Work through the quantum model.

Add first, square afterward

Write z = x + iy, where i² = −1. Its squared magnitude is |z|² = x² + y². The polar form r exp(iθ) separates magnitude r from phase θ, the angle from the positive real axis. These are two descriptions of the same number.

For coherent contributions a and b, |a+b|² = |a|² + |b|² + 2 Re(a* b), where the star means complex conjugation. The final term is interference. Adding probabilities alone discards that term and answers a different question.

With a = 1/2 and b = i/2, the combined amplitude is (1+i)/2 and its squared magnitude is 1/2. Replacing b by −1/2 gives zero. Both choices separately have |a|² = |b|² = 1/4. These are contributions to an output amplitude, not necessarily a complete normalized state.

Normalize without changing relative phase

For a nonzero vector (a₀,…,a₍d−1₎), divide every entry by the same factor √(Σⱼ|aⱼ|²). The vector (3,4i) becomes (3/5,4i/5), not an equal superposition. Taking absolute values, discarding imaginary parts or normalizing entries separately changes the intended state.

Multiplying the entire state by exp(iφ) changes only global phase and leaves every measurement probability unchanged. Multiplying one component changes relative phase and can change a later interference measurement even when the current computational-basis histogram stays fixed.

The all-zero vector has no normalized direction. Nonfinite entries are invalid. Large finite entries need a common scaling factor before numerical squaring to avoid overflow; roundoff tolerance is not a license to repair arbitrary invalid inputs.

Conjugate the first vector

A one-qubit pure state is |ψ⟩ = a|0⟩ + b|1⟩ with |a|² + |b|² = 1. The inner product ⟨φ|ψ⟩ conjugates the entries of φ before multiplying and summing. Without conjugation, imaginary entries can produce an impossible negative self-overlap.

Orthogonal states have zero inner product. Normalized states equal up to global phase have overlap magnitude one. For pure states, |⟨φ|ψ⟩|² is the probability that the projector onto |φ⟩ succeeds on |ψ⟩. It does not measure algorithmic usefulness.

Keep subsystem labels attached

In this ordering, (a,b) ⊗ (c,d) = (ac,ad,bc,bd). Preparing |+⟩ on q₀ and |1⟩ on q₁ gives (0,1/√2,0,1/√2). Reversing the subsystems changes where the nonzero entries appear.

A gate on q₀ is U ⊗ I; a gate on q₁ is I ⊗ U. Applying A first and B second gives BA|ψ⟩. Diagram wire order, amplitude indexing and printed classical strings can follow different conventions even in the same software framework.

Use an asymmetric check: start with |10⟩. CNOT with q₀ controlling q₁ gives |11⟩; CNOT with q₁ controlling q₀ leaves |10⟩ unchanged. A Bell histogram alone can miss reversal because 00 and 11 read the same backward. The three-bit example 110 below provides another check.

An SDK translation requires an explicit permutation from native order to these labeled kets, including the actual classical measurement map. Indiscriminately reversing strings is insufficient when named registers or nontrivial measurement maps are involved. The calculations here do not establish that an SDK adapter was executed.

Specify the observable

A Hermitian operator A describes a finite-dimensional projective observable. Its real eigenvalues are possible outcomes, and its orthogonal eigenspaces define the alternatives. An eigenstate makes this observable certain in the ideal model, not every other observable.

For density matrix ρ, the expectation is Tr(ρA). The variance is Tr(ρA²) − [Tr(ρA)]². A pure state can have nonzero measurement variance.

Pauli Z gives +1 on |0⟩ and −1 on |1⟩. For p₀ = 0.36 and p₁ = 0.64, the expectation is −0.28. Because Z² = I, the variance is 1 − (−0.28)² = 0.9216. A single shot returns +1 or −1, never −0.28.

Separate three kinds of uncertainty

With N independent identical shots, the sample mean estimates the expectation and its variance is the single-shot variance divided by N. Its standard deviation therefore scales as 1/√N. More shots do not remove calibration bias or fix an incorrect model.

