately mixed: Aharonov, Kitaev, and Nisan develop mixed-state circuit semantics; Griffiths and Niu give a canonical measurement/classical-control transformation; and Mermin, Nielsen and Chuang, and Watrous provide standard circuit treatments. The measurement-operation boundary follows Lüders, Davies and Lewis, Kraus, Ozawa, and Busch and collaborators. Hoeffding supplies the bounded-sampling result, while Ristè and collaborators provide a concrete parity-measurement setting. These references support the ideal distinctions here without converting them into hardware performance claims.

No Hamiltonian or $\hbar$ convention is needed for these dimensionless maps. The phase used below is part of an exact prepared state, not a claim about physical time evolution.

Worked Audit: Y-Basis Readout of a Phase State

1. Measurement task and licensed claim
   Predict ideal Y-basis outcomes and Lüders outputs for

   $$|\psi\rangle = \frac{|0\rangle+e^{i\pi/3}|1\rangle}{\sqrt2}.$$

   The record licenses exact circuit semantics only.

2. Quantum registers, classical records, order, and basis
   There is one qubit $Q$ in the computational reference basis and one bit $M$, with $0\leftrightarrow|+y\rangle$ and $1\leftrightarrow|-y\rangle$. The desired-basis column matrix is $V=SH$.

3. Premeasurement state, preparation, and promise
   The displayed state is exact and normalized. There is no promised subspace and no experimental preparation claim.

4. Effects, projectors, and outcome encoding

   $$P_0 = \frac{I+Y}{2} = |+y\rangle\langle+y|, \qquad P_1 = \frac{I-Y}{2} = |-y\rangle\langle-y|.$$

5. Instrument, state update, and destructiveness contract
   Use the ideal Lüders instrument $\mathcal I_a(\rho)=P_a\rho P_a$. The terminal-use branch discards the surviving quantum output; physical detector destructiveness is not modeled. If a reusable original-frame output is requested, rotate the selected reference-basis output back by $V$.

6. Basis-change circuit and time-order convention

   $$V^\dagger = (SH)^\dagger = HS^\dagger.$$

   Physical order is $S^\dagger$ first, then $H$, then fixed-Z readout.

7. Record retention, conditioning, coarse graining, and postprocessing
   Retaining $a$ selects $|\pm y\rangle$ after rotate-back. Forgetting it gives $\sum_aP_a\rho P_a$. There is no additional relabeling or postselection.

8. Shots, resources, and uncertainty model
   Each trial uses one ideal $S^\dagger$, one $H$, and one reference readout. $N$ and sampling uncertainty are explicitly $N/A$ because this is an analytic audit, not a fabricated finite-shot record.

9. Verification metric and evidence
   Here $r_y=\sin(\pi/3)=\sqrt3/2$, so

   $$p_0 = \frac{2+\sqrt3}{4} = 0.9330127019, \qquad p_1 = \frac{2-\sqrt3}{4} = 0.0669872981.$$

   The unread-output purity is

   $$p_0^2+p_1^2 = \frac78.$$

   Applying $H$ before $S^\dagger$ instead measures the X-basis distribution $3/4,1/4$. Probability-sum, state-norm, trace, positivity, and direct-matrix residuals vanish exactly.

10. Conclusion, stopping point, and canonical handoff
    The correct pre-readout circuit is $S^\dagger\rightarrow H\rightarrow M_Z$. It licenses the ideal distribution and declared update, not detector calibration or fidelity. Basis details hand off to Measurement in a Chosen Basis, general disturbance to Quantum Instruments, and physical evidence to Control, Readout, and Calibration.

Worked Audit: Fine Readout and Bell-State Parity

1. Measurement task and licensed claim
   Compare fine computational readout followed by classical parity processing with a direct nondemolition even/odd parity measurement. License outcome and retained-state predictions under the two declared ideal instruments.

2. Quantum registers, classical records, order, and basis
   The register order is $(q_1,q_0)$, displayed as $q_1q_0$. The fine record is $M\in\{00,01,10,11\}$, and the coarse record is $y=q_1\oplus q_0$.

3. Premeasurement state, preparation, and promise

   $$|\Phi^+\rangle = \frac{|00\rangle+|11\rangle}{\sqrt2}.$$

   This is an exact input state with no experimental preparation claim.

