Photon-Subtracted Two-Mode Squeezed Fock States

• PS-TMSFS are non-Gaussian conditional states generated from two-mode squeezed Fock inputs through photon subtraction via beam-splitter taps and heralding.
• They exhibit a polynomial-modulated Gaussian core with enhanced entanglement and structured multiphoton correlations, vital for quantum state engineering.
• Experimental realizations show that the success probability and nonclassical properties are highly sensitive to subtraction order, optical losses, and detector efficiency.

Photon-subtracted two-mode squeezed Fock states (PS-TMSFSs) are non-Gaussian conditional states obtained by combining two-mode squeezing, ancillary vacuum modes, beam-splitter taps, and photon-number-resolving heralding. In the most explicit two-mode formulation, one begins with a two-mode squeezed Fock input $S_2(r)\lvert m_1,m_2\rangle$ and conditions on subtracting $n_1$ and $n_2$ photons from the two modes; in adjacent measurement-based formulations, the same operational motif is used to herald single-mode squeezed Fock outputs from a two-mode entangled Gaussian resource rather than to retain a two-mode entangled output (Ankita et al., 24 Jun 2026, Korolev et al., 2023). The subject therefore sits at the intersection of non-Gaussian state engineering, continuous-variable entanglement manipulation, and conditional photonic measurement.

1. Definition and terminological scope

A two-mode squeezed Fock state is generated by acting with the two-mode squeezing operator

$$S_{2}(r)=\exp\bigl[r\,(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})\bigr]$$

on a two-mode Fock input,

$$\lvert\psi_{\rm TMSFS}\rangle=S_{2}(r)\,\lvert m_{1},m_{2}\rangle_{A_{1}A_{2}}.$$

Photon subtraction is then implemented by mixing each mode with vacuum on ancillary beam splitters and conditioning on photon-number-resolving detections in the tapped ports; successful clicks $(n_1,n_2)$ define the photon-subtracted state (Ankita et al., 24 Jun 2026).

The label PS-TMSFS appears in adjacent but non-identical constructions. Some works use it for an entangled two-mode non-Gaussian resource obtained after subtracting photons from a genuine $S_2(r)\lvert m_1,m_2\rangle$ input. Other works use closely related machinery to generate squeezed Fock states by measurement from a generic two-mode entangled Gaussian state. A further strand uses a two-mode squeezed vacuum plus weak mode mixing and $n$-photon heralding to prepare a small-dimensional “core” state that is later converted into a squeezed coherent-state superposition (Huang et al., 2015, Bashmakova et al., 2024).

A notable boundary case is provided by the general theory of photon subtraction for entangled beams based on two-mode $SU(2)$ coherent states. That framework supplies a compact conditional-output formalism, but it does not itself introduce a two-mode squeezed Fock state $\lvert r;n,m\rangle=S_2(r)\lvert n,m\rangle$ as its working input family; its explicit formulas instead assume a single-mode superposition entering one port of a $n_1$0 beam splitter and vacuum in the other (Rosas-Ortiz et al., 2021).

Work	State or protocol	Relation to PS-TMSFS
(Ankita et al., 24 Jun 2026)	$n_1$1 plus conditional subtraction $n_1$2	Direct two-mode PS-TMSFS resource
(Korolev et al., 2023, Bashmakova et al., 2024)	Two-mode entangled Gaussian resource plus number projection	Measurement-based squeezed-Fock generation
(Huang et al., 2015)	TMSV, weak beam-splitter mixing, $n_1$3-photon heralding	PS-TMSFS “core” for cat-state synthesis
(Rosas-Ortiz et al., 2021)	General subtraction theory for entangled beams	Formal backdrop, not a direct TMSFS construction

2. Conditional generation protocols

In the direct two-mode construction, the input $n_1$4 is first transformed by $n_1$5 and then coupled to ancillary vacua $n_1$6 through beam splitters

$n_1$7

Photon-number-resolving detection projects the ancillas onto $n_1$8, and successful heralds subtract $n_1$9 and $n_2$0 photons from the two system modes. The normalized output is obtained by dividing the unnormalized characteristic function by the success probability

$n_2$1

where $n_2$2 is the differential operator associated with the Fock inputs and subtraction pattern, and $n_2$3 are the real matrices tabulated for the protocol (Ankita et al., 24 Jun 2026).