Outcome randomness describes the specified measurement distribution. Sampling uncertainty describes what finite data establish about it. Model uncertainty asks whether preparation, noise and readout describe the experiment. A narrow statistical interval addresses sampling uncertainty under its assumptions.

Do not multiply estimates of noncommuting observables as though both were jointly measured on each copy. Measuring separate Pauli terms on separate preparations is one way to estimate a Hamiltonian; grouping and covariance can change the sample budget.

Report the observable, eigenvalue convention, counts, estimator and uncertainty procedure. An interval [lo,hi] for p₀ transforms into [2lo−1,2hi−1] for the Z expectation. A probability interval cannot be attached unchanged to a quantity ranging from −1 to +1.

State the matrices and convention

I leaves a qubit unchanged. X exchanges amplitudes. Z changes the sign of the |1⟩ amplitude. Y acts as Y(a,b) = (−ib,ia). X, Y and Z are Hermitian, square to I and have eigenvalues +1 and −1.

For a unit axis vector n and Pauli vector σ = (X,Y,Z), Rₙ(θ) = exp(−iθ n·σ/2) = cos(θ/2)I − i sin(θ/2)n·σ. The angle θ is in radians, with this minus-sign and half-angle convention.

Rz(θ) has diagonal entries exp(−iθ/2), exp(iθ/2). The phase gate P(θ) has entries 1, exp(iθ). They differ by global phase on an isolated target and therefore have the same isolated measurement predictions. Once conditional on another qubit, that phase can become relative between control branches. Controlled-U and controlled-exp(iφ)U need not be equivalent.

Distinguish observable order from global phase

On |0⟩, H then Z produces |−⟩; Z then H produces |+⟩. Their computational-basis histograms match, but an X measurement is deterministically −1 in the first case and +1 in the second.

XZ and ZX differ by a minus sign. On an isolated state, that difference is global and cannot alone be detected by measuring the resulting state. It is not the same demonstration of observable order dependence. A controlled context requires its own phase accounting.

For energy-valued Hamiltonian terms A and B, evolution is exp[−i(A+B)t/ℏ], where ℏ is the reduced Planck constant. Replacing it by exp(−iAt/ℏ) exp(−iBt/ℏ) generally introduces error when A and B do not commute. Shorter repeated steps may reduce approximation error while increasing implementation cost.

A useful circuit explanation names the input, ordered operations, controls, targets and measured observable in text. A diagram must agree with those operations; small gate count and small approximation error are distinct goals.

Keep conditional and averaged states separate

For the ideal Lüders projective measurement with projectors Pⱼ, outcome j has probability pⱼ = Tr(Pⱼρ). If pⱼ is nonzero, its conditional state is PⱼρPⱼ/pⱼ. Discarding the outcome record instead gives ΣⱼPⱼρPⱼ.

Measuring |+⟩ in the computational basis and discarding the record yields I/2. Keeping the record identifies |0⟩ or |1⟩ for that run. The average over records is not one definite branch.

A positive-operator-valued measure consists of positive operators Eⱼ that sum to I and specifies probabilities Tr(ρEⱼ). It does not uniquely determine the state update. A measurement instrument is needed; in a simple case, operators Mⱼ satisfy Eⱼ = Mⱼ†Mⱼ and determine the conditional transformation.

Devices can have the same outcome probabilities on selected states but disturb them differently. A classical readout confusion matrix is not a complete physical quantum measurement model.

Feedback follows the measured record

Mid-circuit measurement produces data, classical logic processes the record, and a later operation depends on it. Delay includes acquisition, processing, transport and actuation. A branch after the whole job finishes is not a real-time branch inside the circuit.

In ideal teleportation, the receiver cannot read the unknown input from the local state before receiving the classical result. The correction conditioned on that record recovers the input. Conditional notation does not create faster-than-light communication.

Postselection keeps only runs meeting a condition. Its acceptance fraction and collection cost remain part of the result. Active correction uses information to modify or reinterpret the state. A protocol may use both, but one cannot replace the other in a claim.