4. Effects, projectors, and outcome encoding
   The fine projectors are $P_b=|b\rangle\langle b|$. The direct parity projectors are

   $$Q_0=P_{00}+P_{11}, \qquad Q_1=P_{01}+P_{10}.$$

5. Instrument, state update, and destructiveness contract
   Compare the fine instrument $\mathcal I_b(\rho)=P_b\rho P_b$ with the direct coarse Lüders instrument $\mathcal J_y(\rho)=Q_y\rho Q_y$. Terminal destructive use discards the quantum output and is a separate contract.

6. Basis-change circuit and time-order convention
   There is no pre-rotation; the computational basis is used throughout. Time-order qualification is $N/A$.

7. Record retention, conditioning, coarse graining, and postprocessing
   The fine case calculates $y=b_1\oplus b_0$ and then erases $b$. The direct case obtains only $y$. They have identical parity probabilities but inequivalent retained states.

8. Shots, resources, and uncertainty model
   This is an analytic audit, so $N$ and sampling uncertainty are $N/A$. Fine readout uses two logical one-qubit readouts; direct parity uses one declared logical parity primitive. No physical-cost or hardware-fidelity equivalence is asserted.

9. Verification metric and evidence

   $$p(00)=p(11)=\frac12, \qquad p(01)=p(10)=0, \qquad p(y=0)=1.$$

   Fine measurement followed by hiding the fine result produces

   $$\rho_{\mathrm{fine}} = \frac12(P_{00}+P_{11}), \qquad \operatorname{Tr}\rho_{\mathrm{fine}}^2 = \frac12.$$

   Direct even parity leaves $|\Phi^+\rangle$ with purity one. Under the squared-fidelity convention,

   $$\langle\Phi^+| \rho_{\mathrm{fine}} |\Phi^+\rangle = \frac12,$$ $$D\!\left( \rho_{\mathrm{fine}}, |\Phi^+\rangle\langle\Phi^+| \right) = \frac12.$$

   Also, $\langle X\otimes X\rangle=0$ after fine-and-forget and $1$ after direct parity, whereas $\langle Z\otimes Z\rangle=1$ in both cases.

10. Conclusion, stopping point, and canonical handoff
    The classical parity distributions agree, but the instruments do not. Multi-Qubit Gates owns parity primitives and ancilla constructions; Quantum Instruments owns the general state-update distinction.

Common Failure Modes

Treating a meter as a gate. A measurement can create a classical record, discard a system, and change the output dimension. It is not a unitary symbol with an omitted matrix.

Applying the basis column matrix instead of its adjoint. If the desired basis vectors are the columns of $V$, fixed-basis readout requires $V^\dagger$. Written products must then be translated into chronological gate order.

Collapsing four different operations into “change basis.” Passive coordinate change, active pre-rotation, measurement, and optional rotate-back are distinct. Only the active operations alter the circuit.

Hiding sign and record conventions. Pauli-Y signs, tensor order, displayed bit order, significance, bit-to-eigenvalue maps, and invalid qudit codes must be declared before interpreting data.

Reading a distribution from one shot. One run returns one outcome. Probabilities and uncertainty require a declared repeated-trial model.

Normalizing an impossible branch. A conditional state exists only when its outcome probability is positive. Postselection must report the success probability and the selected ensemble.

Equating a POVM with an instrument. Effects determine first-outcome probabilities, not disturbance, destructiveness, or the surviving output.

Equating unread measurement with no measurement. Forgetting a record can remove coherence even when no outcome is reported.

Equating fine-and-hidden with directly coarse. Classical relabeling removes fine-subspace cross terms that a direct coarse Lüders measurement can retain.

Calling parity a Bell measurement. One parity bit distinguishes two subspaces, not four Bell states. A complete Bell label requires additional commuting information or a declared fine measurement.

Replacing coherent control by classical feedforward silently. The two processes can agree on computational truth tables while differing on off-diagonal control blocks.

Conflating terminal, destructive, and nondemolition use. Diagram placement does not specify whether a quantum output survives or can be reused.

Treating standard error as total uncertainty. Multinomial formulas do not cover calibration, assignment error, leakage, drift, correlations, or a wrong instrument, and category frequencies are correlated.

Inverting an assignment matrix without an evidence contract. A response matrix is a forward model. Calibration uncertainty, conditioning, regularization, and mitigation require their specialist owners.

Continuing after a validity failure. Negative probabilities, nonpositive branches, unexplained trace loss, a probability sum different from one, or an unmodeled invalid code must stop the claim.

