Parity-Selective Fock Superpositions

• Parity-selective Fock superpositions are bosonic states restricted to a single photon-number parity sector, enabling precise classification via even or odd support.
• State-engineering methods such as cavity QED, optical parametric amplification, and measurement-based schemes achieve high-fidelity preparation and direct parity diagnostics.
• These states are pivotal in quantum error correction and parity-based metrology, offering enhanced phase sensitivity and robust performance in bosonic-code applications.

Searching arXiv for recent and foundational papers on parity-selective Fock superpositions and related optical/bosonic-state methods. Parity-selective Fock superpositions are bosonic states whose Fock expansion is confined to a single photon-number parity sector, or whose coherence across parity sectors is itself the object of study. For a finite superposition

$|\psi\rangle=\sum_{n=0}^{N} c_n|n\rangle,$

even-parity support means $c_n=0$ for all odd $n$, odd-parity support means $c_n=0$ for all even $n$, and mixed-parity support contains both. With the photon-number parity operator $\hat\Pi=(-1)^{\hat n}=e^{i\pi \hat n}$, even-only and odd-only states are parity eigenstates, whereas mixed-parity coherent superpositions are not. In current research, this structure appears in optical tomography, heralded state engineering, ancilla-mediated bosonic control, bosonic quantum error correction, and parity-based metrology (Filippov et al., 2011, Turek et al., 22 May 2026, Kudra et al., 2022).

1. State-space structure and parity classification

The defining algebraic feature of a parity-selective Fock superposition is support on one of the two eigenspaces of $\hat\Pi$. The action

$\hat\Pi|n\rangle = (-1)^n |n\rangle$

implies that every Fock state is already a parity eigenstate, so a superposition restricted to $\{|0\rangle,|2\rangle,|4\rangle,\dots\}$ has eigenvalue $+1$, and one restricted to $c_n=0$0 has eigenvalue $c_n=0$1 (Gerry et al., 2010). This distinction is exact at the level of support, independent of how many Fock components are populated within the chosen sector.

For a general pure state

$c_n=0$2

the parity expectation value is

$c_n=0$3

with $c_n=0$4 and $c_n=0$5 (Gerry et al., 2010). This quantity certifies whether a state is confined to one parity sector, but it is only a sector witness. It does not distinguish different even-only states from one another, different odd-only states from one another, or coherent mixed-parity superpositions from incoherent mixtures with the same diagonal number statistics.

Parity selectivity is preserved by operations that change photon number by multiples of two. In particular, the squeezing operator

$c_n=0$6

preserves Fock parity, so codewords of the form

$c_n=0$7

occupy a single parity sector determined by $c_n=0$8; for the optimal choice $c_n=0$9, both logical states lie entirely in the odd sector (Zeng et al., 5 Oct 2025).

2. Optical tomography and direct parity diagnostics

Optical tomography gives a direct quadrature-space representation of parity-selective superpositions. For finite Fock support, the rotated quadrature amplitudes are proportional to Hermite polynomials,

$n$0

and the crucial identity

$n$1

provides the exact bridge between number parity and quadrature symmetry (Filippov et al., 2011). The measured optical tomogram is

$n$2

The explicit tomogram formula for an arbitrary finite Fock-state superposition shows that the diagonal terms depend on $n$3, whereas off-diagonal coherence terms involve products $n$4 multiplied by phase factors $n$5 (Filippov et al., 2011). Since $n$6 has parity $n$7, interference between same-parity sectors is even in $n$8, while interference between opposite-parity sectors is odd in $n$9. This yields a sharp operational criterion: for even-only and odd-only states,

$c_n=0$0

for every local-oscillator phase $c_n=0$1, whereas coherent mixed-parity states generally satisfy

$c_n=0$2

at fixed $c_n=0$3 (Filippov et al., 2011).

All states obey the more general identity

$c_n=0$4

which the same work proposes as a direct data-consistency check for homodyne experiments (Filippov et al., 2011). For parity-selective states this becomes a stronger fixed-$c_n=0$5 symmetry, so failure of $c_n=0$6 symmetry at fixed $c_n=0$7 indicates mixed-parity contamination, decoherence, or experimental error. The paper further notes that for pure finite-support Fock superpositions, measuring $c_n=0$8 and $c_n=0$9 is sufficient to determine the initial state up to a known twofold ambiguity.