Follow the control chain

The path can include compilation, scheduling, calibrated pulse parameters, digital waveform generation, analog conversion and propagation through a physical control chain. Logical intent, scheduled operation, physical control action and measured performance are related descriptions, not interchangeable ones.

For a microwave-driven qubit, in-phase and quadrature envelopes describe components relative to a carrier reference. Their amplitude and phase determine an effective drive in a suitable rotating-frame model. This account is specific to that control model; it is not a universal wiring diagram.

A pulse amplitude or duration needs the device response, frequency, transfer function and calibration procedure. A square waveform at the generator need not remain square at the qubit: filtering, reflections, distortion and unintended coupling can change it.

Calibration connects the request with a response

Rotation-amplitude sweeps, detuning experiments and Ramsey sequences probe particular parameters. A successful isolated-gate calibration does not establish the same error during a simultaneous many-qubit workload.

Some Z rotations use phase-reference updates instead of a separate physical pulse. Within a supported model, this can reduce direct pulse duration. It still requires correct frame tracking, timing and subsequent phases; the compiler and controller must agree on the accumulated frame.

Superconducting, trapped-ion and neutral-atom implementations need separate platform context. An illustrative control chain omits component specifications and cannot establish an achievable error rate. Neither a gate symbol nor a language-supported instruction guarantees backend support or gate fidelity.

Time is part of the experiment

Operations can overlap, wait for resources or depend on measurement results. Waveform samples, digital logic, acquisition windows and carrier phase may use different clocks whose relationships must be defined.

Sample rate says how often a digital waveform is specified. It is not a measurement of jitter, analog bandwidth, phase noise or gate fidelity. More samples per pulse do not by themselves establish a more accurate physical result.

Make the unit conversion explicit

For carrier frequency f, a time offset Δt gives phase offset Δφ = 2πfΔt. As synthetic arithmetic, 6 GHz × 1 picosecond = 0.006 cycles, or 2.16 degrees. This is a phase offset, not a measured gate error. Infidelity requires the operation, fluctuation statistics and relevant sensitivity model.

A fixed offset may be calibrated. Random fluctuations, drift and correlated offsets have different consequences. A common shift can cancel in one relative measurement but matter elsewhere. A timing result needs its measurement bandwidth and statistical definition.

Respect propagation and decision deadlines

Separate controllers may need shared references, trigger alignment, timestamps and deterministic sequencing. Signals take time to propagate. A measurement-conditioned decision must reach the relevant operation before its deadline; low average network latency does not establish that every decision does.

Optical control raises related synchronization questions but uses different frequencies, components and stabilization methods. The microwave arithmetic above is not an optical phase specification or an invented laboratory tolerance.

A schedule can be syntactically valid without being supported by a real controller. Keep illustrative delays, physical waiting and phase-reference tracking distinct, and distinguish the latency of one result from sustained processing throughput.

Separate the physical signal from its label

A dispersive superconducting-qubit example can include a resonator, amplification, sampling, demodulation and classification. Other platforms can use fluorescence or photon detection. Improving a classifier cannot recover information the physical measurement never distinguished; a strong signal can still be mislabeled through drift or an unsuitable threshold.

Let a be the probability of reporting 1 when 0 was prepared, and b the probability of reporting 0 when 1 was prepared. For true state-1 probability p, the reported rate is y = a + (1−a−b)p. If 1−a−b is nonzero, the unconstrained inverse is p = (y−a)/(1−a−b).

For a = 0.02, b = 0.03 and y = 0.40, p = 0.38/0.95 = 0.40. This coincidence is specific to these inputs and does not make readout errors harmless.

Keep singularity and unphysical estimates visible

If a+b = 1, the reported rate no longer distinguishes p and this inverse is undefined. Near that value, small rate or calibration uncertainties can produce a large change in the inverse. Probabilities must be finite and within their allowed ranges.

Finite data or model mismatch can yield an inverse estimate outside [0,1]. Silent clipping hides this diagnostic. A constrained statistical estimator is a different named procedure, not the same calculation with its result concealed.

