Quantum theory

A quantum fluctuation is a nonzero variance of an observable in a quantum state, even when its mean vanishes. A vacuum field mode has such zero-point variance. In an inflationary background, the expanding geometry transfers short-wavelength vacuum field fluctuations into primordial perturbations as their wavelengths cross the Hubble radius.

For a particle of charge $-e$, the changes $A\mapsto A+\nabla f$ and $\varphi \mapsto \varphi -{\partial }_{t}f$ leave the minimally coupled Schrödinger equation invariant when $\psi \mapsto e^{-ief/\hslash }\psi$.

A quantum measurement assigns probabilities to classical outcomes through positive operators that sum to the identity. A projective measurement uses mutually orthogonal projections.

An ideal measurement preserving the measured eigenspaces. In the projective setting, the selective instrument $\rho \mapsto P_j\rho P_j$ preserves every vector within outcome $j$, including coherence inside a degenerate eigenspace. Repeated measurements agree in the absence of intervening dynamics changing the measured quantity. Measuring a finer observable can destroy degeneracy coherence and need not realize the same Lüders rule instrument.

Choose a shared random axis uniformly from $x,y,z$, perform its entanglement-assisted nondemolition parity measurement with one Bell pair, and accept the anticorrelated parity. The spin singlet state always passes unchanged. The averaged acceptance operator is $\Omega ={\sum }_j(I-{\sigma }_j\otimes {\sigma }_j)/6=P_s+(I-P_s)/3$, so an orthogonal triplet state passes with probability one third. This is a statistical verifier with a nonzero acceptance gap, rather than an exact single-shot singlet-versus-triplet decision. The Bell-pair cost of exact nondemolition singlet verification concerns the stronger zero-error task.

Exact singlet verification accepts the spin singlet state with certainty, rejects every orthogonal state, and preserves the singlet on acceptance. Each nonzero accepting Kraus operator is then a scalar multiple of the singlet projector, whose operator Schmidt rank is four. The local Kraus rank bound from an entangled resource excludes a single shared Bell pair, which has Schmidt rank two. Two shared Bell pairs suffice by a Bell-state nondemolition measurement followed by reporting whether its label is the singlet. This protocol may disturb triplet coherence. The rank bound concerns the resource Schmidt rank, and is not an entropy lower bound of two bits for arbitrary resource states.

Share a Bell state meter pair, apply local system-to-meter CNOT gates, and measure each meter in the computational basis. The parity of the local meter records measures system $Z_A{Z}_B$ while preserving all coherence within either parity sector. The complete tuple of local records has Kraus operator $P_{u\oplus v}/\sqrt{2}$. Each local record is uniform, and the nonlocal result requires local operations and classical communication to compare records. A local Hadamard gate conjugation supplies the corresponding $X_A{X}_B$ measurement.

A finite family of quantum states is antidistinguishable if a quantum measurement has labelled outcomes such that each outcome has zero probability for its corresponding state, while the outcomes form a complete quantum measurement. It can exclude a state without identifying which state was prepared. The PBR exclusion measurement for zero and plus supplies a four-state projective measurement example.

Postselection conditions a quantum measurement on a specified outcome. If its branch acts by a linear operator $K$ on a normalized input $|b⟩$, its probability is $\Vert K|b⟩{\Vert }^2$, and its normalized output is $K|b⟩/\Vert K|b⟩\Vert$. This conditional state is defined only when the probability is nonzero. Physically unsuccessful trials are discarded; postselection is not a deterministic implementation of the nonlinear normalization map.

For an ideal projective measurement with the Lüders rule, condition on both an initial quantum state preparation and a final successful postselection. Its outcome probabilities are proportional to $|⟨b|P_j|a⟩{|}^2$, where $a$ is the forward-evolved initial state and $b$ the backward-evolved final state. The normalization must be nonzero. The expression is symmetric in these two boundary states, although its derivation uses the ordinary Born rule and conditional state update.

