State-Subtraction Methods

• State-subtraction methods are a family of subtraction constructions that remove designated state components to yield finite, non-Gaussian remainders across various disciplines.
• In quantum optics and SU(1,1) interferometry, conditional photon subtraction and coherent operations create nonclassical states that enhance phase sensitivity and enable advanced state engineering.
• In perturbative QCD and kinetic theory, subtraction techniques eliminate double counting and infrared divergences, ensuring accurate modeling of particle interactions and resonances.

State-subtraction method denotes a family of subtraction-based constructions whose meaning depends on disciplinary context. In microwave and continuous-variable quantum optics, it refers to conditional photon-subtraction operations used for non-Gaussian state engineering; in SU(1,1) interferometry, to internal multi-photon subtraction of the probe state; in perturbative QCD, to local subtraction of unresolved radiation or infrared counterterms associated with specific phase-space configurations; and in nonequilibrium kinetic theory, to real intermediate state subtraction that removes on-shell resonant contributions already represented by decay and inverse-decay channels (Govia et al., 2012, Kang et al., 2023, Magnea et al., 2018, Ala-Mattinen et al., 2023). This suggests that the expression is field-dependent, but the common structural motif is the controlled removal of a designated state component in order to expose a finite, non-Gaussian, or non-double-counted remainder.

1. Quantum-optical operator content

In the microwave setting of dichotomic single-photon detection, the central object is the subtraction operator

$$\hat{B}\equiv\sum_{n=1}^\infty |n-1\rangle\langle n|,$$

which acts as $\hat{B}|n\rangle=|n-1\rangle$ for $n\ge 1$ and $\hat{B}|0\rangle=0$. The corresponding click-conditioned state update is

$$\rho \longrightarrow \rho'=\frac{\hat{B}\rho\hat{B}^\dagger}{\mathrm{Tr}[\hat{B}\rho\hat{B}^\dagger]}.$$

The same paper emphasizes that $\hat{B}$ is not the annihilation operator $\hat a$, but is related to it by $\hat a=\hat B\sqrt{\hat n}$; the absence of the $\sqrt n$ weighting is the essential nonlinearity of the measurement back-action in the Josephson Photomultiplier regime (Govia et al., 2012).

For bipartite continuous-variable systems, coherent subtraction generalizes fixed-order subtraction to coherent superpositions of many subtraction patterns. The subtraction operator is written as

$$C_S=\sum_{m_1,m_2=0}^{\infty} A_{m_1m_2}\,a_1^{m_1}a_2^{m_2}, \qquad \rho_S=C_S\rho_G C_S^\dagger,$$

with $\hat{B}|n\rangle=|n-1\rangle$0 a Gaussian kernel. Jiang et al. introduce the still more general operator

$\hat{B}|n\rangle=|n-1\rangle$1

and note that any two-mode CV state can be written in this form (Jiang et al., 2012).

The resulting distinction is technically important. Single-mode subtraction, fixed local subtraction $\hat{B}|n\rangle=|n-1\rangle$2, coherent subtraction $\hat{B}|n\rangle=|n-1\rangle$3, and annihilation-based subtraction $\hat{B}|n\rangle=|n-1\rangle$4 are different operations with different number-basis nonlinearities, separability properties, and implementation assumptions. In the cited literature, “state-subtraction method” therefore does not mean a unique operator identity, but a class of conditional or coherent subtraction maps.

2. Measurement-based state engineering in microwave cavities

The measurement-based version starts from a coherent state

$\hat{B}|n\rangle=|n-1\rangle$5

and applies one or more click-conditioned subtraction events. After one subtraction,

$\hat{B}|n\rangle=|n-1\rangle$6

which is not a coherent state; the photon-number distribution is no longer Poissonian, and the resulting state is non-Gaussian and nonclassical. After $\hat{B}|n\rangle=|n-1\rangle$7 clicks, the unnormalized state is $\hat{B}|n\rangle=|n-1\rangle$8, and the success probability is

$\hat{B}|n\rangle=|n-1\rangle$9

The protocol then exploits the noncommutativity of subtraction and coherent displacement,

$n\ge 1$0

to engineer families of target states by sequences of displacements and $n\ge 1$1-photon subtraction steps (Govia et al., 2012).