Basis-state calibration can include preparation error. The model may miss leakage, simultaneous-readout correlations or drift between calibration and experiment. Multiplying single-qubit assignment matrices assumes a structure that must be tested.

Evaluate on separate data

Training and evaluating a classifier on the same labeled records can overstate performance. Use independent evaluation data, record acquisition conditions and check drift. Retain the confusion matrix when class imbalance matters and carry calibration uncertainty into the corrected estimate.

This arithmetic is a stated classical assignment model, not a full quantum measurement instrument or a hardware calibration. Keep raw rates, assumptions and any correction visible together.

Different experiments answer different questions

Relaxation probes how populations approach equilibrium under a stated model. Ramsey probes phase evolution and dephasing. Echo changes sensitivity to some slowly varying disturbances. Gate benchmarking and application tests measure other behavior; one number cannot replace all these experiments.

In a simple zero-temperature relaxation model, excited-state population decays as exp(−t/T₁). A compatible two-level Markovian model can give a coherence envelope exp(−t/T₂) and 1/T₂ = 1/(2T₁) + 1/Tφ. With nonnegative pure-dephasing rate, T₂ ≤ 2T₁.

That relation is not a universal audit rule for every fitted envelope or non-Markovian experiment. Noise can produce non-exponential decay. State the sequence, fit function, uncertainty, operating conditions and measurement time.

Operating context can change

Parameters can drift between calibration and a workload. Recalibration consumes time and resources. Inspect residuals: a poor fit can reflect the wrong model, insufficient data or actual instability, not simply a parameter that needs more digits.

A gate measured alone can behave differently when neighbors are active because of crosstalk, frequency collisions, shared resources or schedule changes. Distinguish isolated and simultaneous tests and state whether the operating point was maintained.

Randomized benchmarking estimates a modeled average under its protocol. It does not certify the worst-case error of every algorithm or reveal every coherent and correlated mechanism. Long coherence does not by itself establish accurate entangling gates, measurement or scalable control.

Separate rate from delay

A decoder uses syndrome records and a model to infer a correction or update a tracked frame. Throughput is sustained processing rate; latency is the delay before a particular item or decision becomes available. A pipeline can have high throughput and substantial individual latency.

A fast decision on one small case does not show that a decoder sustains a long experiment with many concurrent logical blocks. Some decisions can wait; others must arrive before a dependent operation.

Calculate only the declared fluid model

As synthetic constant-rate arithmetic, suppose an initially empty queue receives 1,000 units per second and processes 900 per second. After five seconds, the backlog is (1,000−900) × 5 = 500 units. With initial backlog Q₀ and constant rates λ and μ, the nonnegative fluid result is max(0,Q₀+(λ−μ)t).

This is a fluid toy model, not a decoder simulation. Real arrivals can burst, syndrome patterns can require different work, and communication, memory movement and dependencies can dominate. Mean service above mean arrivals does not by itself establish acceptable tail latency or remove every risk of growth.

Tie a deadline to a dependency

Tracking a Pauli frame can avoid physically applying every inferred correction immediately. A later operation or interpretation may still depend on that frame. Its deadline comes from the architecture and dependency, not a universal rule that every decode must finish within one syndrome cycle.

A timing benchmark needs the code, distance, concurrent blocks, noise model, syndrome volume, processing hardware and included work. Separate offline replay from live closed-loop operation. Omitting transfer or preprocessing may suit a component benchmark, but those omissions remain relevant to a system claim.

Classical information flow belongs in a logical-computing resource budget alongside qubits and quantum gates. The arithmetic here establishes no operational decoder performance.

Identify the platform

Many superconducting-qubit systems use dilution refrigeration. Trapped-ion and neutral-atom systems use vacuum and optical control, with implementation-dependent details. Photonic systems involve sources, interference, switching and detection. Spin systems can combine semiconductor devices with demanding control and cryogenic requirements.

A system explanation must identify which platform it describes. Illustrative component relationships cannot establish exact vendor dimensions, an engineering bill of materials or attainable error rates.