The vacuum-seeded sector, $n_2$4, reduces to the two-mode squeezed vacuum (TMSV),

$n_2$5

Simultaneous subtraction of $n_2$6 photons from both modes produces the unnormalized state

$n_2$7

up to the normalization factor $n_2$8 (Magana-Loaiza et al., 2019). This sector is important because it is both analytically tractable and experimentally established.

A small-mixing approximation leads to a particularly compact heralded state. After weak beam-splitter mixing of TMSV signal and idler, $n_2$9-photon detection in the heralding arm produces

$$S_{2}(r)=\exp\bigl[r\,(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})\bigr]$$0

with approximate success probability

$$S_{2}(r)=\exp\bigl[r\,(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})\bigr]$$1

This two-component non-Gaussian “core” is the state later converted into a squeezed even coherent-state superposition (Huang et al., 2015).

3. Representations in Fock space, quadrature space, and phase space

The PS-TMSFS admits a natural phase-space description in terms of a Gaussian kernel dressed by derivatives. In the two-mode formulation, the unnormalized output characteristic function is

$$S_{2}(r)=\exp\bigl[r\,(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})\bigr]$$2

and the normalized characteristic function is

$$S_{2}(r)=\exp\bigl[r\,(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})\bigr]$$3

Because the differential operator $$S_{2}(r)=\exp\bigl[r\,(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})\bigr]$$4 acts on the Gaussian kernel, the output is not merely Gaussian: it carries a polynomial prefactor in the phase-space variables, with highest total degree $$S_{2}(r)=\exp\bigl[r\,(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})\bigr]$$5 whenever subtraction is present (Ankita et al., 24 Jun 2026). This is the direct phase-space signature of non-Gaussianity.

An equivalent quadrature-space representation is available for measurement-generated squeezed-Fock states derived from a generic two-mode Gaussian wavefunction with parameters $$S_{2}(r)=\exp\bigl[r\,(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})\bigr]$$6. After number projection on tapped ancillas, the heralded wavefunction assumes the form

$$S_{2}(r)=\exp\bigl[r\,(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})\bigr]$$7

where $$S_{2}(r)=\exp\bigl[r\,(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})\bigr]$$8 and $$S_{2}(r)=\exp\bigl[r\,(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})\bigr]$$9 are Hermite polynomials, and the effective squeezings $$\lvert\psi_{\rm TMSFS}\rangle=S_{2}(r)\,\lvert m_{1},m_{2}\rangle_{A_{1}A_{2}}.$$0 are fixed by the universal-solution conditions (Bashmakova et al., 2024). The Hermite structure makes explicit that photon subtraction promotes the Gaussian core into a polynomial-times-Gaussian state, which is precisely the squeezed-Fock pattern.

For the vacuum-seeded case, the Wigner function can be generated by applying differential operators to the known TMSV Wigner function,

$$\lvert\psi_{\rm TMSFS}\rangle=S_{2}(r)\,\lvert m_{1},m_{2}\rangle_{A_{1}A_{2}}.$$1

so that each subtraction order introduces additional oscillations, negative lobes, and interference fringes in phase space (Magana-Loaiza et al., 2019). This suggests a common structural picture across formulations: subtraction converts a Gaussian entangled resource into a non-Gaussian state whose analytic expressions remain controlled by derivatives of Gaussian objects.

4. Universal-solution regime and optical implementations

A central analytical result in the measurement-based literature is the “universal-solution” regime, in which the heralded output exactly matches the ideal $$\lvert\psi_{\rm TMSFS}\rangle=S_{2}(r)\,\lvert m_{1},m_{2}\rangle_{A_{1}A_{2}}.$$2th squeezed-Fock wavefunction for arbitrary $$\lvert\psi_{\rm TMSFS}\rangle=S_{2}(r)\,\lvert m_{1},m_{2}\rangle_{A_{1}A_{2}}.$$3. For a non-displaced two-mode entangled Gaussian state with wavefunction parameters $$\lvert\psi_{\rm TMSFS}\rangle=S_{2}(r)\,\lvert m_{1},m_{2}\rangle_{A_{1}A_{2}}.$$4, exact equality holds if and only if

$$\lvert\psi_{\rm TMSFS}\rangle=S_{2}(r)\,\lvert m_{1},m_{2}\rangle_{A_{1}A_{2}}.$$5

Eliminating $$\lvert\psi_{\rm TMSFS}\rangle=S_{2}(r)\,\lvert m_{1},m_{2}\rangle_{A_{1}A_{2}}.$$6 gives the real constraint

$$\lvert\psi_{\rm TMSFS}\rangle=S_{2}(r)\,\lvert m_{1},m_{2}\rangle_{A_{1}A_{2}}.$$7

In that regime, the heralded state coincides with $$\lvert\psi_{\rm TMSFS}\rangle=S_{2}(r)\,\lvert m_{1},m_{2}\rangle_{A_{1}A_{2}}.$$8 exactly for any $$\lvert\psi_{\rm TMSFS}\rangle=S_{2}(r)\,\lvert m_{1},m_{2}\rangle_{A_{1}A_{2}}.$$9 (Korolev et al., 2023).

