Conditional Squeezing in Quantum Systems

• Conditional squeezing is defined as the reduction of state uncertainty by conditioning on measurement records or heralding events, yielding a posterior state with lower variances than its unconditional counterpart.
• In quantum-optical and atomic spin systems, protocols such as homodyne detection and feedback stabilize unstable quadratures and achieve superior scaling (e.g., d⁻¹ versus d⁻¹/²) compared to traditional QND measurements.
• The concept extends to heralded finite-dimensional systems, bosonic control operations, and metrological criteria, offering enhanced precision and novel insights in both experimental and theoretical frameworks.

Conditional squeezing denotes a family of procedures, states, and quantitative criteria in which squeezing is defined relative to conditioning information. In the canonical quantum-optical meaning, the conditioning variable is a continuous measurement record, and the squeezed state is the posterior Gaussian state inferred from that record rather than the unconditional state obtained by tracing over the probe or averaging over outcomes. In other settings, the conditioning may be a heralding event, a no-jump trajectory, a remote party’s measurement outcome, or a quantum memory system. Across these usages, the central distinction is between a conditioned object—state, variance, or entropy—and its unconditional counterpart (Vasilyev et al., 2013, Chen et al., 2011).

1. Measurement-conditioned squeezing in Gaussian spin systems

A standard formulation treats a polarized collective spin in the Holstein–Primakoff or Gaussian approximation. For large mean polarization along $x$, the transverse components are mapped to canonical variables

$$X=\frac{J_y}{\sqrt{\langle J_x\rangle}},\qquad P=\frac{J_z}{\sqrt{\langle J_x\rangle}}, \qquad [X,P]=i,$$

and the optical probe is described by field quadratures $x(t)$ and $p(t)$. In this setting, conditional squeezing means that the atomic state is updated using the actual homodyne photocurrent of the outgoing light, so the posterior covariance is smaller than the covariance of the unconditional state obtained by tracing over the light. The general Gaussian light–matter interaction is parametrized as

$$H^{\mathrm{int}}=\sqrt g\big[\alpha X\,x(t)+\beta P\,p(t)\big], \qquad \alpha=\cos\theta,\quad \beta=\sin\theta,$$

with $g=d\gamma$, where $d$ is the optical depth. Within this framework, the conditional covariance obeys a Riccati-type filtering equation, and the optimized steady-state conditional squeezing is not obtained at the QND point $\theta=0$, but near the two-mode-squeezing point $\theta=-\pi/4$. In the large-$d$ regime the optimized conditional variance scales as

$$X=\frac{J_y}{\sqrt{\langle J_x\rangle}},\qquad P=\frac{J_z}{\sqrt{\langle J_x\rangle}}, \qquad [X,P]=i,$$0

whereas the optimized unconditional dissipative scheme yields

$$X=\frac{J_y}{\sqrt{\langle J_x\rangle}},\qquad P=\frac{J_z}{\sqrt{\langle J_x\rangle}}, \qquad [X,P]=i,$$1

Thus the conditional protocol achieves $$X=\frac{J_y}{\sqrt{\langle J_x\rangle}},\qquad P=\frac{J_z}{\sqrt{\langle J_x\rangle}}, \qquad [X,P]=i,$$2 scaling, while the unconditional protocol scales only as $$X=\frac{J_y}{\sqrt{\langle J_x\rangle}},\qquad P=\frac{J_z}{\sqrt{\langle J_x\rangle}}, \qquad [X,P]=i,$$3 (Vasilyev et al., 2013).