Trace heat and information separately

In a cryogenic example, cables and components can carry signals and heat. Temperature stages have different purposes and cooling capacities. More channels can create thermal and packaging challenges, while nearby electronics or multiplexing introduce their own tradeoffs. There is no universal cable-per-qubit ratio.

In an optical example, light must be generated, stabilized, delivered, shaped and detected. Loss, mode matching, alignment and detection matter. Encoding information in photons differs from using photons to connect modules of another platform; specify the optical subsystem’s role.

Keep the operational footprint in view

Power supplies, vibration isolation, vacuum equipment, timing distribution, classical processors and calibration can be part of the operating system. A chip photograph is not its complete footprint. A chip metric does not directly establish datacenter energy, cost or performance.

Supporting resources need not all scale linearly with qubit count. Shared equipment, modularity and architectural changes can alter scaling. Identify what is shared, what is per module and what remains unspecified.

Follow storage, control delivery, readout, environmental support and classical feedback. At each step ask what the component does, which limits it introduces and how that behavior is measured. Exact dimensions, component specifications and achieved performance require device-specific evidence.

Begin with a task and an error budget

A resource estimate needs a target result, precision, acceptable failure probability, circuit construction, error-correction approach and scheduling model. Data storage is only part of the resources: ancillas, routing, resource-state factories, buffers and spare capacity can matter.

T count is the number of T operations. T depth is the number of sequential layers under a stated scheduling and ancilla model. A resource-state factory has an output rate, footprint, error characteristics and startup behavior. Enough data qubits do not guarantee that states arrive when needed.

Keep separate lower bounds separate

As synthetic planning arithmetic, one million required resource states and a combined supply of 1,000 acceptable states per second imply at least 1,000 seconds for supply alone, ignoring startup and buffering. If dependency-constrained execution requires 2,000 seconds, that is a separate lower bound.

The larger bound is 2,000 seconds, but taking the maximum is not a complete schedule. Overlap, contention, interruptions and state availability still need modeling. This example is not a device runtime prediction.

Carry the assumptions through the conclusion

Allocate failure probability among memory, gates, preparation, measurement, approximation and other mechanisms. The union bound gives P(any failure) ≤ Σⱼpⱼ without requiring independence. Multiplying survival probabilities requires additional assumptions and is not automatic.

Code distance depends on the workload and error model. A phenomenological scaling formula teaches conditional trends, not decoder- and circuit-specific performance. Correlated faults and architecture constraints can invalidate extrapolation from one physical-error rate.

Vary dominant assumptions such as operation time, logical failure model, factory rate, routing overhead, precision and algorithm construction. Preserve the baseline and distinguish a lower bound, a modeled schedule and measured execution. Historical resource estimates retain their dates and assumptions; they are not automatically the current best estimate.

Normalize and read the probabilities

The conjugate of 4i is −4i, so |4i|² = 16. The squared norm is 9 + 16 = 25, and the norm is 5. Dividing both entries by 5 gives |ψ⟩ = (3/5)|0⟩ + (4i/5)|1⟩.

The amplitudes are (0.6,0.8i); computational-basis probabilities are (0.36,0.64). Their magnitude ratio is 3:4, and the second amplitude has relative phase π/2. Replacing 0.8i with 0.8 keeps these probabilities but changes the state.

Multiplying the whole vector by i gives (0.6i,−0.8), an equivalent pure state. Imaginary amplitudes still matter: a common phase is unobservable, while a phase difference can alter later interference.

Construct the density matrix

The outer product |ψ⟩⟨ψ| is [[0.36,−0.48i],[0.48i,0.64]]. Its trace is one and determinant zero. In ρ = (I+xX+yY+zZ)/2, the upper off-diagonal entry is (x−iy)/2. Thus (x,y,z) = (0,0.96,−0.28). Its squared length is 0.9216 + 0.0784 = 1.

The sign matters: y = −2 Im(ρ₀₁), not +2 Im(ρ₀₁). A computational-basis histogram cannot reveal the wrong sign.