The paper identifies three principal output families: ordinary squeezed vacuum states ($n\ge 1$2), generalized squeezed states with $n\ge 1$3-fold rotational symmetry ($n\ge 1$4), and squeezed multi-component Schrödinger cat states. Their phase-space diagnostics are the Wigner and Husimi–$n\ge 1$5 functions; the cited simulations show $n\ge 1$6-fold rotational symmetry, strong squeezing along specific quadratures, multiple coherent peaks for cat-like states, and interference fringes with negativity. The same work reports generalized squeezed states ($n\ge 1$7) with fidelities $n\ge 1$8 with respect to

$n\ge 1$9

and success probabilities $\hat{B}|0\rangle=0$0 between $\hat{B}|0\rangle=0$1 and $\hat{B}|0\rangle=0$2 for modest $\hat{B}|0\rangle=0$3. For squeezed multi-component cat states, the paper gives fidelities $\hat{B}|0\rangle=0$4 and success probabilities typically between $\hat{B}|0\rangle=0$5 and $\hat{B}|0\rangle=0$6 in the reported examples (Govia et al., 2012).

Implementation is discussed in the setting of a high-$\hat{B}|0\rangle=0$7 superconducting microwave resonator strongly coupled to a Josephson Photomultiplier. In the operating regime of interest, the click Kraus operator is effectively $\hat{B}|0\rangle=0$8, while relaxation and dark counts are treated as deviations from the ideal model. Superconducting resonators with $\hat{B}|0\rangle=0$9-factors approaching $$\rho \longrightarrow \rho'=\frac{\hat{B}\rho\hat{B}^\dagger}{\mathrm{Tr}[\hat{B}\rho\hat{B}^\dagger]}.$$0 and photon lifetimes $$\rho \longrightarrow \rho'=\frac{\hat{B}\rho\hat{B}^\dagger}{\mathrm{Tr}[\hat{B}\rho\hat{B}^\dagger]}.$$1 ns, together with JPM measurement times $$\rho \longrightarrow \rho'=\frac{\hat{B}\rho\hat{B}^\dagger}{\mathrm{Tr}[\hat{B}\rho\hat{B}^\dagger]}.$$2 ns or less, imply that on the order of $$\rho \longrightarrow \rho'=\frac{\hat{B}\rho\hat{B}^\dagger}{\mathrm{Tr}[\hat{B}\rho\hat{B}^\dagger]}.$$3 detection operations can be performed within a single cavity lifetime (Govia et al., 2012).

3. Coherent subtraction and entanglement in bipartite continuous variables

For bipartite Gaussian kernels, coherent subtraction is analyzed primarily as an entanglement-engineering and entanglement-detection procedure. If $$\rho \longrightarrow \rho'=\frac{\hat{B}\rho\hat{B}^\dagger}{\mathrm{Tr}[\hat{B}\rho\hat{B}^\dagger]}.$$4 is a separable Gaussian kernel in second standard form, then its coherent-state decomposition yields

$$\rho \longrightarrow \rho'=\frac{\hat{B}\rho\hat{B}^\dagger}{\mathrm{Tr}[\hat{B}\rho\hat{B}^\dagger]}.$$5

with $$\rho \longrightarrow \rho'=\frac{\hat{B}\rho\hat{B}^\dagger}{\mathrm{Tr}[\hat{B}\rho\hat{B}^\dagger]}.$$6 Gaussian and positive-definite. Acting with coherent subtraction gives

$$\rho \longrightarrow \rho'=\frac{\hat{B}\rho\hat{B}^\dagger}{\mathrm{Tr}[\hat{B}\rho\hat{B}^\dagger]}.$$7

which remains separable. Accordingly, coherent subtraction cannot create entanglement from a separable second-standard-form Gaussian kernel (Jiang et al., 2012).