This formulation also clarifies why “conditional” is not synonymous with “QND.” The standard QND choice is contained as a special case, but the global Gaussian optimization shows that stronger atom–light correlations, generated by an interaction with pronounced two-mode-squeezing character, improve the information extraction rate enough to outperform pure QND in the moderate-optical-depth regime. The tradeoff is that the antisqueezed quadrature becomes unstable. For a single detected optical quadrature, the squeezed variance is best, but the conjugate variance diverges; a second homodyne channel, introduced by splitting off a fraction $$X=\frac{J_y}{\sqrt{\langle J_x\rangle}},\qquad P=\frac{J_z}{\sqrt{\langle J_x\rangle}}, \qquad [X,P]=i,$$4 of the output light, supplies information about the unstable quadrature and allows feedback stabilization. In that stabilized regime, the favorable $$X=\frac{J_y}{\sqrt{\langle J_x\rangle}},\qquad P=\frac{J_z}{\sqrt{\langle J_x\rangle}}, \qquad [X,P]=i,$$5 scaling persists up to

$$X=\frac{J_y}{\sqrt{\langle J_x\rangle}},\qquad P=\frac{J_z}{\sqrt{\langle J_x\rangle}}, \qquad [X,P]=i,$$6

after which the scaling reverts to $$X=\frac{J_y}{\sqrt{\langle J_x\rangle}},\qquad P=\frac{J_z}{\sqrt{\langle J_x\rangle}}, \qquad [X,P]=i,$$7 (Vasilyev et al., 2013).

The same formalism extends directly to two ensembles in cascaded geometry. When the dynamics factorize into EPR modes $$X=\frac{J_y}{\sqrt{\langle J_x\rangle}},\qquad P=\frac{J_z}{\sqrt{\langle J_x\rangle}}, \qquad [X,P]=i,$$8 and $$X=\frac{J_y}{\sqrt{\langle J_x\rangle}},\qquad P=\frac{J_z}{\sqrt{\langle J_x\rangle}}, \qquad [X,P]=i,$$9, the reduced conditional equations become equivalent to the single-oscillator case, so the same distinction between conditional and dissipative generation, and the same $x(t)$0 versus $x(t)$1 scaling, apply to nonlocal spin squeezing and Gaussian entanglement (Vasilyev et al., 2013).

2. Collective-spin implementations and scalable trajectory models

An experimental archetype of conditional spin squeezing is the cavity-QED measurement of a large $x(t)$2Rb ensemble by vacuum Rabi splitting. In that experiment, the two clock states $x(t)$3 and $x(t)$4 define a collective pseudo-spin, and the cavity normal-mode splitting measures $x(t)$5 and, after a microwave $x(t)$6-pulse swap, $x(t)$7, thereby yielding

$x(t)$8

The first QND measurement prepares a posterior state with reduced $x(t)$9 uncertainty, and a second measurement verifies the resulting correlations. For nearly $p(t)$0 atoms in a low-finesse cavity with $p(t)$1, the reported conditional spin-noise reduction is $p(t)$2 relative to the CSS projection noise, with an inferred spectroscopic gain of $p(t)$3 below the SQL and a directly observed $p(t)$4 spectroscopic gain (Chen et al., 2011).

The same conditional-squeezing paradigm becomes numerically nontrivial for large monitored ensembles, because the full stochastic master equation scales exponentially in the atom number. An exact route for identical atoms is to compress all density-matrix elements related by permutation symmetry into collective variables labeled by occupation counts $p(t)$5. This reduces the number of independent variables to

$p(t)$6

and permits exact stochastic-master-equation simulations up to roughly $p(t)$7 atoms in a bad cavity under homodyne detection. In that representation, the conditional squeezing of $p(t)$8 appears as a narrowing Gaussian-like distribution of collective density-matrix elements, while probe strength, polarization, detection efficiency, spontaneous emission, and atom number determine the balance between information gain and decoherence (Zhang et al., 2024).