Exercises

1. Basis-Rotation Direction and Time Order

Let $|f_a\rangle=V|e_a\rangle$. Prove which operation must precede fixed reference-basis readout. Apply the result to X and Y measurements, and use $|+y\rangle$ to diagnose the error made by applying $V$ instead.

Solution

The amplitude for reference outcome $a$ after applying $V^\dagger$ is

$$\langle e_a|V^\dagger|\psi\rangle = \langle f_a|\psi\rangle,$$

which is exactly the desired-basis amplitude. For X readout, $V=H=V^\dagger$. For Y readout,

$$V=SH, \qquad V^\dagger=HS^\dagger.$$

The rightmost operator acts first, so the chronological sequence is $S^\dagger$, then $H$, then reference readout. With

$$|+y\rangle = \frac{|0\rangle+i|1\rangle}{\sqrt2},$$

we have

$$S^\dagger|+y\rangle = |+\rangle, \qquad HS^\dagger|+y\rangle = H|+\rangle = |0\rangle.$$

Thus the $+y$ state produces outcome zero with certainty. If $V=SH$ is applied instead, direct multiplication gives

$$SH|+y\rangle = \frac{1+i}{2} \left( |0\rangle+|1\rangle \right),$$

which has computational probabilities $1/2,1/2$. The wrong operation fails the eigenstate test.

2. A Reset Instrument and Its cq Record

Consider

$$\rho= \begin{pmatrix} 3/4&1/4\\ 1/4&1/4 \end{pmatrix}, \qquad K_0=|0\rangle\langle0|, \qquad K_1=|0\rangle\langle1|.$$

Treat $K_m:\mathcal H_{Q_{\mathrm{in}}}\to \mathcal H_{Q_{\mathrm{out}}}$ as a same-dimensional reset instrument. The output label denotes the reset register after the interface; the map by itself does not claim that the physical input subsystem was destroyed.

Find the POVM, outcome branches and probabilities, selected outputs, unread output, and normalized classical-quantum state. Compare the result with the computational Lüders instrument.

Solution

The effects are

$$K_0^\dagger K_0 = |0\rangle\langle0| = P_0, \qquad K_1^\dagger K_1 = |1\rangle\langle1| = P_1,$$

and $P_0+P_1=I$. Thus the POVM is the computational PVM. The subnormalized branches are

$$\mathcal I_0(\rho) = K_0\rho K_0^\dagger = \frac34|0\rangle\langle0|,$$ $$\mathcal I_1(\rho) = K_1\rho K_1^\dagger = \frac14|0\rangle\langle0|.$$

Their traces give $p_0=3/4$ and $p_1=1/4$. Both normalized conditional outputs equal $|0\rangle\langle0|$, and the unread output is therefore

$$\mathcal I_0(\rho)+\mathcal I_1(\rho) = |0\rangle\langle0|.$$

The normalized cq state is

$$\omega_{M Q_{\mathrm{out}}} = \frac34 |0\rangle\langle0|_M \otimes |0\rangle\langle0|_{Q_{\mathrm{out}}} +\frac14 |1\rangle\langle1|_M \otimes |0\rangle\langle0|_{Q_{\mathrm{out}}}.$$

A computational Lüders instrument has the same effects and probabilities, but its selected outputs are $|0\rangle\langle0|$ and $|1\rangle\langle1|$, with unread output $\operatorname{diag}(3/4,1/4)$. Equal POVMs therefore do not imply equal instruments.

3. Bitstring Order, Marginals, and Re-encoding

For the following distribution displayed in $q_2q_1q_0$ order:

• $000:1/8$, $001:1/8$, $010:1/4$, $011:0$;
• $100:0$, $101:1/4$, $110:1/8$, $111:1/8$.

Rewrite only the displayed labels in $q_0q_1q_2$ order, compute all one-bit marginals, and find $p(q_2\oplus q_0=1)$. Explain why relabeling is not a quantum wire permutation.

Solution

Reversing each displayed label, while leaving its probability attached to the same physical outcome, gives:

• $000:1/8$, $100:1/8$, $010:1/4$, $110:0$;
• $001:0$, $101:1/4$, $011:1/8$, $111:1/8$.

