Single-Photon-Subtracted Squeezed Vacuum

• The paper establishes that SPSSV, produced by single-photon subtraction, is exactly equivalent (up to a global phase) to a squeezed one-photon Fock state.
• SPSSV uniquely shifts the Fock structure from even to odd parity, introducing non-Gaussianity essential for boson sampling and advanced quantum encoding.
• Heralding techniques via beam-splitter detection and generalized photon subtraction models ensure precise state preparation and higher generation rates.

Single-photon-subtracted squeezed vacuum (SPSSV) is the single-mode state obtained by acting once with the annihilation operator on a squeezed vacuum, $\hat a \hat S(\xi)\ket{0}$. For $r\neq 0$, with $\xi=re^{i\theta}$, its normalized form is $$\ket{\psi_{\mathrm{SPSSV}}}=(\sinh r)^{-1}\hat a \hat S(\xi)\ket0$$, and it is exactly equivalent, up to a global phase, to a squeezed one-photon Fock state $\hat S(\xi)\ket1$ (Olson et al., 2014). Because squeezed vacuum has only even Fock components, single-photon subtraction moves the state into the odd sector and thereby produces a non-Gaussian, odd-parity squeezed state. In the literature the same object also appears under the notation SPSSVS, and as the $m=1$ member of broader PSSVS or PASSV families (Yuanbek et al., 2024, Samantaray et al., 2018).

1. Definition and exact state identity

The single-mode squeezing operator is written in closely related sign conventions. One convention is

$$\hat S(\xi)=\exp\!\left[\frac{1}{2}(\xi^*\hat a^2-\xi\hat a^{\dagger 2})\right],$$

while another is

$$\hat S(\xi)=\exp\!\left[\frac{1}{2}\xi \hat a^{\dagger 2}-\frac{1}{2}\xi^* \hat a^2\right].$$

In either convention, the squeezed vacuum is $\ket{\xi}=\hat S(\xi)\ket0$, and its mean photon number is $$\langle \xi|\hat a^\dagger\hat a|\xi\rangle=\sinh^2 r$$ (Olson et al., 2014, Yuanbek et al., 2024).

With the convention used in “Sampling arbitrary photon-added or photon-subtracted squeezed states is in the same complexity class as boson sampling,” the key identity is

$r\neq 0$0

After normalization, the prefactor drops out, leaving

$r\neq 0$1

up to a physically irrelevant global phase (Olson et al., 2014). The weak-measurement paper reaches the same conclusion in the alternative sign convention: there too the normalized SPSSV is $r\neq 0$2, again up to a global phase (Yuanbek et al., 2024).

This equivalence has a sharpened counterpart for pure squeezed vacua: “Adding or Subtracting a single Photon is the same for Pure Squeezed Vacuum States” proves that, for a pure squeezed vacuum input, normalized one-photon subtraction and normalized one-photon addition give the same output state, and argues that for pure states this identity occurs essentially only for squeezed vacua (Steuernagel et al., 2024). A useful consequence is that SPSSV is not merely analogous to a squeezed single-photon state; it is exactly that state after normalization.

2. Fock structure, parity, and non-Gaussianity

The Fock expansion of a squeezed vacuum makes the parity structure explicit: $r\neq 0$3 Only even photon numbers occur. Applying $r\neq 0$4 therefore produces a state with only odd photon numbers. In the notation of the weak-to-strong measurement study,

$r\neq 0$5

with

$r\neq 0$6

The odd-only support is therefore exact, not approximate (Yuanbek et al., 29 Jul 2025).

This even-to-odd parity flip is the most basic structural distinction between squeezed vacuum and SPSSV. Several papers make this the operational centerpiece of the state. In the boson-sampling construction, parity is the logical degree of freedom that replaces ordinary photon-number occupancy; in multimode encoding, parity orthogonality makes the basis states distinguishable; in weak-measurement settings, the same odd-parity structure underlies the phase-space interference patterns of the conditional pointer state (Olson et al., 2014, Arzani et al., 2018, Yuanbek et al., 2024).

Photon subtraction is also the non-Gaussian step. The parent squeezed vacuum is Gaussian, whereas SPSSV is not. The 2020 generalized PSSVS paper places SPSSV in the odd family

$r\neq 0$7

so that the generalized single-photon-subtracted squeezed vacuum is the $r\neq 0$8 odd state $r\neq 0$9. In the harmonic-oscillator limit this reduces to the standard SPSSV, while the generalized construction is used to analyze quadrature squeezing, photon-number squeezing, and Wigner negativity beyond the ordinary oscillator model (Dey et al., 2020).

