We derive quantum Rényi-2 entropy power inequalities for Gaussian operations of the beam-splitting and squeezing type. We first show that known quantum von Neumann entropy power inequalities generalize straightforwardly to quantum Rényi-2 entropy power inequalities for Gaussian states but fail to do so for non-Gaussian states. We then derive quantum Rényi-2 entropy power inequalities that provide lower bounds for the Gaussian operations for any state. The inequality for the squeezing operation is shown to have applications in the generation and detection of quantum entanglement.
We derive quantum Rényi-2 entropy power inequalities for Gaussian operations of the beam-splitting and squeezing type. We first show that known quantum von Neumann entropy power inequalities generalize straightforwardly to quantum Rényi-2 entropy power inequalities for Gaussian states but fail to do so for non-Gaussian states. We then derive quantum Rényi-2 entropy power inequalities that provide lower bounds for the Gaussian operations for any state. The inequality for the squeezing operation is shown to have applications in the generation and detection of quantum entanglement.
We propose a generalization of the quantum entropy power inequality involving condi-tional entropies. For the special case of Gaussian states, we give a proof based on perturba-tion theory for symplectic spectra. We discuss some implications for entanglement-assisted classical communication over additive bosonic noise channels. 1 Classical and quantum entropy-power inequalities The entropy power inequality, proposed by Shannon [27] and later established with increasing rigor by Stam [29] and Blachman [5], has become a fundamental tool in classical informa-tion theory. Shannon’s original application of the entropy power inequality is a lower bound on the capacity of an additive (but potentially non-Gaussian) noise channel [27, Theorem 18]. However, the usefulness of the entropy power inequality is especially evident in multi-terminal information theory. Among the most well-known applications are the characterization of the Gaussian broadcast channel by Bergman [4], the Gaussian two-description problem by Ozarow [25] and the quadratic Gaussian CEO problem by Oohama [24]. In these multi-user settings, Fano’s inequality by itself is insufficient to characterize different tradeoffs. A more recent application proposed by Liu and Viswanath [23] uses entropy power inequalities to solve certain optimization problems. The entropy power inequality lower bounds the differential entropy of the convolution of two independent random variables X, Y taking values in Rn. Its covariance-preserving version states that H( λX + 1 − λY) ≥ λH(X) + (1 − λ)H(Y) for all 0 ≤ λ ≤ 1. (1) Eq. (1) can be shown [22, 32] to be equivalent to the more commonly used statement e2H(X+Y)/n ≥ e2H(X)/n + e2H(Y)/n. The latter explains the terminology as e2H(X)/n is the power, i.e., variance of a Gaussian ran-dom variable with identical entropy as H(X). Inequalities such as (1) are closely related to Log-Sobolev inequalities (see e.g., [31]) as well as Brunn-Minkowski-type inequalities [6]. Gen-eralizations to free probability [30] and quantum states have been considered. 1 ar
We prove the multimode conditional quantum Entropy Power Inequality for bosonic quantum systems. This inequality determines the minimum conditional von Neumann entropy of the output of the most general linear mixing of bosonic quantum modes among all the input states of the modes with given conditional entropies. Bosonic quantum systems constitute the mathematical model for the electromagnetic radiation in the quantum regime, which provides the most promising platform for quantum communication and quantum key distribution. We apply our multimode conditional quantum Entropy Power Inequality to determine new lower bounds to the squashed entanglement of a large family of bosonic quantum Gaussian states. The squashed entanglement is one of the main entanglement measures in quantum communication theory, providing the best known upper bound to the distillable key. Exploiting this result, we determine a new lower bound to the squashed entanglement of the multimode bosonic Gaussian channels that are extreme,i.e., that cannot be decomposed as a non-trivial convex combination of quantum channels. The squashed entanglement of a quantum channel provides an upper bound to its secret-key capacity,i.e., the capacity to generate a secret key shared between the sender and the receiver. Lower bounds to the squashed entanglement are notoriously hard to prove. Our results contribute to break this barrier and will stimulate further research in the field of quantum communication with bosonic quantum systems.
