Entropy and Variance Squeezing in Quantum Systems

• Entropy and variance squeezing are quantitative measures that reduce uncertainty in quantum states, with entropy squeezing comparing state entropy to classical benchmarks and variance squeezing quantifying observable fluctuations.
• They rely on fundamental uncertainty principles, such as Heisenberg and BBM bounds, to enforce trade-offs between localized variance reduction and compensatory increases in conjugate variables.
• These concepts drive advances in quantum metrology and information processing by enabling lower error bounds and enhanced control in systems ranging from isotonic oscillators to spin ensembles.

Entropy and variance squeezing refer to rigorous quantitative manifestations of quantum fluctuation suppression, uncertainty reduction, and information concentration, with deep relevance across quantum physics, information theory, metrology, and control. They quantify, respectively, reductions in global entropy (e.g., Shannon or von Neumann) and in lower moments such as variances (e.g., quadrature, spin, or photon-number fluctuations), with precise relationships and operational significance that differ by context but are fundamentally linked by uncertainty principles and performance bounds in both continuous and discrete settings.

1. Definitions and Foundational Distinction

Variance squeezing designates reduction of the variance of an observable—below a reference value set by quantum uncertainty principles or classical optima. For a canonical pair $(x, p)$, variance squeezing in $x$ is achieved if $\operatorname{Var}(x) < \frac{1}{2}$, maintaining $$(\operatorname{Var}(x)\operatorname{Var}(p)) \geq \frac{1}{4}$$ (Heisenberg). The notion generalizes to $S_j$ operators (e.g., atomic pseudospin or angular momentum), with the variance-squeezing factor $$V(S_j) = \Delta S_j - \sqrt{\big|\langle S_k\rangle\big|/2}$$, and $V(S_j) < 0$ signifying squeezing in $S_j$ (Liang et al., 4 Jan 2026).

Entropy squeezing refers to the reduction, below a quantum or classical bound, of a (possibly multidimensional) entropy—Shannon, von Neumann, Rényi, or other forms—associated to a state or observable. For position or momentum densities, entropy squeezing in $x$ is defined as $S_x < S_\text{x}^{(\text{HO},g.s.)}$, where the latter is the harmonic oscillator ground-state entropy (Ghasemi et al., 2011). More generally, entropy squeezing is assessed against established uncertainty inequalities, such as the Beckner–Bialynicki-Birula–Mycielski (BBM) bound for continuous distributions, or combinatorial analogues for discrete settings (Aravinda, 2022, Chung et al., 2015).

2. Squeezing in Quantum Systems: Continuous Variables

a) Isotonic Oscillator Eigenstates

In the isotonic oscillator, both entropy and variance squeezing are exhibited with precise quantification. For eigenstates $\psi(x)$ with densities $\rho(x)$ and $\xi(p)$:

• Shannon entropies: $S_x = -\int_0^\infty \rho(x)\log \rho(x)dx$, $S_p = -\int \xi(p)\log \xi(p)dp$.
• BBM relation: $S_x + S_p \geq \ln(e\pi) \approx 2.1447$ (Ghasemi et al., 2011).
• Entropy squeezing in $x$ if $S_x < 1.07236$.
• Amplitude (variance) squeezing in $x$ if $(\Delta x)^2 < 0.5$.

Only ground states and, in this system, first excited states exhibit position-entropy squeezing; variance squeezing is more restrictive, limited to ground states. When present, entropy squeezing is compensated by increased $S_p$, preserving the BBM bound. Similarly, for variance, the Heisenberg bound is preserved as squeezing in $x$ is offset by increased variance in $p$.

b) Jaynes–Cummings Model with Squeezed Fields

When a two-level atom interacts with a squeezed coherent state, the photon-number variance, $(\Delta n)^2$, and the atomic field linear entropy, $S_L(t)$, display synchronized minima. Mild squeezing (modest $r$) sharply localizes the photon-number distribution and minimizes $(\Delta n)^2$, inducing strong time-dependent entropy squeezing (near-purity "spikes" in $S_L$) in the atomic subsystem, particularly in the collapse regions. Excessive squeezing destroys these effects—variance and entropy both increase, and entanglement reasserts itself (Subeesh et al., 2012).

3. Squeezing and Uncertainty Principles: Entropic vs. Variance Bounds

Quantum uncertainty bounds provide the rigorous context for both forms of squeezing. Entropy and variance inequalities act as dual constraints:

• BBM entropic uncertainty for pure states: $S_x + S_p \geq \ln(e\pi)$.
• Heisenberg: $\Delta x\,\Delta p \geq \tfrac12$.
• For spins or finite-dimensional systems with observables $S_x, S_y, S_z$ (for qutrits), the entropic relation is $H(S_x) + H(S_y) + H(S_z) \geq 2$, leading to entropy squeezing diagnostics such as $E(S_j) = \exp[H(S_j)] - e/\sqrt{\exp[H(S_z)]}$ (Liang et al., 4 Jan 2026).

In multivariable feedback and estimation, negentropy and spectral flatness (Wiener entropy) set information-theoretic lower bounds for achievable error variance, establishing "uncertainty principles" for feedback architectures—the best one can achieve is determined by the entropy of the disturbance or noise process (Fang, 2014).