For much larger ensembles, a stochastic cumulant mean-field theory provides a scalable approximation. Applied to tens of thousands of $p(t)$9Rb atoms in an optical cavity, it models conditional spin squeezing generated solely by homodyne probing of atom–cavity dressed states. The central transduction mechanism is that the cavity transmission amplitude near a dressed-state resonance depends sensitively on $$H^{\mathrm{int}}=\sqrt g\big[\alpha X\,x(t)+\beta P\,p(t)\big], \qquad \alpha=\cos\theta,\quad \beta=\sin\theta,$$0, so probing near the upper dressed-state resonance, with $$H^{\mathrm{int}}=\sqrt g\big[\alpha X\,x(t)+\beta P\,p(t)\big], \qquad \alpha=\cos\theta,\quad \beta=\sin\theta,$$1, maximizes the squeezing rate. The theory reproduces the expected decrease of $$H^{\mathrm{int}}=\sqrt g\big[\alpha X\,x(t)+\beta P\,p(t)\big], \qquad \alpha=\cos\theta,\quad \beta=\sin\theta,$$2, the probe-induced rotation of the mean spin in the equatorial plane, and the strong dependence of conditional squeezing on probe power, efficiency, atom-cavity detuning, and atom number (Zhang et al., 2023).

3. Mechanical conditional squeezing and the conditional–unconditional divide

In optomechanics, conditional squeezing refers to the posterior motional state inferred from a continuous position measurement. The fundamental object is again a conditional covariance matrix, but the distinction between conditional and unconditional states becomes especially sharp because the conditional state center executes a stochastic trajectory in phase space. A measurement-conditioned Gaussian state can therefore be squeezed even when the unconditional ensemble state remains broad.

A direct verification of this principle was carried out for a $$H^{\mathrm{int}}=\sqrt g\big[\alpha X\,x(t)+\beta P\,p(t)\big], \qquad \alpha=\cos\theta,\quad \beta=\sin\theta,$$3 suspended mirror in a detuned optical cavity. The reflected-light photocurrent was processed with causal and anti-causal Wiener filters, yielding verified conditional standard deviations of $$H^{\mathrm{int}}=\sqrt g\big[\alpha X\,x(t)+\beta P\,p(t)\big], \qquad \alpha=\cos\theta,\quad \beta=\sin\theta,$$4 for position and $$H^{\mathrm{int}}=\sqrt g\big[\alpha X\,x(t)+\beta P\,p(t)\big], \qquad \alpha=\cos\theta,\quad \beta=\sin\theta,$$5 for momentum. The result established conditional mechanical squeezing for a mg-scale pendulum near quantum regimes, while making explicit that the verified object was the measurement-conditioned state rather than an unconditional squeezed steady state (Santiago-Condori et al., 2020).

A later optomechanical experiment approached the zero-point threshold much more closely by continuous measurement and post-processing of a $$H^{\mathrm{int}}=\sqrt g\big[\alpha X\,x(t)+\beta P\,p(t)\big], \qquad \alpha=\cos\theta,\quad \beta=\sin\theta,$$6 GaAs cantilever with an optically stiffened resonance. The normalized conditional covariance matrix had minimum eigenvalue $$H^{\mathrm{int}}=\sqrt g\big[\alpha X\,x(t)+\beta P\,p(t)\big], \qquad \alpha=\cos\theta,\quad \beta=\sin\theta,$$7, corresponding to $$H^{\mathrm{int}}=\sqrt g\big[\alpha X\,x(t)+\beta P\,p(t)\big], \qquad \alpha=\cos\theta,\quad \beta=\sin\theta,$$8 times the zero-point variance, or $$H^{\mathrm{int}}=\sqrt g\big[\alpha X\,x(t)+\beta P\,p(t)\big], \qquad \alpha=\cos\theta,\quad \beta=\sin\theta,$$9 above the zero-point fluctuations. This remained a classically squeezed state, not yet a quantum squeezed state, but it was reported as $g=d\gamma$0 orders of magnitude closer to the quantum regime in variance than the previous record (Lane et al., 2024).

Theoretical analyses of conditional mechanical squeezing clarify both the ideal mechanism and its limits. In two-tone backaction-evading optomechanical measurements, continuous monitoring of the cavity output generates conditional squeezing of the measured mechanical quadrature $g=d\gamma$1, with an exact steady-state conditional covariance available in the QND/RWA regime. Beyond the adiabatic approximation, the exact theory predicts an optimal cavity linewidth,

$g=d\gamma$2

rather than monotonic improvement toward smaller $g=d\gamma$3. Beyond the rotating-wave approximation, the same monitoring can generate conditional mechanical squeezing, intra-cavity squeezing, and optomechanical entanglement simultaneously (Brunelli et al., 2019).