The symmetric coherent subtraction of a symmetric two-mode squeezed thermal state,

$$\rho \longrightarrow \rho'=\frac{\hat{B}\rho\hat{B}^\dagger}{\mathrm{Tr}[\hat{B}\rho\hat{B}^\dagger]}.$$8

has a sharper result. For the Gaussian kernel, the necessary and sufficient separability condition is

$$\rho \longrightarrow \rho'=\frac{\hat{B}\rho\hat{B}^\dagger}{\mathrm{Tr}[\hat{B}\rho\hat{B}^\dagger]}.$$9

Jiang et al. show that the separability condition of $\hat{B}$0 is exactly the same as that of its Gaussian kernel. In this case, symmetric coherent subtraction preserves the separability boundary rather than shifting it (Jiang et al., 2012).

A different behavior occurs for a generic first-standard-form Gaussian kernel at the boundary $\hat{B}$1, with

$\hat{B}$2

For

$\hat{B}$3

the realignment criterion leads to a sufficient entanglement condition in terms of

$\hat{B}$4

If $\hat{B}$5 for all $\hat{B}$6 and $\hat{B}$7 for at least one pair $\hat{B}$8, then the symmetric coherent subtraction of the boundary-separable Gaussian kernel is entangled. In phase language, constructive interference among subtraction amplitudes is the relevant design rule (Jiang et al., 2012).

This body of results places coherent subtraction between two limits. In one limit it is a non-entangling dressing of a separable Gaussian mixture; in the other it is an entanglement-generating non-Gaussian operation whose efficacy depends on the coefficient structure of the coherent superposition.

4. Internal multi-photon subtraction in SU(1,1) interferometry

In SU(1,1) interferometry, the subtraction step is placed inside the interferometer, after the first two-mode parametric amplifier and before the phase shifter and second amplifier. With input

$\hat{B}$9

the subtraction operator is

$\hat a$0

so that the normalized post-selected state is

$\hat a$1

The phase is encoded only on mode $\hat a$2 through $\hat a$3, and homodyne detection measures

$\hat a$4

Phase sensitivity is evaluated by

$\hat a$5

The paper analyzes phase sensitivity, quantum Fisher information, and quantum Cramér–Rao bound for this internally photon-subtracted probe (Kang et al., 2023).

The principal findings are systematic. Internal photon subtraction always improves phase sensitivity relative to the standard SU(1,1) interferometer without subtraction, and increasing the number of subtracted photons improves sensitivity further. The QFI,

$\hat a$6

also increases with gain $\hat a$7, coherent amplitude $\hat a$8, and subtraction number. For fixed $\hat a$9, the standard SU(1,1) interferometer cannot surpass SQL, whereas symmetric multi-photon subtraction on both modes can surpass SQL over a broad parameter range, even under significant internal losses such as $\hat a=\hat B\sqrt{\hat n}$0 (Kang et al., 2023).

Mode asymmetry is intrinsic because the phase shift and homodyne measurement both act on mode $\hat a=\hat B\sqrt{\hat n}$1. As a result, single-mode subtraction on $\hat a=\hat B\sqrt{\hat n}$2 and on $\hat a=\hat B\sqrt{\hat n}$3 has different metrological consequences. For homodyne-based phase sensitivity, subtraction on mode $\hat a=\hat B\sqrt{\hat n}$4 tends to be better at small $\hat a=\hat B\sqrt{\hat n}$5 or small $\hat a=\hat B\sqrt{\hat n}$6, while subtraction on mode $\hat a=\hat B\sqrt{\hat n}$7 becomes superior at larger $\hat a=\hat B\sqrt{\hat n}$8 or larger $\hat a=\hat B\sqrt{\hat n}$9. For QFI and QCRB, subtraction on mode $\sqrt n$0 is generally slightly better overall. The same analysis under loss yields

$\sqrt n$1

and the photon-subtracted schemes retain improved robustness against internal photon losses (Kang et al., 2023).

In this metrological usage, state subtraction is not primarily a state-generation primitive for standalone nonclassicality, but a controlled non-Gaussian transformation of the intracavity probe state.