In the original order,

$$p(q_0=1) = p(001)+p(011)+p(101)+p(111) = \frac12,$$ $$p(q_1=1) = p(010)+p(011)+p(110)+p(111) = \frac12,$$ $$p(q_2=1) = p(100)+p(101)+p(110)+p(111) = \frac12.$$

Odd $q_2\oplus q_0$ occurs for $001,011,100,110$. Only $001$ and $110$ carry nonzero probability here, so

$$p(q_2\oplus q_0=1) = \frac18+\frac18 = \frac14.$$

The operation changed notation in the classical record only. A physical wire permutation would apply a SWAP network to quantum subsystems and could affect later gates.

4. Fine Readout versus Coarse Parity Measurement

For $|\Phi^+\rangle$, compare fine computational measurement followed by hiding the fine record with a direct even-parity Lüders measurement. Compute purity, squared fidelity, trace distance, and the $X\otimes X$ and $Z\otimes Z$ expectations.

Solution

The input density operator is

$$\rho_\Phi = \frac12 \left( P_{00}+P_{11} +|00\rangle\langle11| +|11\rangle\langle00| \right).$$

Fine computational measurement removes the two cross terms:

$$\rho_{\mathrm{fine}} = \frac12(P_{00}+P_{11}).$$

By contrast, $Q_0|\Phi^+\rangle=|\Phi^+\rangle$, so direct even-parity Lüders measurement leaves $\rho_\Phi$ unchanged. Since

$$\rho_{\mathrm{fine}}^2 = \frac14(P_{00}+P_{11}),$$

its purity is $1/2$. Under the squared-fidelity convention,

$$\langle\Phi^+| \rho_{\mathrm{fine}} |\Phi^+\rangle = \frac12.$$

The difference

$$\rho_\Phi-\rho_{\mathrm{fine}} = \frac12 \left( |00\rangle\langle11| +|11\rangle\langle00| \right)$$

has eigenvalues $+1/2$ and $-1/2$ in the even subspace. Its trace norm is one, so the trace distance is $1/2$. Because $X\otimes X$ exchanges $|00\rangle$ and $|11\rangle$,

$$\langle X\otimes X\rangle_{\mathrm{fine}}=0, \qquad \langle X\otimes X\rangle_{\Phi}=1.$$

Both states lie entirely in the even computational-parity subspace, so $\langle Z\otimes Z\rangle=1$ for both. Their parity statistics agree even though their retained coherence does not.

5. Shot Counts and Estimator Uncertainty

Let $K\sim\operatorname{Binomial}(400,3/4)$ and $\widehat p=K/400$. Compute the mean, variance, and standard deviation of the count and estimator, then state what these quantities exclude.

Solution

For a binomial count,

$$\mathbb E K = Np = 400\frac34 = 300,$$ $$\operatorname{Var}K = Np(1-p) = 400\frac34\frac14 = 75.$$

Therefore

$$\sigma_K = \sqrt{75} = 8.660254.$$

Dividing the count by $400$ gives

$$\operatorname{Var}\widehat p = \frac{\operatorname{Var}K}{400^2} = \frac3{6400},$$ $$\sigma_{\widehat p} = \frac{\sqrt3}{80} = 0.0216506.$$

These are model standard deviations under stable independent Bernoulli trials with $p=3/4$. They are not confidence intervals and do not include preparation error, detector calibration, assignment error, leakage, drift, shot correlations, or a wrong state-update model.

6. Unread Measurement Is Not a Unitary

Start from $\rho_+=|+\rangle\langle+|$. Compute the unread computational Lüders output, its purity, its squared fidelity and trace distance from the input, and explain why no qubit-only unitary implements the map.

Solution

In the computational basis,

$$\rho_+ = \frac12 \begin{pmatrix} 1&1\\ 1&1 \end{pmatrix}.$$

The unread Lüders output is

$$P_0\rho_+P_0+P_1\rho_+P_1 = \frac12 \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} = \frac I2.$$

Its purity is

$$\operatorname{Tr} \left( \frac I2 \right)^2 = \frac12.$$

Because the input is pure, the squared fidelity is

$$\langle+| \frac I2 |+\rangle = \frac12.$$

Also,

$$\rho_+-\frac I2 = \frac X2,$$

whose eigenvalues are $\pm1/2$. Hence the trace distance is $\tfrac12\|\rho_+-I/2\|_1=1/2$. A qubit-only unitary preserves the spectrum and purity, so it cannot map a pure state to $I/2$. The lost coherence is carried by a measurement record or environment that has been ignored.