When arbitrary finite local dimensions are allowed, bosonic Gaussian states and nondegenerate classical Gaussian block vectors satisfy exactly the same unconditional balanced linear entropy inequalities. We establish this equivalence for von Neumann entropy, extending the known Rényi-2 correspondence. Spectral entropy integrals and compatible realification transfer classical determinant inequalities to bosonic states and, in the forward direction, to even fermionic Gaussian states under complete-mode canonical anticommutation relation (CAR) restrictions. A high-temperature limit gives the bosonic converse. Both transfers include pure modes and imply the Zhang-Yeung non-Shannon inequality. We also derive quantitative independence bounds from conditional mutual information ε = I(A:C | B). For bosons, bounded covariance or total photon number gives a trace-distance bound C√ε for ρAC relative to ρA ⊗ ρC, with spectral and matrix Gram refinements of the prefactor. For fermions, an explicit square-root bound requires no energy or spectral gap assumption, and a complementary bound of order √[ε log(1/ε)] is uniform in the number of conditioning modes with A, C fixed. Known Gaussian continuity theorems convert the new covariance estimates to state distance. The fixed-mode square-root exponents are optimal. Combining fermionic rigidity with the established entropy-deficit identity yields an operational lower bound on non-Gaussianity from CMI and a probability difference for an even measurement event on AC. Version 1.0. Complete manuscript with Appendices A–G.
Recently, many experiments have been conducted with the goal of demonstrating a quantum advantage over classical computation. One popular framework for these experiments is Gaussian boson sampling, where quadratic photonic input states are interfered via a linear optical unitary and subsequently measured in the Fock basis. In this paper, we study the modal entanglement of the output states in this framework just before the measurement stage. Specifically, we compute Page curves as measured by various Rényi- α entropies, where the Page curve describes the entanglement between two partitioned groups of output modes averaged over all linear optical unitaries. We derive these formulas for α = 1 (i.e., the von Neumann entropy) and, more generally, for all positive integer α , in the asymptotic limit of infinite number of modes and for input states that are composed of single-mode-squeezed-vacuum state with equal squeezing strength. We then analyze the limiting behaviors when the squeezing is small and large. Having determined the averages, we then explicitly calculate the Rényi- α variance for integers α > 1 and are able to show that these entropies are weakly typical.
Characterizing information-carrying capacities of bosonic communication channels is of significant practical interest. For thermal noise channels, using coherent (product) states yields an achievable rate for classical communication which is conjectured to be optimal. However, it is not known whether coding strategies using entanglement may perform better. Here we discuss upper bounds on classical capacities of thermal noise channels. These imply that coherent-state coding is close to optimal. Our main tool is a quantum analog of the entropy power inequality introduced Shannon. It gives a lower bound on the output von Neumann entropy when two independent signals combine at a beamsplitter.
We develop a method for deriving Gaussian optimality results for bosonic quantum systems from inequalities for qubit systems. Inspired by Gross's derivation of the Gaussian logarithmic Sobolev inequality from an inequality on the two-point space, we use qubits as quantum two-point spaces and lift bosonic Weyl operators to pseudo-Weyl operators on large qubit systems. By controlling weighted norms and relative entropy in the fluctuation limit, we establish two transfer theorems that convert tensorized qubit inequalities into inequalities for bosonic Gaussian channels. As a first application, we prove that the sharp 2-logarithmic Sobolev inequality for the bosonic Ornstein--Uhlenbeck semigroup tensorizes without loss in the constant. We also establish a sharp output entropy inequality for tensor powers of a class of qubit channels. This includes a quantum Mrs.~Gerber's lemma determining the minimum output entropy of tensor powers of the qubit depolarizing channel at fixed input entropy. Applying our relative entropy transfer theorem, we obtain multimode constrained minimum output entropy bounds for Gaussian additive-noise channels and thermal attenuators in the heating range.