Equalities and inequalities from entanglement, loss, and beam splitters

TL;DR

The paper studies how optical loss and beam splitter interference shape entanglement, nonclassicality, and phase-space representations in continuous-variable quantum optics. Building on a companion result that balanced beam splitters maximize entanglement when interfering a state with vacuum, it derives monotonicity and convexity properties of purity, plus corollaries for similarity measures under loss, beyond Cauchy–Schwarz. It then provides a suite of equalities and inequalities for quasiprobability distributions and for ladder-operator relations, linking them to the quadrature coherence scale and to two-copy observables, and establishes a precise link between nonclassicality and entanglement under loss through the QCS. Collectively, these results connect phase-space formalisms, operator inequalities, and information-theoretic quantities, offering rigorous bounds and tools for certifying nonclassicality and guiding photonic quantum technologies.

Abstract

Quantum optics bridges esoteric notions of entanglement and superposition with practical applications like metrology and communication. Throughout, there is an interplay between information theoretic concepts such as entropy and physical considerations such as quantum system design, noise, and loss. Therefore, a fundamental result at the heart of these fields has numerous ramifications in development of applications and advancing our understanding of quantum physics. Our recent proof for the entanglement properties of states interfering with the vacuum on a beam splitter led to monotonicity and convexity properties for quantum states undergoing photon loss [Lupu-Gladstein et al., arXiv:2411.03423 (2024)] by breathing life into a decades-old conjecture. In this work, we extend these fundamental properties to measures of similarity between states, provide inequalities for creation and annihilation operators beyond the Cauchy-Schwarz inequality, prove a conjecture [Hertz et al., PRA 110, 012408 (2024)] dictating that nonclassicality through the quadrature coherence scale is uncertifiable beyond a loss of 50%, and place constraints on quasiprobability distributions of all physical states. These ideas can now circulate afresh throughout quantum optics.

Figures (3)

• Figure 1: Top: zero-momentum slice of the Wigner function $W(x,0)$ of a Fock state $\vert 1 \rangle$ undergoing some loss $T=1$ (solid), $1/2$ (long dashed), $0$ (dashed). Bottom: its von Neumann entropy $H_1(\rho_T)$ (solid) and purity $\mathcal{P}(\rho_T)$ (dashed).
• Figure 2: Two-copy circuit used to measure the purity of a state $\rho$. The left scheme is the usual setup used to measure purity using two copies of a state Daleyetal2012Islametal2015, the other two are equivalent setups when the same loss $T$ is applied on both copies of the state and are essential to proving the convexity properties of lossy states and the entanglement-generation properties of beam splitters.
• Figure 3: On the left is the Husimi quasiprobability distribution of the vacuum $P_{\vert 0 \rangle}(\alpha ,-1)=\frac{|\alpha |^{2n}}{\pi n!}{\mathrm e}^{-|\alpha |^2}$. The middle distribution is defined as $\frac{1}{2}P_{\vert 0 \rangle}(\alpha /\sqrt{2},-1)$, a dilated version of the vacuum Husimi distribution, and the right distribution as $2 P_{\vert 0 \rangle}(\sqrt{2}\alpha ,-1)$, a compressed version of the vacuum Husimi distribution.

Theorems & Definitions (28)

• Lemma 1: Symmetry of purity (Lemma 1 in Ref. CompanionShortarXiv)
• Theorem 2: Convexity of purity (Theorem 7 in Ref. CompanionShortarXiv)
• Lemma 3: Positive polynomial expansion of Hilbert-Schmidt norm (Lemma 6 in Ref. CompanionShortarXiv)
• Theorem 4: Concavity of entropy (Theorem 4 in Ref. CompanionShortarXiv)
• Corollary 5: Mutual information
• proof
• Corollary 6: Hilbert-Schmidt inner-product and loss
• proof
• Conjecture 7
• Corollary 8