5. Infrared subtraction in perturbative QCD

In perturbative QCD, subtraction methods are introduced because real and virtual corrections are separately infrared divergent. The subtraction term locally reproduces the soft and collinear behavior of the real matrix element in phase space, is simple enough to be integrated analytically over unresolved degrees of freedom, and is then added back in integrated form so that pole cancellation with virtual terms becomes explicit. The cited literature explicitly interprets this logic as a state-subtraction method in the sense that one subtracts local counterterms associated with a given color-ordered or sector-defined phase-space configuration (Daleo et al., 2010, Magnea et al., 2018).

In NNLO antenna subtraction with one hadronic initial state, the required ingredients are initial–final three-parton tree antennae $\sqrt n$2, four-parton tree antennae $\sqrt n$3, and one-loop three-parton antennae $\sqrt n$4. The framework organizes double-real, real–virtual, and double-virtual pieces by subtracting antenna counterterms built from color-connected radiators and reduced matrix elements with mapped momenta. For the integration of initial–final antennae, the phase-space integrals are rewritten as cut loop integrals and reduced by IBP and Lorentz-invariance identities to master integrals; the paper reports $\sqrt n$5 master integrals for NNLO double-real initial–final antennae and $\sqrt n$6 master integrals for NNLO one-loop initial–final antennae. The integrated antennae are distributions in the Bjorken-like variable $\sqrt n$7, involving $\sqrt n$8 and plus-distributions, and were cross-checked against known NNLO DIS coefficient functions (Daleo et al., 2010).

Local analytic sector subtraction reorganizes the same infrared problem with sector functions and Catani–Seymour-type mappings. At NNLO for final-state radiation, the method is defined as local, sectorized, and analytic: counterterms match singular behavior point-by-point in phase space, singular regions are isolated by sector partitions, and the counterterms are integrated analytically over unresolved radiation. The construction uses NLO sector functions $\sqrt n$9, NNLO sector functions $$C_S=\sum_{m_1,m_2=0}^{\infty} A_{m_1m_2}\,a_1^{m_1}a_2^{m_2}, \qquad \rho_S=C_S\rho_G C_S^\dagger,$$0, projectors such as $$C_S=\sum_{m_1,m_2=0}^{\infty} A_{m_1m_2}\,a_1^{m_1}a_2^{m_2}, \qquad \rho_S=C_S\rho_G C_S^\dagger,$$1, $$C_S=\sum_{m_1,m_2=0}^{\infty} A_{m_1m_2}\,a_1^{m_1}a_2^{m_2}, \qquad \rho_S=C_S\rho_G C_S^\dagger,$$2, $$C_S=\sum_{m_1,m_2=0}^{\infty} A_{m_1m_2}\,a_1^{m_1}a_2^{m_2}, \qquad \rho_S=C_S\rho_G C_S^\dagger,$$3, $$C_S=\sum_{m_1,m_2=0}^{\infty} A_{m_1m_2}\,a_1^{m_1}a_2^{m_2}, \qquad \rho_S=C_S\rho_G C_S^\dagger,$$4, and $$C_S=\sum_{m_1,m_2=0}^{\infty} A_{m_1m_2}\,a_1^{m_1}a_2^{m_2}, \qquad \rho_S=C_S\rho_G C_S^\dagger,$$5, and produces finite combinations of the schematic form

$$C_S=\sum_{m_1,m_2=0}^{\infty} A_{m_1m_2}\,a_1^{m_1}a_2^{m_2}, \qquad \rho_S=C_S\rho_G C_S^\dagger,$$6

The corresponding integrated counterterms cancel the poles of real–virtual and double-virtual contributions (Magnea et al., 2018, Magnea et al., 2019).