The case $(n_1,n_2)$0 is exceptional. Because $(n_1,n_2)$1 is linear, one-photon subtraction from an arbitrary non-displaced two-mode entangled Gaussian state always yields a first squeezed-Fock state $(n_1,n_2)$2, with

$(n_1,n_2)$3

The corresponding heralding probability satisfies $(n_1,n_2)$4, and that maximum is attained precisely in the universal-solution regime (Korolev et al., 2023). This is a sharp statement about the privileged status of first-order subtraction.

Two optical implementations are treated in detail. In the beam-splitter (BS) approach, two single-mode squeezed vacua of parameters $(n_1,n_2)$5 are mixed with power transmission $(n_1,n_2)$6, producing

$(n_1,n_2)$7

In the controlled-$(n_1,n_2)$8 (CZ) approach,

$(n_1,n_2)$9

The BS protocol has energy cost

$S_2(r)\lvert m_1,m_2\rangle$0

whereas the CZ protocol requires

$S_2(r)\lvert m_1,m_2\rangle$1

because it uses two auxiliary squeezed ancilla modes. The ideal success probability peaks at approximately $S_2(r)\lvert m_1,m_2\rangle$2 for $S_2(r)\lvert m_1,m_2\rangle$3 or $S_2(r)\lvert m_1,m_2\rangle$4; BS requires only vacuum taps, while CZ requires two ancilla squeezers plus passive optics and becomes competitive only if large-$S_2(r)\lvert m_1,m_2\rangle$5 CZ gates and high ancilla squeezing are available (Bashmakova et al., 2024).

5. Nonclassicality, statistics, and entanglement

For the vacuum-seeded PS-TMSFS sector, subtraction changes both local and joint photon statistics in a highly structured way. The single-mode second-order correlation function

$S_2(r)\lvert m_1,m_2\rangle$6

tends toward $S_2(r)\lvert m_1,m_2\rangle$7 with increasing subtraction order $S_2(r)\lvert m_1,m_2\rangle$8, indicating that each marginal becomes closer to Poissonian (Magana-Loaiza et al., 2019). Experimentally, the marginals evolve from thermal at $S_2(r)\lvert m_1,m_2\rangle$9 to nearly Poisson distributions of mean $n$0 by $n$1, which was described as a first step toward an “entangled laser” (Magana-Loaiza et al., 2019).

Entanglement and nonclassical correlations can increase under symmetric subtraction. For pure symmetric photon-subtracted TMSV states, the Schmidt coefficients are explicit, and both the negativity

$n$2

and the logarithmic negativity

$n$3

increase with $n$4 at fixed $n$5 (Magana-Loaiza et al., 2019). At the level of moment criteria, the Agarwal parameter

$n$6

is negative for all $n$7, and the second-order moment matrix $n$8 has negative determinant and a negative minimal eigenvalue for all $n$9 and $SU(2)$0, certifying non-classical correlations between the modes (Magana-Loaiza et al., 2019).

A general theoretical restriction is that photon subtraction does not produce nonclassical fields from classical fields (Rosas-Ortiz et al., 2021). This corrects a common overgeneralization: subtraction is a non-Gaussian filter acting on pre-existing quantum correlations and fluctuations, not a universal generator of nonclassicality from arbitrary classical inputs.

Model dependence remains important. In a one-sided lossy-channel analysis restricted to the subspace $SU(2)$1, one-photon subtraction leaves only populations in $SU(2)$2 and $SU(2)$3, with no off-diagonal coherence in that truncated description and $SU(2)$4 (Zhang et al., 2019). This does not contradict the symmetric ideal-state results; it instead shows that entanglement retention depends sensitively on whether subtraction is one-sided or symmetric, on whether loss is present, and on whether one studies the full state or a truncated sector.