A complementary control-theoretic treatment shows that even optimal feedback does not generally collapse the distinction between conditional and unconditional states. In a full linear-quadratic-Gaussian treatment without the standard rotating-wave and adiabatic approximations, the exact conditional covariance can differ substantially from approximate predictions, and the unconditional state prepared by feedback remains broader than the conditional posterior state. The paper emphasizes that the two cannot coincide in a typical control scheme, even with infinite feedback strength (Isaksen et al., 2022). A further extension to PID control argues that derivative feedback, unlike standard proportional feedback, affects both the conditional and the unconditional squeezing of a mechanical quadrature (Hijano et al., 17 Apr 2026).

4. Heralding, no-jump evolution, and postselected finite-dimensional squeezing

A second major usage of conditional squeezing is postselected rather than continuously monitored. In one realization, the conditioning variable is the absence of a decay event. For the non-Hermitian one-axis twisting model,

$g=d\gamma$4

the “conditional” state is the normalized no-jump trajectory of an open spin ensemble. In this postselected dynamics, the non-Hermitian term filters the spin wavefunction and can generate squeezing substantially better than Hermitian OAT; for $g=d\gamma$5, the steady-state squeezing scales toward the Hermitian TACT law $g=d\gamma$6. The practical drawback is the success probability of no decay,

$g=d\gamma$7

which decreases rapidly with system size and evolution time, making the best steady-state regime mainly relevant to small systems. More practical finite-time regimes, such as $g=d\gamma$8 or $g=d\gamma$9, still improve substantially on Hermitian OAT while reducing the success-cost penalty (Wu et al., 2015).

In linear-optical conditional measurement, heralding on a photon-count outcome after a beam splitter produces a non-Gaussian output whose squeezing can be analyzed exactly once it is rewritten as a displaced finite-dimensional Fock superposition. With coherent input $d$0, number-state input $d$1, and heralding on $d$2 detected photons, the conditional output takes the form

$d$3

Because the displacement does not affect quadrature squeezing, the problem reduces to squeezing of a truncated Fock superposition. In this way the scheme can saturate the finite-dimensional squeezing bound exactly for $d$4, with $d$5, and for $d$6, with $d$7. For higher $d$8, the best performance is obtained by fixing the heralding event to single-photon detection $d$9, but exact saturation is generally lost because the available parameters do not independently control all coefficients of the larger superposition. Under nonideal source and detector models, the optimal states remain accessible with success probabilities around $\theta=0$0–$\theta=0$1 in the tabulated examples (Esakkimuthu et al., 2024).

In continuous-variable gate synthesis, the phrase also appears historically as a contrast class. A teleportation-like deterministic squeezing gate was introduced explicitly against earlier heralded conditional squeezing gates, which the paper described as the best prior measurement-based implementations but with low success probabilities; it quotes a target squeezing of $\theta=0$2 with success probability $\theta=0$3 for the conditional approach it seeks to replace (Sun et al., 2022).

5. Conditional Gaussian operations in bosonic control and communication

Conditional squeezing can also refer to a Gaussian operation whose sign or application depends on a control system. A recent example is generalized cross-resonant driving in qubit–bosonic systems. For a two-photon qubit–oscillator interaction,

$\theta=0$4

combined with an appropriate qubit drive, the effective dressed-frame unitary becomes

$\theta=0$5

so the sign of the squeezing parameter is controlled by the dressed qubit state. The same framework yields multimode qubit-conditional two-mode squeezing from interactions of the form $\theta=0$6, and the paper emphasizes applications to bosonic quantum computing and quantum simulation in superconducting circuits (Ayyash, 20 Mar 2025).