Apply H before squaring

H(a,b) = ((a+b)/√2,(a−b)/√2). Here the output is ((3+4i)/(5√2),(3−4i)/(5√2)); both squared magnitudes are 25/50 = 1/2. Balanced probabilities do not imply equal or real amplitudes.

Normalizing (3,4) instead gives the same initial 0.36:0.64 histogram. After H, its probabilities are 49/50 and 1/50. The difference comes from relative phase.

Normalization and applying the Born rule to an already normalized state are separate operations. An all-zero vector cannot be normalized, and nonfinite inputs are invalid. An ideal arithmetic result is not evidence of a physical gate or a tested SDK implementation.

Form the projector with a conjugated row

Let |ψ⟩ = (√3/2)|0⟩ + (i/2)|1⟩. Its probabilities are 3/4 and 1/4. The outer product gives ρ = [[3/4,−i√3/4],[i√3/4,1/4]]. Use the conjugate transpose, not an ordinary transpose.

For ρ = (I+xX+yY+zZ)/2, x = 2 Re(ρ₀₁), y = −2 Im(ρ₀₁), and z = ρ₀₀−ρ₁₁. The Bloch vector is (0,√3/2,1/2). Its length is one, and the Y expectation is positive √3/2.

For a valid one-qubit state with Bloch length r, eigenvalues are (1+r)/2 and (1−r)/2. Here they are 1 and 0. Purity is Tr(ρ²) = (1+r²)/2 = 1. Base-two von Neumann entropy is −Σⱼλⱼ log₂λⱼ = 0 bits, using the continuous convention 0 log₂0 = 0.

Remove coherence without changing the histogram

Setting the off-diagonal entries to zero gives diag(3/4,1/4). Its Bloch vector is (0,0,1/2), purity 5/8 and entropy approximately 0.811278 bits. The computational-basis probabilities are unchanged, but the ideal Y expectation is now zero.

For I/2, the Bloch vector is zero, eigenvalues are 1/2 and 1/2, purity is 1/2 and entropy is one bit. Many pure-state ensembles give this same density matrix; it does not carry a uniquely determined hidden surface point.

A globally pure Bell state has local density matrix I/2 on either subsystem. Local entropy does not mean that the global state was a classical mixture.

Require positivity

The matrix [[1/2,0.6],[0.6,1/2]] is Hermitian with trace one, yet its eigenvalues are 1.1 and −0.1. It is not a physical density matrix. Plausible diagonal probabilities are insufficient: the matrix must also be positive semidefinite.

Small numerical excesses within a stated tolerance can be roundoff. Substantial violations cannot be silently repaired as though the input were unchanged. Any projection or state-reconstruction procedure is a separate estimator.

Multiply rows by the state vector

In the basis |0⟩, |1⟩: X = [[0,1],[1,0]], Y = [[0,−i],[i,0]], Z = [[1,0],[0,−1]], and H = [[1,1],[1,−1]]/√2. Multiply the matrix by the amplitudes before taking squared magnitudes.

X|ψ⟩ = (4i/5,3/5), exchanging the computational-basis probabilities. Y|ψ⟩ = (4/5,3i/5); the first entry is positive because (−i)(4i/5) = 4/5. Z|ψ⟩ = (3/5,−4i/5), changing relative phase while preserving the current histogram.

H|ψ⟩ = ((3+4i)/(5√2),(3−4i)/(5√2)), with outcome probabilities 1/2 and 1/2. Comparing only those probabilities can miss a wrong phase.

Check invariants and order

X² = Y² = Z² = H² = I. Applying any one twice recovers the state. Unitaries preserve normalization and inner products; numerical checks require a stated tolerance.

On |0⟩, H then Z gives |−⟩, while Z then H gives |+⟩. Both have 50:50 computational-basis probabilities. An X measurement gives −1 for the first and +1 for the second.

XZ versus ZX alone gives only a global minus sign on an isolated target. The operator algebra is noncommuting, but that particular output difference alone is not observable. Adding control changes the phase accounting.

Separate arithmetic from SDK or hardware evidence

An SDK comparison must map its basis ordering explicitly and compare states up to global phase where appropriate. Component-by-component comparison can reject equivalent vectors; probability-only comparison can accept inequivalent vectors.