6. Experimental realizations and neighboring state engineering

The experimentally most mature instance is the simultaneous subtraction from bright two-mode squeezed vacuum, which constitutes the $SU(2)$5 member of the broader PS-TMSFS family. In that implementation, TMSV states were generated in an $SU(2)$6 ppKTP waveguide (type-II SPDC) pumped at $SU(2)$7 with $SU(2)$8 pulses from a Ti:Sapphire laser of repetition rate $SU(2)$9, pulse picked to $\lvert r;n,m\rangle=S_2(r)\lvert n,m\rangle$0. The signal and idler at $\lvert r;n,m\rangle=S_2(r)\lvert n,m\rangle$1 were separated on a PBS, each passed through a $\lvert r;n,m\rangle=S_2(r)\lvert n,m\rangle$2 fiber coupler, and the tapped fractions were monitored by transition-edge sensors (TESs) with approximately $\lvert r;n,m\rangle=S_2(r)\lvert n,m\rangle$3 detection efficiency; the remaining beams were sent to a second TES pair for conditioned joint-photon-number measurements. The fitted detection parameters were $\lvert r;n,m\rangle=S_2(r)\lvert n,m\rangle$4 and $\lvert r;n,m\rangle=S_2(r)\lvert n,m\rangle$5 per pulse, and the measured distributions agreed with the lossy theoretical model at the $\lvert r;n,m\rangle=S_2(r)\lvert n,m\rangle$6-confidence level (Magana-Loaiza et al., 2019).

A distinct but closely related experiment used a two-mode squeezed state, linear mixing, and $\lvert r;n,m\rangle=S_2(r)\lvert n,m\rangle$7-photon detection to synthesize heralded optical coherent-state superpositions. For two-photon detection with high-efficiency superconducting nanowire single-photon detectors, the freely propagating state exhibited a $\lvert r;n,m\rangle=S_2(r)\lvert n,m\rangle$8 fidelity with a squeezed even coherent-state superposition of size $\lvert r;n,m\rangle=S_2(r)\lvert n,m\rangle$9 (Huang et al., 2015). In the technical formulation underlying that experiment, the heralded non-Gaussian state is the two-component core

$n_1$00

which is then converted by single-mode squeezing into a larger-amplitude squeezed coherent-state superposition (Huang et al., 2015).

These experimental results indicate that the practical value of PS-TMSFS-type constructions is not confined to the exact entangled two-mode resource of the most literal definition. The same conditional subtraction architecture supports high-dimensional multiphoton correlation engineering, heralded squeezed-Fock preparation, and cat-state synthesis, with the precise output class determined by the input seed, the entangling transformation, and which modes are retained after conditioning.

7. Applications, performance trade-offs, and limitations

A recent phase-space analysis of continuous-variable teleportation with a PS-TMSFS resource examined Braunstein-Kimble teleportation using the characteristic function of the photon-subtracted resource. For coherent-state inputs, the teleportation fidelity takes the form

$n_1$01

and for squeezed-vacuum inputs,

$n_1$02

with $n_1$03 and $n_1$04 specified by the phase-space construction (Ankita et al., 24 Jun 2026). The main performance result is restrictive: no substantial enhancement of teleportation fidelity is detected, and fidelity above the classical coherent-state benchmark occurs only for the symmetric $n_1$05 subtraction configuration in the low-squeezing regime, with that region shrinking as squeezing increases (Ankita et al., 24 Jun 2026).

This limitation is consistent with the broader resource trade-offs identified in measurement-based squeezed-Fock generation. Photon subtraction increases non-Gaussianity, but it also reduces success probability at higher subtraction orders and can be highly sensitive to optical loss and detector imperfections. In the BS-versus-CZ comparison, uniform photon loss degrades both schemes, with the BS approach remaining more robust to uniform losses, while CZ is more tolerant of detector inefficiency but pays a larger resource cost and requires ancillary squeezing of at least $n_1$06 for $n_1$07 fidelity in the reported feasibility estimate (Bashmakova et al., 2024).

The resulting picture is nuanced rather than uniform. PS-TMSFS-type states are effective for non-Gaussian state engineering, heralded squeezed-Fock generation, and the manipulation of multiphoton correlations. They do not, however, provide a blanket advantage for all continuous-variable protocols. A plausible implication is that their utility is task-specific: protocols that directly exploit Hermite-structured non-Gaussianity and conditional state preparation can benefit substantially, whereas protocols optimized for Gaussian entanglement, such as standard Braunstein-Kimble teleportation, may not.