Low-lying examples make the distinction transparent. The mixed-parity state $n$0 has a tomogram with a linear term in $n$1,

$n$2

so its quadrature distribution is generically asymmetric in $n$3. By contrast, the even-parity state $n$4 yields

$n$5

which is pointwise even in $n$6 for all $n$7 (Filippov et al., 2011).

Thermal admixture degrades these signatures in a parity-specific way. For

$n$8

the tomogram is

$n$9

and $\hat\Pi=(-1)^{\hat n}=e^{i\pi \hat n}$0 is a centered Gaussian, independent of $\hat\Pi=(-1)^{\hat n}=e^{i\pi \hat n}$1 and even in $\hat\Pi=(-1)^{\hat n}=e^{i\pi \hat n}$2 (Filippov et al., 2011). As a result, odd-in-$\hat\Pi=(-1)^{\hat n}=e^{i\pi \hat n}$3 mixed-parity interference is directly suppressed by the factor $\hat\Pi=(-1)^{\hat n}=e^{i\pi \hat n}$4, while same-parity interference retains evenness but loses contrast. The corresponding purity depends only on $\hat\Pi=(-1)^{\hat n}=e^{i\pi \hat n}$5, not on relative phases, so purity by itself does not diagnose parity coherence.

3. State-engineering routes

A direct experimental route to parity-selective superpositions is amplitude sculpting from a coherent state in a dispersive superconducting cavity. In that method, the cavity is first prepared in

$\hat\Pi=(-1)^{\hat n}=e^{i\pi \hat n}$6

and photon-number-selective multitone qubit rotations modulate each Fock amplitude independently. Post-selection on the qubit excited state projects the cavity into

$\hat\Pi=(-1)^{\hat n}=e^{i\pi \hat n}$7

so parity-selective targets are obtained simply by setting $\hat\Pi=(-1)^{\hat n}=e^{i\pi \hat n}$8 on the unwanted parity sector (Wang et al., 2017). The paper experimentally demonstrates this capability with squeezed states, whose Fock expansion contains only even photon numbers,

$\hat\Pi=(-1)^{\hat n}=e^{i\pi \hat n}$9

and reports fidelities $\hat\Pi$0, $\hat\Pi$1, and $\hat\Pi$2 for $\hat\Pi$3, with success probabilities $\hat\Pi$4, $\hat\Pi$5, and $\hat\Pi$6 (Wang et al., 2017).

A measurement-based alternative starts from a coherent state in a resonator coupled to an ancillary qubit by the Jaynes–Cummings interaction. Repeated rounds of free evolution and projective ancilla measurement induce a diagonal number-space filter,

$\hat\Pi$7

which can preserve selected Fock sectors while exponentially suppressing the rest (Zhang et al., 2024). With ground-state conditioning at resonance,

$\hat\Pi$8

the vacuum is automatically protected, and choosing

$\hat\Pi$9

also protects $\hat\Pi|n\rangle = (-1)^n |n\rangle$0, enabling preparation of

$\hat\Pi|n\rangle = (-1)^n |n\rangle$1

For even $\hat\Pi|n\rangle = (-1)^n |n\rangle$2, these are even-parity superpositions; for odd $\hat\Pi|n\rangle = (-1)^n |n\rangle$3, they are coherent superpositions across parity sectors. The protocol generates $\hat\Pi|n\rangle = (-1)^n |n\rangle$4 with $\hat\Pi|n\rangle = (-1)^n |n\rangle$5, fidelity over $\hat\Pi|n\rangle = (-1)^n |n\rangle$6, and fewer than $\hat\Pi|n\rangle = (-1)^n |n\rangle$7 measurements, while the sign alternates with the parity of the successful cycle number $\hat\Pi|n\rangle = (-1)^n |n\rangle$8 (Zhang et al., 2024).