These hand calculations and reference arithmetic describe ideal mathematical gates. They do not certify device noise, compilation or an arbitrary backend. No physical-device execution or executed SDK adapter is demonstrated in this lesson.

Define the expression before evaluating it

Use |Φ+⟩ = (|00⟩+|11⟩)/√2. Restrict local observables to the X–Z plane: A(a) = cos(a)Z + sin(a)X and B(b) = cos(b)Z + sin(b)X, with outcomes +1 or −1. In this model E(a,b) = cos(a−b), with angles in radians.

Choose S = E(a₀,b₀) + E(a₀,b₁) + E(a₁,b₀) − E(a₁,b₁). Local hidden-variable models under the relevant locality and setting-independence assumptions obey |S| ≤ 2.

Set a₀ = 0, a₁ = π/2, b₀ = π/4 and b₁ = −π/4. The correlations are, respectively, 1/√2, 1/√2, 1/√2 and −1/√2. Therefore S = 2√2 ≈ 2.828427. Each correlation is within [−1,1]; this is below the algebraic maximum 4 and reaches the quantum bound.

Add one named noise assumption

Let ρ(v) = v|Φ+⟩⟨Φ+| + (1−v)I₄/4, where 0 ≤ v ≤ 1. The maximally mixed term has zero correlation for these traceless observables, so E is multiplied by v. At the chosen settings S = 2√2v.

For v = 0.8, S ≈ 2.262742. The chosen expression exceeds 2 when v > 1/√2. This statement is confined to this state family and these measurements. No violation here does not prove separability or certify a device as classical.

Nonfinite angles and visibility outside [0,1] are invalid. The stated model cannot exceed |S| = 2√2. At v = 0 all four correlations vanish. The sign convention for S must remain explicit.

Identify the missing experimental evidence

A physical estimate needs counts for all setting pairs, a setting-selection protocol and a statistical analysis. Detection, timing, independence and data selection affect the interpretation. Favorable postselection without accounting for its procedure can alter the claim.

Matching 00 and 11 in one basis can also arise from a classical mixture. Choosing a visibility value sets a model assumption; it does not measure entanglement. Exact or synthetic model output does not establish loophole closure, device independence or communication security.

Apply the proposed copier to a superposition

Suppose U copies computational-basis inputs into a blank target: |0⟩|0⟩ becomes |00⟩ and |1⟩|0⟩ becomes |11⟩. CNOT with q₀ controlling q₁ does this for that orthogonal set.

For |+⟩|0⟩ = (|00⟩+|10⟩)/√2, linearity gives (|00⟩+|11⟩)/√2. Two independent copies would be |+⟩|+⟩ = (|00⟩+|01⟩+|10⟩+|11⟩)/2. The actual output lacks the |01⟩ and |10⟩ terms.

Their overlap is 1/√2 and squared pure-state fidelity is 1/2. They are not related by global phase. For general a|0⟩+b|1⟩, encoding gives a|00⟩+b|11⟩; two copies would require coefficients a², ab, ab and b². A fixed linear map cannot produce that nonlinear dependence for every unknown input.

State the boundaries of the theorem

Known states can be prepared repeatedly from a classical description. A specified orthogonal set can be distinguished and copied. Approximate cloners can optimize a stated approximation. None is a universal perfect copier of an arbitrary unknown state.

Classical bits encoded in orthogonal states can be copied within that basis. The obstruction is maintaining every coherent relationship for unrestricted unknown quantum states; it is not a temporary hardware imperfection.

Encoding and transfer are different operations

Repetition encoding maps a|0⟩+b|1⟩ to a|000⟩+b|111⟩, not three independent copies. Suitable parity measurements can diagnose a restricted class of errors without learning the logical amplitudes. Measuring each physical qubit to recover a and b would destroy the protected coherent state.

Teleportation consumes the original input in its measurement protocol and uses shared entanglement plus classical communication. It does not leave a second intact copy at the sender. Copying, encoding, preparation and transfer must remain distinct.