In the ABL rule, probabilities depend on the full intermediate quantum measurement instrument. A binary Lüders rule quantum measurement preserves coherence within its unresolved complement, whereas a fully resolved projective measurement removes it. Summing fine-grained probabilities after quantum measurement does not generally reproduce the coarse-grained experiment. Different postselection success rates are part of the distinction.

For a uniform prestate in $N+1$ dimensions and poststate proportional to ${\sum }_{i=1}^N|i⟩-(N-1)|N+1⟩$, each binary question $P_i$ versus $I-P_i$ has conditional probability one for $i\le N$. A fully resolved quantum measurement instead assigns $1/D$ to each of the first $N$ outcomes and $(N-1{)}^2/D$ to the last. The context dependence of pre- and post-selected measurements explains why these alternative certainties do not imply simultaneous properties.

A measurement interaction correlates a measured system's alternatives with distinguishable states of an apparatus. Reading the apparatus then implements the associated quantum measurement.

A generalized quantum measurement is specified by operators $M_i$ satisfying ${\sum }_i{M}_i^{†}{M}_i=I$. On input $\rho$, outcome $i$ has probability $\mathrm{Tr}(M_i^{†}{M}_i\rho )$ and conditional state $M_i\rho M_i^{†}/\mathrm{Tr}(M_i^{†}{M}_i\rho )$.

A positive operator-valued measure is a family of positive operators $E_i$ satisfying ${\sum }_i{E}_i=I$. It determines measurement probabilities $\mathrm{Pr}(i)=\mathrm{Tr}(E_i\rho )$; choosing measurement operators with $E_i=M_i^{†}{M}_i$ additionally determines state updates.

A projective measurement is specified by mutually orthogonal projections $P_i$ with ${\sum }_i{P}_i=I$. On a state $\rho$, outcome $i$ has probability $\mathrm{Tr}(P_i\rho )$.

The orthonormal quantum states $|{\alpha }_s⟩=(|0⟩+(-1{)}^s{e}^{-i\alpha }|1⟩)/\sqrt{2}$, $s=0,1$, define an equatorial projective measurement. The measured observable is $\cos \alpha X-\sin \alpha Y$, with eigenvalue $(-1{)}^s$. Thus $\alpha =0$ measures $X$, and $\alpha =\pi /2$ measures $-Y$ with these outcome labels. The sign convention matters when comparing different definitions of equatorial bases.

Conditional on outcome $i$ of a projective measurement, the Lüders rule updates the state to

The post-measurement state is the conditional quantum state after a specified measurement outcome. For a projective measurement, it is given by the Lüders rule.

If the outcome of a projective measurement $\{P_i\}$ is discarded, its nonselective state update is the pinching channelIt removes coherence between distinct measurement subspaces.

In an orthonormal basis, rank-one dephasing is the quantum channel $\Delta (\rho )={\sum }_i|{\psi }_i⟩⟨{\psi }_i|\rho |{\psi }_i⟩⟨{\psi }_i|$. It removes off-diagonal entries while preserving the basis probabilities. Its output Von Neumann entropy is their Shannon entropy, and the relative-entropy identity for rank-one dephasing quantifies the entropy increase.

Because $\log \Delta (\rho )$ is diagonal, $\mathrm{Tr}(\rho \log \Delta (\rho ))=\mathrm{Tr}(\Delta (\rho )\log \Delta (\rho ))$. Hence $D_{\mathrm{rel}}(\rho \Vert \Delta (\rho ))=S(\Delta (\rho ))-S(\rho )$. Support inclusion under rank-one dephasing handles zero probabilities, and Klein's inequality makes the difference nonnegative. Equality holds exactly when the state was already diagonal.

If $p_i=⟨{\psi }_i|\rho |{\psi }_i⟩=0$, positivity gives $\sqrt{\rho }|{\psi }_i⟩=0$, so the corresponding basis vector is in $\ker \rho$. Therefore $\ker \Delta (\rho )\subseteq \ker \rho$, and $\mathrm{supp}\rho \subseteq \mathrm{supp}\Delta (\rho )$. This makes the quantum relative entropy of $\rho$ relative to its rank-one dephasing finite.