4. Information-Theoretic and Statistical Squeezing: Discrete and Log-Concave Cases

For unimodal continuous distributions, the variance is sandwiched between entropy powers: $$N(X) \leq \operatorname{Var}(X) \leq C \cdot N(X), \quad N(X) = \frac{1}{2\pi e}e^{2h(X)}$$ where $h(X)$ is the differential entropy. As entropy decreases, variance decays exponentially, making entropy reduction a proxy for distributional concentration (Chung et al., 2015).

In discrete log-concave distributions, for the min-entropy $H_\infty(X) = -\log \max_k p(k)$ and variance $\operatorname{Var}(X)$,

$$H_\infty(X) \geq \tfrac{1}{2}\ln(1 + \operatorname{Var}(X)).$$

Here the geometric law extremizes the entropy-variance relationship, with equality asymptotically (Aravinda, 2022). This "entropy–variance squeezing curve" quantifies the minimal global disorder required to realize a target variance.

For Rényi entropy $H_\alpha$, minimization under fixed variance among log-concave laws exhibits a phase transition: for $\alpha \leq \alpha^* \approx 1.241$, the symmetric uniform minimizes entropy; for $\alpha \geq \alpha^*$, the (two-sided) exponential does. In general log-concave cases, for $\alpha \geq 2$, the one-sided exponential law is extremal (Białobrzeski et al., 2021).

5. Operational Manifestations in Quantum Technologies

a) Atomic and Spin-Systems

Analysis of squeezing in structured atomic systems, such as the V-type atom in a dissipative cavity, demonstrates that entropy squeezing (via $E(S_x)<0$) is a strictly more informative and robust indicator of quantum noise suppression than variance squeezing. Entropy-based indicators detect quantum uncertainty reduction, even if the second-moment (variance-based) criteria are blind due to vanishing commutator expectations (Liang et al., 4 Jan 2026). Physically, this difference emerges most starkly on resonance, where $\langle S_z\rangle = 0$ disables variance squeezing in $S_x$, yet entropy squeezing remains fully present.

b) Entanglement and Many-Body Systems

In bipartite spin-$\tfrac12$ systems, explicit analytic links relate von Neumann entropy of reduced subsystems to variance-based collective spin-squeezing (Kitagawa–Ueda, Wineland parameters). Near maximal entanglement, the entropy achieves its maximum and the variance in the perpendicular collective spin vanishes, representing perfect spin-squeezing (Deb, 8 Feb 2025).

In Bose–Einstein condensates and lattice-based cold-atom platforms, engineered local entropy reduction, via controlled tunneling, allows simultaneous reduction of local entropy and enhancement of operational (spin) squeezing, improving metrological utility without trade-offs (Cattani et al., 2013).

c) Thermodynamics and Stochastic Processes

In thermodynamic machines such as Maxwell's refrigerator subject to squeezed thermal reservoirs, squeezing modifies the mean and variance of stochastic entropy production. Here, squeezing reduces both mean and relative fluctuations of entropy production, enhancing cooling rates and enabling violations of canonical thermodynamic bounds (e.g., Landauer's)—the system becomes more "efficient" at entropy reduction, with ensemble and trajectory-level squeezing of fluctuations (Manzano, 2017).

6. Squeezing in Information Processing and Diffusion Inference

Recent advances in diffusion generative models exploit entropy-aware (variance-optimized) inference, explicitly minimizing the conditional entropy $H(x_{t-1}|x_t)$ at each denoising step. The EVODiff algorithm directly "squeezes" the conditional variance (and hence conditional entropy) at each reverse iteration, achieving strict decreases in both uncertainty and sample variance relative to prior solvers (Li et al., 30 Sep 2025). This directly tightens the evidence lower bound (ELBO), improves mutual information recovery, and reduces reconstruction error in image generation.

7. Comparative Synthesis and Significance

Entropy squeezing and variance squeezing, though fundamentally distinct—global disorder vs. second-moment spread—are deeply interlinked by uncertainty relations and operational constraints in quantum, classical, and stochastic settings:

• Entropy squeezing is universally sensitive, detecting all forms of concentration and quantum coherence; it is preserved (or detectable) even in cases where variance cannot register squeezing, e.g., when the mean commutator vanishes.
• Variance squeezing provides sharper physical intuition and direct operational meaning in common quantum protocols (measurement precision, metrology), but is sometimes insensitive to higher-order or non-Gaussian signature squeezing.
• Both forms translate, via explicit bounds, into practical performance guarantees for estimation, control, error suppression, metrology, and non-equilibrium thermodynamics.

They form complementary but non-equivalent measures of quantum fluctuation suppression, each with precise and sometimes non-overlapping operational domains. Their interplay is essential for the design and assessment of advanced quantum technologies, uncertainty-limited measurement, feedback control, and information-theoretic optimality in both continuous and discrete systems (Ghasemi et al., 2011, Chung et al., 2015, Aravinda, 2022, Liang et al., 4 Jan 2026, Manzano, 2017, Li et al., 30 Sep 2025, Deb, 8 Feb 2025, Cattani et al., 2013).