3. Preparation and heralding models

At the idealized state-vector level, SPSSV is written as $\xi=re^{i\theta}$0. Several papers emphasize that an experimentally faithful description is instead a conditional beam-splitter-and-detection process. In “Theory of Photon Subtraction for Two-Mode Entangled Light Beams,” the output of a 50/50 lossless beam splitter is decomposed into idler Fock states correlated with normalized $\xi=re^{i\theta}$1-subtracted signal states

$\xi=re^{i\theta}$2

with exact heralding probability

$\xi=re^{i\theta}$3

For a squeezed-vacuum input, the $\xi=re^{i\theta}$4 branch is the odd one-photon-subtracted state $\xi=re^{i\theta}$5, and in the weak-squeezing limit heralding on one idler photon approximately prepares $\xi=re^{i\theta}$6 in the signal (Rosas-Ortiz et al., 2021).

The same preparation logic extends to multimode fields. “Tomography of a Mode-Tunable Coherent Single-Photon Subtractor” implements a subtraction operator

$\xi=re^{i\theta}$7

by sum-frequency generation, rather than an incoherent “subtract somewhere in the beam” process. The corresponding subtraction map is

$\xi=re^{i\theta}$8

and the experimentally reconstructed realistic operation is

$\xi=re^{i\theta}$9

with reported values

$$\ket{\psi_{\mathrm{SPSSV}}}=(\sinh r)^{-1}\hat a \hat S(\xi)\ket0$$0

and typical mode selectivity $$\ket{\psi_{\mathrm{SPSSV}}}=(\sinh r)^{-1}\hat a \hat S(\xi)\ket0$$1 (Ra et al., 2017). This matters directly for multimode SPSSV generation, because it distinguishes coherent subtraction in a chosen supermode from an incoherent mixture over modes.

A later generation-focused development is generalized photon subtraction (GPS). In that scheme, conventional photon subtraction is extended by mixing the target squeezed vacuum with an orthogonally squeezed ancillary input before on/off heralding. With ideal one-photon heralding, the target remains a squeezed single-photon state, i.e. the normalized SPSSV. The practical gain is a higher heralding rate at comparable state quality: the paper reports generation-rate improvement for 2-dB- and 4-dB-squeezed single-photon states, by more than one order of magnitude particularly for 2 dB (Tomoda et al., 2024).

4. Sampling, encoding, and computational roles

SPSSV enters linear-optical complexity theory through a parity-based reformulation of boson sampling. In the construction of (Olson et al., 2014), $$\ket{\psi_{\mathrm{SPSSV}}}=(\sinh r)^{-1}\hat a \hat S(\xi)\ket0$$2 modes are prepared in photon-subtracted squeezed vacuum states, the remaining $$\ket{\psi_{\mathrm{SPSSV}}}=(\sinh r)^{-1}\hat a \hat S(\xi)\ket0$$3 modes in ordinary squeezed vacuum, and the whole register is sent through a passive real interferometer $$\ket{\psi_{\mathrm{SPSSV}}}=(\sinh r)^{-1}\hat a \hat S(\xi)\ket0$$4. The output is measured mode by mode with parity POVM elements

$$\ket{\psi_{\mathrm{SPSSV}}}=(\sinh r)^{-1}\hat a \hat S(\xi)\ket0$$5

Because unsullied squeezed vacuum is even and a single subtraction flips parity to odd, parity detection exactly recovers the output pattern. The resulting amplitudes are controlled by submatrix permanents, and the sampling problem is in the same complexity class as Aaronson–Arkhipov boson sampling. The equivalence is exact and independent of the squeezing parameter (Olson et al., 2014).

A different computational role appears in multimode encoding. “High-dimensional quantum encoding via photon-subtracted squeezed states” takes the elementary single-mode SPSSV basis state

$$\ket{\psi_{\mathrm{SPSSV}}}=(\sinh r)^{-1}\hat a \hat S(\xi)\ket0$$6

and embeds it into a multimode product of squeezed vacua. The resulting basis states are

$$\ket{\psi_{\mathrm{SPSSV}}}=(\sinh r)^{-1}\hat a \hat S(\xi)\ket0$$7

so exactly one mode carries odd parity while the rest remain even. Orthogonality follows from parity, and an arbitrary encoded state is

$$\ket{\psi_{\mathrm{SPSSV}}}=(\sinh r)^{-1}\hat a \hat S(\xi)\ket0$$8

The subtraction mode itself is programmable,

$$\ket{\psi_{\mathrm{SPSSV}}}=(\sinh r)^{-1}\hat a \hat S(\xi)\ket0$$9

so the encoded amplitudes are controlled by classical gate fields rather than by reconfiguring interferometers (Arzani et al., 2018).

These two uses share the same structural fact: SPSSV is an odd-parity excitation sitting on an even-parity squeezed background. In one case that parity encodes a boson-sampling output string; in the other it defines an orthogonal multimode code space.