Encode all four messages

Start with |Φ+⟩ = (|00⟩+|11⟩)/√2. Alice applies X on q₀ when m₁ = 1, then Z on q₀ when m₀ = 1. Thus messages 00, 01, 10 and 11 use I, X, Z and ZX, respectively.

The resulting states are: 00 → (|00⟩+|11⟩)/√2; 01 → (|01⟩+|10⟩)/√2; 10 → (|00⟩−|11⟩)/√2; 11 → (|01⟩−|10⟩)/√2. These are mutually orthogonal Bell states. Reversing X and Z in the last isolated encoding changes only global phase.

Bob’s local reduced state is I/2 for every message. He cannot read Alice’s choice from q₁ alone. Alice must transmit q₀ before he can make the joint decoding measurement.

Decode after transmission

Once Bob holds both qubits, he applies CNOT from q₀ to q₁, then H to q₀. The four encoded states become |00⟩, |01⟩, |10⟩ and |11⟩, respectively. Computational-basis measurement returns m₀m₁ in this ideal model.

Checking only 00 and 11 can hide reversed labels. The asymmetric messages 01 and 10 distinguish the encoding rules and output ordering.

Count the full communication resources

The protocol uses one previously distributed entangled pair, one transmitted qubit after the message is chosen, local operations and a joint measurement. “Two classical bits with one qubit” requires the entanglement assistance to be stated. Creating and distributing the shared pair was not free.

If establishing that pair is part of the same physical-channel workflow, include its cost together with storage, loss, synchronization and error handling. The elementary result does not give two classical bits per unassisted one-qubit channel.

The distant marginal before transmission does not reveal a readable message, so the protocol does not signal faster than light. A noisy implementation induces a four-message classical channel; retain per-message errors and preparation costs alongside an average success rate. Ideal arithmetic does not measure that channel.

Define the oracle and ordering

A hidden n-bit string s defines fₛ(x) = s·x modulo two. A standard bit oracle maps |x⟩|b⟩ to |x⟩|b xor fₛ(x)⟩. Preparing the target in |−⟩ gives phase kickback: |x⟩ receives (−1) raised to fₛ(x), while the target factor remains unchanged.

In the exact deterministic classical query model, n independent queries are sufficient and necessary to identify an arbitrary n-bit secret: query strings with a single 1 to reveal its bits. The ideal quantum construction uses one oracle query under the corresponding quantum-access assumptions.

An explicit known-secret teaching circuit has gates determined by s. Giving that secret to the compiler is not discovering it. Oracle construction, target preparation, other gates and physical execution are distinct from query count.

Follow an asymmetric three-bit example

Set n = 3 and s = 110, with the leftmost bit labeled q₀. Start in |000⟩ and apply H to each qubit. All eight basis amplitudes become 1/√8.

In basis order 000, 001, 010, 011, 100, 101, 110, 111, the phase oracle gives signs +, +, −, −, −, −, +, +. Reversing the secret would change this pattern and the final state.

After the final Hadamards, the amplitude of |y⟩ is 2⁻ⁿ times the sum over all x of (−1) raised to [(s xor y)·x]. For y = s, all terms are +1 and the amplitude is 1. If y differs in any bit, the terms pair with opposite signs and cancel.

The output is |110⟩, with probability one in the ideal model. Predict the string before revealing it, and inspect the phase signs before the last Hadamards: the cancellation in amplitudes explains the result.

Bound the conclusion

The query comparison must use the same problem and compatible oracle access, rather than charging unlike implementation work to only one side. The explicit known-secret oracle has O(n) gate structure; one query is not one operation for the whole workflow.

The n-versus-one comparison here is not exponential and does not solve arbitrary hidden-data problems. It uses the special linear form of fₛ. A tiny ideal example or locally checked reference calculation does not demonstrate practical advantage over a classical computer.

No physical-device execution or executed SDK adapter is demonstrated here. A software translation must retain the named bit ordering and test asymmetric strings such as 110, not rely only on symmetric examples.