In optical parametric amplification, parity selectivity can be fixed at the heralding stage. A two-mode OPA with squeezed-vacuum signal input and idler injection/detection $\hat\Pi|n\rangle = (-1)^n |n\rangle$9 yields a heralded signal whose parity is

$\{|0\rangle,|2\rangle,|4\rangle,\dots\}$0

because the signal input is even and the total process conserves parity (Turek et al., 22 May 2026). This produces two distinct families: large squeezed cat states for some $\{|0\rangle,|2\rangle,|4\rangle,\dots\}$1, and parity-selective few-Fock-state superpositions for others. The explicit examples

$\{|0\rangle,|2\rangle,|4\rangle,\dots\}$2

and

$\{|0\rangle,|2\rangle,|4\rangle,\dots\}$3

realize even- and odd-parity low-order superpositions, respectively (Turek et al., 22 May 2026). For $\{|0\rangle,|2\rangle,|4\rangle,\dots\}$4, $\{|0\rangle,|2\rangle,|4\rangle,\dots\}$5, $\{|0\rangle,|2\rangle,|4\rangle,\dots\}$6, and $\{|0\rangle,|2\rangle,|4\rangle,\dots\}$7, the paper reports $\{|0\rangle,|2\rangle,|4\rangle,\dots\}$8, $\{|0\rangle,|2\rangle,|4\rangle,\dots\}$9 for $+1$0, and $+1$1, $+1$2 for $+1$3.

A cavity-QED filtering protocol for traveling light pulses provides a particularly explicit low-photon even-parity example. With a controlled phase flip

$+1$4

followed by an atomic rotation and atomic measurement, a displaced squeezed vacuum can be mapped in one round onto

$+1$5

The balanced target $+1$6 is reported with success probability $+1$7, and its coherence exceeds the ultimate QNG threshold $+1$8 (Teja et al., 5 Sep 2025). A plausible implication is that the same parity-splitting mechanism can also isolate the odd branch $+1$9, although that branch is not developed in the same detail.

4. Parity-selective operations, transfer, and recovery maps

Parity-selective control is not limited to state preparation; it also appears as an operational primitive. In circuit QED, the Selective Number-dependent Arbitrary-Phase Photon-Addition gate realizes simultaneous transitions

$c_n=0$00

for chosen $c_n=0$01 (Kudra et al., 2022). Because adding one photon flips number parity, driving only odd $c_n=0$02 implements a coherent map from the odd-parity cavity sector to the even-parity sector while leaving even states approximately unchanged. The paper demonstrates

$c_n=0$03

with bosonic fidelities $c_n=0$04, $c_n=0$05, and $c_n=0$06 for three phases, and shows that the same drives leave the even input $c_n=0$07 nearly unchanged, with cavity-state fidelities around $c_n=0$08–$c_n=0$09 (Kudra et al., 2022). With qubit reset, this becomes an effective parity-restoring recovery primitive for cat and binomial codes.

A different use of parity appears in constrained bosonic state transfer. For two oscillators coupled with opposite signs to one ancilla, transfer is exactly equivalent to synthesizing parity on the antisymmetric bright mode

$c_n=0$10

The transfer criterion on a finite cutoff is

$c_n=0$11

so even bright-mode occupations must acquire phase $c_n=0$12 and odd occupations phase $c_n=0$13 (Laha et al., 28 Jun 2026). Resonant single-ancilla transfer is exact only for the one-photon sector; detuned Jaynes–Cummings evolution supplies a two-parameter approximation route. For cutoff $c_n=0$14, the best-found parameters

$c_n=0$15

give

$c_n=0$16

separating coherent parity-synthesis error from residual ancilla excitation (Laha et al., 28 Jun 2026).

Ancilla-assisted generalized-parity measurement extends the notion from ordinary parity to modular classes. After $c_n=0$17 rounds, the generalized-parity subspace is

$c_n=0$18

with $c_n=0$19 reducing to ordinary even/odd parity (Zhang et al., 31 May 2026). The protocol retains every trajectory by adapting the next interrogation time to the last ancilla outcome, and for a resonant JC model it can isolate target photon numbers $c_n=0$20 within $c_n=0$21 rounds, with averaged fidelity about $c_n=0$22 and about $c_n=0$23 probability of fidelity above $c_n=0$24 in ensemble sampling (Zhang et al., 31 May 2026). This suggests a measurement backbone for parity-sector filtering, although the same work is explicit that its primary final target is a single large Fock state rather than a stable parity-selective superposition.