A different operational use appears in protocols that synthesize nonlinear gates from conditional Gaussian resources. There, squeezing is not applied conditionally to the probe mode itself; rather, an ancilla is first squeezed, then coupled linearly to the probe, and finally measured. The conditioned Kraus map on the probe contains a Kerr-like term $\theta=0$7, measurement-dependent linear phases, and finite-squeezing attenuation terms. In the strong-squeezing limit, the nonunitary factors vanish and the effective conditional map approaches an almost deterministic nonlinear gate, enabling cat-state and compass-state generation and super-Heisenberg sensing proposals (Johnsson et al., 2021).

In quantum optical communication, inverse squeezing can appear inside a conditional receiver architecture without the squeezing operation itself being adaptive. In squeezed-state pulse-position modulation, empty slots are $\theta=0$8 and the pulse slot is $\theta=0$9. A fixed inverse-squeezing stage $\theta=-\pi/4$0 maps the ideal codeword to coherent-state PPM with enhanced effective pulse amplitude $\theta=-\pi/4$1, after which a conditional pulse-nulling tree acts on the resulting state. The paper is careful that this is not “conditional squeezing” in the narrow sense of adaptively choosing squeezing based on intermediate outcomes; rather, it is inverse squeezing embedded in a genuinely conditional receiver (Bai et al., 30 Jun 2026).

6. Metrological, steering, entropic, and mathematical generalizations

Beyond state preparation, conditional squeezing has been generalized into a family of metrological and information-theoretic quantities. For bipartite spin systems, one can define a conditional spin-squeezing parameter by averaging the metrological squeezing of Bob’s conditional states $\theta=-\pi/4$2, prepared by Alice’s outcome $\theta=-\pi/4$3 for setting $\theta=-\pi/4$4: $\theta=-\pi/4$5 This quantity measures the metrological usefulness of the assemblage rather than of Bob’s reduced state alone, and it yields experimentally practical steering criteria. The higher-order hierarchy based on nonlinear spin observables approaches the conditional quantum-Fisher-information witness and is particularly useful for non-Gaussian spin states (Guo et al., 2021).

In a two-qubit entropic setting, the phrase “conditional entropy squeezing” denotes quantities built from conditional Shannon entropies of Pauli measurements,

$\theta=-\pi/4$6

and the paper studies the average of two conditional entropy squeezing factors as an indicator of one-way normalized entropic-uncertainty-relation steering. For maximally entangled states, the average conditional entropy squeezing and the normalized steering quantity show complete similarity; for partially entangled states they remain close but not identical, and under acceleration, amplitude damping, dephasing, and entanglement swapping they typically change in parallel (Obada et al., 2023).

A more abstract usage concerns the two-mode squeezing channel itself in the presence of side information. The conditional entropy power inequality for bosonic quantum systems states that if two inputs $\theta=-\pi/4$7 and $\theta=-\pi/4$8 are conditionally independent given a memory $\theta=-\pi/4$9, then after a squeezing channel $d$0 with $d$1,

$d$2

Here “conditional” refers to conditioning on a quantum memory rather than to measurement-conditioned state preparation, and the result characterizes the minimum conditional output entropy achievable under a two-mode squeezer (Palma et al., 2017).

In a distinct mathematical usage, the related phrase “conditional global non-squeezing” appears in the study of the cubic nonlinear Klein–Gordon equation on $d$3. There, “conditional” refers not to quantum-state conditioning but to a theorem that global-in-time symplectic non-squeezing follows if one assumes a uniform a priori Strichartz bound,

$d$4

The terminology is therefore adjacent rather than identical: it concerns a conditional non-squeezing statement in Hamiltonian PDE rather than conditional preparation of squeezed quantum states (Mendelson, 2014).

Taken together, these usages define a broad but coherent theme. In experimental quantum optics, atomic physics, and optomechanics, conditional squeezing is primarily a posterior variance reduction generated by continuous monitoring or heralding. In bosonic control, it becomes a controlled Gaussian operation. In metrology and steering theory, it becomes a conditional performance functional. In information theory, it enters as a conditional entropic constraint for the squeezing channel. The common structure is that squeezing is no longer an absolute property of an unconditional state alone, but of a state, operation, or uncertainty measure evaluated relative to explicitly retained side information.