5. Measurement, metrology, and state engineering

A substantial recent literature uses SPSSV as a pointer state in postselected von Neumann measurement. In “Single-Photon-Subtracted-Squeezed-Vacuum-State Based Postselected Weak Measurement and its Applications,” the initial pointer is

$\hat S(\xi)\ket1$0

and after postselection the normalized pointer becomes

$\hat S(\xi)\ket1$1

Here $\hat S(\xi)\ket1$2 is the dimensionless coupling strength and

$\hat S(\xi)\ket1$3

is the weak value. The reported conclusions are selective rather than uniform: in the weak regime $\hat S(\xi)\ket1$4, large anomalous weak values enhance quadrature squeezing, enlarge Wigner-negative regions over much of parameter space, and produce $\hat S(\xi)\ket1$5 for the ratio of postselected to non-postselected signal-to-noise ratio, while the Mandel factor is not positively improved; the decreasing fidelity between the initial and final pointer states is interpreted as state engineering (Yuanbek et al., 2024).

The 2025 weak-to-strong measurement study retains SPSSV as the single-mode benchmark and tracks its crossover from quantum to classical behavior. The postselected SPSSV pointer is again a superposition of oppositely displaced components, and the weak-to-strong transition is encoded in a parameter $\hat S(\xi)\ket1$6 that moves from the weak value $\hat S(\xi)\ket1$7 at $\hat S(\xi)\ket1$8 to the classical conditional value $\hat S(\xi)\ket1$9 at $m=1$0. The paper reports enhancement of Wigner–Yanase skew information in low-squeezing regimes, exponential decay of amplitude-squared squeezing as $m=1$1 increases, reshaping of photon-number distributions, and a Husimi–Kano $m=1$2-function transition from quantum non-Gaussianity to classical single-peak separability; for $m=1$3, the postselection success probability tends to $m=1$4 (Yuanbek et al., 29 Jul 2025).

Metrological conclusions depend strongly on the task. In the single-phase Mach–Zehnder scenario studied in “Single phase and correlated phase estimation with multi-photon annihilated squeezed vacuum states,” the $m=1$5 PASSV state is exactly SPSSV. The paper finds that at fixed squeezing parameter photon subtraction can look advantageous because the mean photon number rises, but when the total input energies is held fixed, the advantage due to photon subtraction is completely lost for single phase estimation. In correlated phase estimation, by contrast, some advantage appears to come from both the energy rise and improvement in photon statistics, with an advantage of about $m=1$6 under realistic detection efficiency (Samantaray et al., 2018).

6. Related states, generalizations, and conceptual caveats

A recurring source of confusion is the difference between single-mode SPSSV and its two-mode or multimode analogues. “Optical vortex produced by Single Photon Subtraction from two mode squeezed state Produces Maximum Entanglement” studies

$m=1$7

not the single-mode state $m=1$8. In that two-mode setting, subtracting one photon from one arm is equivalent to adding one photon to the other arm, and the normalized state becomes

$m=1$9

Among the subtraction numbers studied there, the single-photon-subtracted case yields the largest entanglement enhancement relative to the original TMSV, but the state is a two-mode analogue rather than canonical single-mode SPSSV (Bandyopadhyay et al., 2012).

Another distinct object is the photon-added-then-subtracted squeezed coherent state of the QKD paper,

$$\hat S(\xi)=\exp\!\left[\frac{1}{2}(\xi^*\hat a^2-\xi\hat a^{\dagger 2})\right],$$0

which is not SPSSV. Likewise, the $$\hat S(\xi)=\exp\!\left[\frac{1}{2}(\xi^*\hat a^2-\xi\hat a^{\dagger 2})\right],$$1 limit of that construction gives $$\hat S(\xi)=\exp\!\left[\frac{1}{2}(\xi^*\hat a^2-\xi\hat a^{\dagger 2})\right],$$2, not $$\hat S(\xi)=\exp\!\left[\frac{1}{2}(\xi^*\hat a^2-\xi\hat a^{\dagger 2})\right],$$3 (Oulouda et al., 2022). By contrast, the generalized PSSVS framework of (Dey et al., 2020) really does contain SPSSV as the $$\hat S(\xi)=\exp\!\left[\frac{1}{2}(\xi^*\hat a^2-\xi\hat a^{\dagger 2})\right],$$4 odd state in the family

$$\hat S(\xi)=\exp\!\left[\frac{1}{2}(\xi^*\hat a^2-\xi\hat a^{\dagger 2})\right],$$5

and uses quadrature squeezing, photon-number squeezing, and Wigner negativity to compare generalized and harmonic-oscillator cases.

The literature also shows that subtraction-vs-addition intuition is context dependent. For pure single-mode squeezed vacuum, normalized addition and subtraction give the same state (Steuernagel et al., 2024). For the two-mode squeezed-vacuum case, single-photon subtraction enhances entanglement and generates a first-order elliptical quantum optical vortex (Bandyopadhyay et al., 2012). For four-mode squeezed vacuum, there are bipartitions in which photon subtraction yields more entanglement than photon addition, contrary to the familiar two-mode ordering (Das et al., 2015). A plausible implication is that SPSSV is best understood not as an isolated odd squeezed state alone, but as the elementary non-Gaussian excitation underlying a much wider family of parity-structured and heralded squeezed-light constructions.