7. Terminal Readout versus Deferred Measurement

Let $P_m=|m\rangle\langle m|_c$. Compare measurement of $c$ followed by classically applying $X^m$ to target $t$ with a CNOT followed by later computational measurement of $c$. State the exact equivalence and its limitations.

Solution

The measured-and-conditioned branch for outcome $m$ is

$$(P_m\otimes X^m) \rho (P_m\otimes X^m).$$

The coherent operation is

$$U_{\mathrm{CNOT}} = \sum_mP_m\otimes X^m.$$

Projecting its control output gives

$$(P_m\otimes I)U_{\mathrm{CNOT}} = P_m\otimes X^m.$$

Therefore applying CNOT and eventually producing the same computational control record yields exactly the same subnormalized branch and hence the same final cq state as measuring first and applying the classically selected $X^m$.

Before that final measurement, however, CNOT retains off-diagonal control blocks

$$(P_m\otimes X^m) \rho (P_n\otimes X^n), \qquad m\ne n.$$

The early measured process removes them. Later interference on the control can therefore distinguish the processes. Detector destructiveness, reset, controller latency, noisy classically selected gates, and logical or physical resource accounting also prevent an unrestricted equivalence claim.

8. Complete Ten-Field Qutrit Fourier Readout Record

Let $\omega=e^{2\pi i/3}$ and

$$|f_a\rangle = \frac1{\sqrt3} \sum_{j=0}^2 \omega^{ja}|j\rangle, \qquad V=F_3.$$

Complete the full circuit-measurement record for terminal Fourier-basis measurement of $|0\rangle$, including a two-bit outcome code and $N=900$ trials.

Solution

1. Measurement task and licensed claim
   The task is ideal terminal measurement of $|0\rangle$ in the qutrit Fourier basis. The claim covers exact probabilities and elementary shot statistics only.

2. Quantum registers, classical records, order, and basis
   There is one qutrit $Q$ and a two-bit classical record. Encode $0\to00$, $1\to01$, and $2\to10$; $11$ is invalid. The reference basis order is $|0\rangle,|1\rangle,|2\rangle$, and the desired basis is $\{|f_0\rangle,|f_1\rangle,|f_2\rangle\}$.

3. Premeasurement state, preparation, and promise
   The exact input is $|0\rangle$. There is no subspace promise and no experimental preparation claim.

4. Effects, projectors, and outcome encoding
   The effects are $P_a=|f_a\rangle\langle f_a|$, with $\sum_{a=0}^2P_a=I_3$, and outcomes use the declared two-bit code.

5. Instrument, state update, and destructiveness contract
   Use the terminal destructive map

   $$\mathcal I_a(\rho) = \operatorname{Tr}(P_a\rho)\tau,$$

   where $\tau$ is a fixed trivial output state. No reusable qutrit output is licensed.

6. Basis-change circuit and time-order convention
   Apply $F_3^\dagger$ before computational qutrit readout. There is one pre-rotation, so no further product-order ambiguity arises.

7. Record retention, conditioning, coarse graining, and postprocessing
   Retain $a$, merge no outcomes, perform no postselection, and require zero ideal probability for invalid code $11$.

8. Shots, resources, and uncertainty model
   Each of $N=900$ trials uses one exact ideal $F_3^\dagger$, one qutrit readout, and one fresh preparation. Compilation, calibration, drift, and assignment error are $N/A$ for this ideal record.

9. Verification metric and evidence
   Since

   $$\langle f_a|0\rangle = \frac1{\sqrt3},$$

   every valid outcome has $p_a=1/3$. Expected counts are $300$. For each count,

   $$\operatorname{Var}(n_a) = 900\frac13\frac23 = 200, \qquad \sigma_{n_a} = 10\sqrt2.$$

   Each frequency has

   $$\sigma_{\widehat p_a} = \frac{\sqrt{200}}{900} = \frac{\sqrt2}{90} = 0.0157135.$$

   For $a\ne b$,

   $$\operatorname{Cov}(\widehat p_a,\widehat p_b) = -\frac{p_ap_b}{N} = -\frac1{8100}.$$

   The three valid probabilities sum to one and the forbidden code has probability zero.

10. Conclusion, stopping point, and canonical handoff
    The licensed ideal record is uniform over $00,01,10$, with $11$ forbidden. The analysis stops before qutrit detector realization, calibration, compilation, or mitigation and hands those questions to the relevant hardware owners.

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