Quantum state tomography estimates enough expectation values on many identically prepared systems to reconstruct their density operator. A single ordinary specimen cannot provide arbitrarily many independent samples without disturbance.

If a quantum instrument leaves every input state unchanged even after conditioning on its classical output, then its output distribution is independent of the input state. Thus an ordinary quantum process cannot learn arbitrary state information while preserving every state.

A quantum instrument associates each classical outcome with a completely positive trace-nonincreasing map; summing the outcome maps gives a quantum channel. It specifies both outcome probabilities and conditional output states.

If a state has good amplitude $\sin \theta$, the product of the reflection in the bad axis and the reflection in the state line rotates its good-bad plane by $2\theta$. After $k$ iterations the good probability isso $O(1/\sin \theta )$ iterations raise a small good amplitude close to one.

If the initial good amplitude $p=\sin \theta$ is known, an ancillary qubit can reduce it to $\sin {\theta }^{\prime }=pc$ with ${\theta }^{\prime }=\pi /(4k+2)$ and $0<c\le 1$. After $k$ ordinary amplitude-amplification iterations the good amplitude is $\sin ((2k+1){\theta }^{\prime })=1$, so the target state is prepared exactly.

For known good probability $p={\sin }^2\theta$, choose $j=\lceil \pi /(4\theta )-1/2\rceil$. A known reversible preparation of one state with good probability $p_{\ast }$ permits $j$ ordinary amplitude amplification iterations to reach the good subspace exactly. With ancillary marking access, prepare a flag with one-probability $q=p_{\ast }/p$ and define joint success as good data and flag one. Reflection about the prepared product state is known, while the joint phase test requires a controlled marking oracle or a computed Boolean output. This changes the success projector and enlarges the state space; it does not evade the obstruction to uniform probability lowering by a unitary. Query costs of preparation, its inverse and marking must all be counted.

Phase-matched amplitude amplification replaces the two sign-flip reflections by selective phase rotations. Choosing their phases from the known initial overlap can rotate the state exactly onto the target even when an integer number of ordinary Grover rotations would overshoot it.

A geometric amplitude-amplification schedule tries iteration counts $1,2,4,8,\dots$ when the initial good amplitude is unknown. It reaches the unknown optimal scale with only a geometric-series overhead.

For a unit vector $|\psi ⟩$, the reflection $I-2|\psi ⟩⟨\psi |$ negates the component parallel to $|\psi ⟩$ and fixes its orthogonal complement.

For a unit vector $|\psi ⟩$ and real $\varphi$,multiplies the component parallel to $|\psi ⟩$ by $e^{i\varphi }$ and fixes its orthogonal complement. The case $\varphi =\pi$ is a reflection operator.

A reflection operator is a unitary involution that changes sign on one orthogonal subspace and fixes its orthogonal complement. For a unit vector $|\psi ⟩$, $I-2|\psi ⟩⟨\psi |$ reflects in the hyperplane perpendicular to $|\psi ⟩$.

The HHL algorithm prepares a quantum state proportional to $A^{-1}b$ for a sparse, efficiently accessible, well-conditioned linear system $Ax=b$, provided that $|b⟩$ can be prepared efficiently. Its runtime depends polynomially on the sparsity, condition number, inverse precision, and $\log N$.

On a positive eigenvalue label $\lambda$ and a clean quantum ancilla, the HHL algorithm applies a quantum variable rotation with angle ${\theta }_{\lambda }=\arcsin (c/\lambda )$, where $0<c\le {\lambda }_{\min }$:Uncomputation of the eigenvalue register followed by conditioning on flag one multiplies each input eigenvector amplitude by $c/\lambda$. If $|b⟩$ is normalized, the success probability is $c^2\Vert A^{-1}|b⟩{\Vert }^2$. Choosing $c={\lambda }_{\max }/\kappa$ from a valid condition number bound gives success probability at least $1/{\kappa }^2$.