The NLO extension to initial- and final-state radiation preserves the same philosophy. The real phase space is partitioned by positive sector functions $$C_S=\sum_{m_1,m_2=0}^{\infty} A_{m_1m_2}\,a_1^{m_1}a_2^{m_2}, \qquad \rho_S=C_S\rho_G C_S^\dagger,$$7 satisfying $$C_S=\sum_{m_1,m_2=0}^{\infty} A_{m_1m_2}\,a_1^{m_1}a_2^{m_2}, \qquad \rho_S=C_S\rho_G C_S^\dagger,$$8, with each sector containing at most one soft leg and one collinear pair. The local counterterm is

$$C_S=\sum_{m_1,m_2=0}^{\infty} A_{m_1m_2}\,a_1^{m_1}a_2^{m_2}, \qquad \rho_S=C_S\rho_G C_S^\dagger,$$9

and its analytical integral yields $\hat{B}|n\rangle=|n-1\rangle$00 and $\hat{B}|n\rangle=|n-1\rangle$01 terms that cancel, respectively, the poles of virtual corrections and PDF counterterms. The same work introduces smooth damping factors with tunable exponents $\hat{B}|n\rangle=|n-1\rangle$02 to improve numerical stability without changing the pole structure, and numerically validates the method for processes including $\hat{B}|n\rangle=|n-1\rangle$03, $\hat{B}|n\rangle=|n-1\rangle$04, $\hat{B}|n\rangle=|n-1\rangle$05, $\hat{B}|n\rangle=|n-1\rangle$06, and $\hat{B}|n\rangle=|n-1\rangle$07 (Bertolotti et al., 2022).

Within QCD, therefore, state subtraction is neither measurement-based nor heralded. It is an analytic device for removing unresolved radiation contributions at the integrand level and restoring them in a form where infrared cancellation is manifest.

6. Real intermediate state subtraction in kinetic theory

In nonequilibrium kinetic theory, the subtraction target is an on-shell resonance already included as an explicit species in the Boltzmann network. For a process

$\hat{B}|n\rangle=|n-1\rangle$08

with unstable $\hat{B}|n\rangle=|n-1\rangle$09, the propagator is

$\hat{B}|n\rangle=|n-1\rangle$10

If the kinetic equations also contain $\hat{B}|n\rangle=|n-1\rangle$11 and $\hat{B}|n\rangle=|n-1\rangle$12, then the resonant contribution to $\hat{B}|n\rangle=|n-1\rangle$13 is double-counted unless the real intermediate state is subtracted from the $\hat{B}|n\rangle=|n-1\rangle$14 channel (Ala-Mattinen et al., 2023).

The cited analysis traces the ambiguity of RIS subtraction to the on-shell approximation that reduces the Schwinger–Dyson and Kadanoff–Baym description to the Boltzmann limit. Standard RIS subtraction and principal-value subtraction share the same on-shell limit,

$\hat{B}|n\rangle=|n-1\rangle$15

but differ in the off-shell Hermitian part of the propagator. The paper argues that both SRS and PVS can yield negative effective scattering rates because they overestimate the on-shell weight and force the off-shell remainder to become negative in parts of the resonance neighborhood (Ala-Mattinen et al., 2023).

To avoid this, the paper introduces a cut-subtraction scheme. With a cut function $\hat{B}|n\rangle=|n-1\rangle$16, the off-shell propagator is defined as

$\hat{B}|n\rangle=|n-1\rangle$17

and the on-shell weight is measured by

$\hat{B}|n\rangle=|n-1\rangle$18

Here $\hat{B}|n\rangle=|n-1\rangle$19 is an effective one-particle weight function: $\hat{B}|n\rangle=|n-1\rangle$20 indicates that the near-resonant region is well approximated by an on-shell particle description, whereas $\hat{B}|n\rangle=|n-1\rangle$21 signals that off-shell contributions dominate and the Boltzmann treatment of the resonance as an independent thermal species is not reliable (Ala-Mattinen et al., 2023).

This version of state subtraction has a different objective from both quantum-optical subtraction and QCD infrared subtraction. It is a consistency prescription for kinetic equations, ensuring that decay/inverse-decay terms and resonant $\hat{B}|n\rangle=|n-1\rangle$22 channels do not represent the same physical intermediate state twice.