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arXiv:2403.18890 · Physical Review Research

Average Rényi entanglement entropy in Gaussian boson sampling

Jason Youm, Joseph T. Iosue, Adam Ehrenberg, Yu-Xin Wang, Alexey Vyacheslavovich Gorshkov

Published May 8, 2025• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevresearch.7.023125

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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.
Bosonic Gaussian states are a special class of quantum states in an infinite dimensional Hilbert space that are relevant to universal continuous-variable quantum computation as well as to near-term quantum sampling tasks such as Gaussian Boson Sampling. In this work, we study entanglement within a set of squeezed modes that have been evolved by a random linear optical unitary. We first derive formulas that are asymptotically exact in the number of modes for the Rényi-2 Page curve (the average Rényi-2 entropy of a subsystem of a pure bosonic Gaussian state) and the corresponding Page correction (the average information of the subsystem) in certain squeezing regimes. We then prove various results on the typicality of entanglement as measured by the Rényi-2 entropy by studying its variance. Using the aforementioned results for the Rényi-2 entropy, we upper and lower bound the von Neumann entropy Page curve and prove certain regimes of entanglement typicality as measured by the von Neumann entropy. Our main proofs make use of a symmetry property obeyed by the average and the variance of the entropy that dramatically simplifies the averaging over unitaries. In this light, we propose future research directions where this symmetry might also be exploited. We conclude by discussing potential applications of our results and their generalizations to Gaussian Boson Sampling and to illuminating the relationship between entanglement and computational complexity.
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We study the von Neumann entanglement entropy generated by a Haar distributed passive interferometer acting on nnn equally squeezed input modes with fixed nonzero squeezing strength sss. Previous work established proportional weak typicality for integer R'enyi orders α≥2α\geq 2α≥2 and stated a sublinear von Neumann result, while the proportional von Neumann case remained open. For a subsystem of knk_nkn​ modes satisfying kn/n→r∈(0,1)k_n/n\to r\in(0,1)kn​/n→r∈(0,1), we prove that, for every ε>0\varepsilon>0ε>0 and all sufficiently large nnn, P(∣S1,nES1,n−1∣≥ε)≤2exp⁡[−cs,rε2n2log⁡2(en)].\mathbb{P}\left(\left|\frac{S_{1,n}}{\mathbb{E}S_{1,n}}-1\right|\geq\varepsilon\right)\leq2\exp\left[-\frac{c_{s,r}\varepsilon^2n^2}{\log^2(en)}\right].P(​ES1,n​S1,n​​−1​≥ε)≤2exp[−log2(en)cs,r​ε2n2​]. The proof represents the entropy as a singular value statistic of a principal block of UUTUU^{\mathsf T}UUT, where UUU denotes the unitary interferometer. It regularizes the logarithmic singularity at the endpoint corresponding to a pure Gaussian mode and applies concentration on the unitary group. The result establishes proportional von Neumann weak typicality and further implies almost sure convergence of S1,n/ES1,nS_{1,n}/\mathbb{E}S_{1,n}S1,n​/ES1,n​ to 111, a typical volume law, and the variance bound Var(S1,n)=Os(log⁡2n)\mathrm{Var}(S_{1,n})=O_s(\log^2 n)Var(S1,n​)=Os​(log2n). An accompanying Lean 4 development verifies the proof chain.
We study the tri-partite genuine Rényi multi-entropy of fully symmetric Gaussian states as a function of the squeezing parameter for different total number of bosonic modes under different subsystem configurations. Exact results for a few modes are obtained, and we conjecture an asymptotic formula at the limit of large squeezing parameter as a function of the replica index nnn, while also finding subleading subsystem-size dependent terms for n=3n=3n=3 under a specific tripartition. We analytically show that the genuine n=2n=2n=2 Rényi multi-entropy of any pure bosonic Gaussian states is zero for any number of modes and for any tripartition. Furthermore, to investigate the dependence on subsystem size, we plot the curve of genuine Rényi multi-entropy as a function of the number of modes in the subsystems while keeping the total number of modes fixed. We discuss differences and similarities between this result and studies of the multi-entropy curve for Haar random states. In particular, we identify a subsystem size configuration related with the multi-entropy time in an evaporating black hole for which the behavior of the genuine multi-entropy changes.
The full state vector of boson sampling is generated by passing S S single photons through beam splitters of M M modes. We express the initial Fock state in terms of 2 ^{S-1} S − 1 generalized coherent states, making possible the exact application of the unitary evolution. Due to the favorable polynomial scaling of numerical effort in M M , we can investigate Rényi entanglement entropies for moderate particle and huge mode numbers. We find symmetric Page curves with a maximum entropy at equal partition, which is almost independent on Rényi index. Furthermore, the maximum entropy as a function of mode index saturates for M\geq S^2 M ≥ S 2 in the collision-free subspace case. The asymptotic value of the entropy increases linearly with S S . In addition, we show that the build-up of the entanglement leads to a cusp in the asymmetric entanglement curve. Maximum entanglement is reached well before the mode population is distributed over the whole system.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number NNN and allowed average decoding error εεε equals the vanishing-error capacity at mean photon number N/(1−ε)N/(1-ε)N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.