5. Metrological and coding significance

Parity is a natural observable in interferometry precisely because it resolves the even/odd redistribution induced by nonclassical Fock superpositions. For N00N inputs, parity measurement on one output mode yields

$c_n=0$25

with phase uncertainty

$c_n=0$26

i.e. the Heisenberg limit (Gerry et al., 2010). For twin-Fock inputs, the first beam splitter produces an even-only superposition in each arm,

$c_n=0$27

and the output parity signal becomes

$c_n=0$28

with sensitivity near $c_n=0$29 around $c_n=0$30 (Gerry et al., 2010). In both cases parity does not resolve the detailed in-sector coefficients; rather, it converts them into a phase-sensitive binary readout.

Not all parity-structured Fock superpositions are equally effective metrologically. Two-mode squeezed vacuum and pair coherent states both occupy only the number-correlated diagonal subspace

$c_n=0$31

hence only even total photon numbers, yet their parity-interferometric behavior differs sharply (Gerry et al., 2010). Two-mode squeezed vacuum achieves Heisenberg-limited sensitivity but not super-resolution, whereas pair coherent states yield both. The decisive distinction is the weighting $c_n=0$32: broad, thermal-like support washes out oscillatory parity structure, while narrow, sub-Poissonian support preserves it.

Parity-selective few-Fock-state superpositions also function as bosonic-code resources. The superposition-of-squeezed-Fock-states code keeps both logical codewords in one parity sector, remains exactly orthogonal for any squeezing, and for the optimal odd-parity choice $c_n=0$33 achieves residual Knill–Laflamme violations scaling as $c_n=0$34 (Zeng et al., 5 Oct 2025). The same work reports that, at about $c_n=0$35 dB squeezing and $c_n=0$36, the autonomous QEC protocol surpasses break-even. In a different platform, the even-parity $c_n=0$37 superposition generated by cavity-QED filtering reaches maximal phase-sensing quantum Fisher information $c_n=0$38 at $c_n=0$39, exceeding the $c_n=0$40 limit of any superposition up to a single photon in the same model (Teja et al., 5 Sep 2025). OPA-generated parity-selective states $c_n=0$41, $c_n=0$42, and related configurations likewise show strong Wigner negativity, substantial phase-space complexity, and QFI performance above both the SQL and the paper’s adopted Heisenberg reference for several parameter regimes (Turek et al., 22 May 2026).

6. Limitations, fragility, and interpretive cautions

Parity is structurally robust but diagnostically coarse. The expectation value $c_n=0$43 depends only on diagonal number statistics, so it cannot by itself certify parity coherence or distinguish coherent parity mixing from incoherent sector mixing (Gerry et al., 2010). Optical tomography refines that picture by resolving odd-in-$c_n=0$44 interference terms, but thermal admixture suppresses precisely those terms and purity remains insensitive to relative phases, depending only on $c_n=0$45 (Filippov et al., 2011). A parity label is therefore reliable as a sector classification, but not as a complete descriptor of state structure.

A second caution is that global parity conservation need not imply clean bosonic parity selectivity in reduced descriptions. In the quantum Rabi model, the exact conserved symmetry is

$c_n=0$46

yet in the strong-coupling superradiant regime the bosonic part of a single eigenstate can develop comparable odd- and even-Fock populations (Yang et al., 2022). The same study reports that parity expectation values of individual low-lying eigenstates become irregular, while the pairwise parity sum within each nearly degenerate level pair remains zero. This distinction between global light-matter parity and reduced bosonic Fock-parity content is essential when interpreting parity-selective signatures in hybrid systems.

A third caution is that parity selectivity alone does not determine performance. The comparison between two-mode squeezed vacuum and pair coherent states shows that identical support restrictions in total photon-number parity can yield qualitatively different parity signals because the decisive variable is the detailed weighting across Fock sectors (Gerry et al., 2010). Likewise, adaptive generalized-parity protocols can create modularly selected Fock combs at intermediate stages, but those same schemes may still be optimized for eventual single-Fock-state isolation rather than for preserving a parity-selective superposition as the final resource (Zhang et al., 31 May 2026). In practice, parity-selective Fock superpositions are best characterized by combining support restrictions, coherence diagnostics, and the specific dynamical map that prepared them.