Papers

We show that operator dynamics in U(1) symmetric systems are constrained by two independent conserved charges and construct a non-Abelian operator size basis that respects both, enabling a symmetry-resolved characterization of operator growth. The non-Abelian operator size depends on the operator's nonlocal structure and is organized by an SU(2) algebra. Operators associated with large total angular momentum are relatively simple, while those with small angular momentum are more complex. Operator growth is thus characterized by a reduction in angular momentum and can be probed using out-of-time-ordered correlators. Using the charge-conserving Brownian Sachdev-Ye-Kitaev model, we derive an exact classical master equation that governs the size distribution, the distribution of an operator expanded in this basis, for arbitrary system sizes. The resulting dynamics reveal that the size distribution follows a chi-squared form, with the two conserved charges jointly determining the overall time scale and the shape of the distribution. In particular, single-particle operators retain a divergent peak at large angular momentum throughout the time evolution.

Neutral atom quantum computers and to a lesser extent trapped ions may suffer from atom loss. In this work, we investigate the impact of atom loss in long chains of trapped ions. Even though this is a relatively rare event, ion loss in long chains must be addressed because it destabilizes the entire chain resulting in the loss of all the qubits of the chain. We propose a solution to the chain loss problem based on (1) a quantum error correction code distributed over multiple long chains, (2) beacon qubits within each long chain to detect the loss of a chain, and (3) a decoder adapted to correct a combination of circuit faults and erasures after beacon qubits convert chain losses into erasures. We verify the chain loss correction capability of our scheme through circuit level simulations with a distributed $[[72,12,6]]$ BB code with beacon qubits.

The integration of sensing and communication functionalities within a common system is one of the main innovation drivers for next-generation networks. In this paper, we introduce a quantum integrated sensing and communication (QISAC) protocol that leverages entanglement in quantum carriers of information to enable both superdense coding and quantum sensing. The proposed approach adaptively optimizes encoding and quantum measurement via variational circuit learning, while employing classical machine learning-based decoders and estimators to process the measurement outcomes. Numerical results for qudit systems demonstrate that the proposed QISAC protocol can achieve a flexible trade-off between classical communication rate and accuracy of parameter estimation.

We present a detailed analysis of slowly driven quantum thermal machines based on interacting qubits within the framework of the Lindblad master equation. By implementing a systematic expansion in the driving rate, we derive explicit expressions for the rate of work of the driving forces, the heat currents exchanged with the reservoirs, and the entropy production up to second order, ensuring full thermodynamic consistency in the linear-response regime. The formalism naturally separates geometric and dissipative contributions, identified by a Berry curvature and a metric in parameter space, respectively. Analytical results show that the geometric heat pumped per cycle is bounded by $k_B T N_q \ln 2$ for $N_q$ non-interacting qubits, in direct analogy with the Landauer limit for entropy change. This bound can be surpassed when qubit interactions and asymmetric couplings to the baths are introduced. Numerical results for the interacting two-qubit system reveal a non-trivial role of the interaction between qubits and the coupling between the qubits and the baths in the behavior of the dissipated power. The approach provides a general platform for studying dissipation, pumping, and performance optimization in driven quantum devices operating as heat engines.

High-performance numerical quantum compilers rely on classical optimization, but are limited by slow numerical evaluations and a design that makes extending them with new instructions a difficult, error-prone task for domain experts. This paper introduces OpenQudit, a compilation framework that solves these problems by allowing users to define quantum operations symbolically in the Qudit Gate Language (QGL), a mathematically natural DSL. OpenQudit's ahead-of-time compiler uses a tensor network representation and an e-graph-based pass for symbolic simplification before a runtime tensor network virtual machine (TNVM) JIT-compiles the expressions into high-performance native code. The evaluation shows that this symbolic approach is highly effective, accelerating the core instantiation task by up to $\mathtt{\sim}20\times$ on common quantum circuit synthesis problems compared to state-of-the-art tools.

Quantum chemistry and optimization are two of the most prominent applications of quantum computers. Variational quantum algorithms have been proposed for solving problems in these domains. However, the design of the quantum circuit ansatz remains a challenge. Of particular interest is developing a method to generate circuits for any given instance of a problem, not merely a circuit tailored to a specific instance of the problem. To this end, we present a reinforcement learning (RL) approach to learning a problem-dependent quantum circuit mapping, which outputs a circuit for the ground state of a Hamiltonian from a given family of parameterized Hamiltonians. For quantum chemistry, our RL framework takes as input a molecule and a discrete set of bond distances, and it outputs a bond-distance-dependent quantum circuit for arbitrary bond distances along the potential energy curve. The inherently non-greedy approach of our RL method contrasts with existing greedy approaches to adaptive, problem-tailored circuit constructions. We demonstrate its effectiveness for the four-qubit and six-qubit lithium hydride molecules, as well as an eight-qubit H$_4$ chain. Our learned circuits are interpretable in a physically meaningful manner, thus paving the way for applying RL to the development of novel quantum circuits for the ground states of large-scale molecular systems.

We show that a distribution related to Gaussian Boson Sampling (GBS) on graphs can be sampled classically in polynomial time. Graphical applications of GBS typically sample from this distribution, and thus quantum algorithms do not provide exponential speedup for these applications. We also show that another distribution related to Boson sampling can be sampled classically in polynomial time.

Two-mode squeezing is central to entangled-photon generation and nonlinear interferometry, yet standard perturbative low-gain treatments and Gaussian formalisms can obscure the interference of photon-number amplitudes, especially in nonlinear interferometers and at high gain. Here we derive a closed-form Fock-basis expression for the action of the two-mode squeezing operator on arbitrary number states, enabling the direct analysis of nonlinear interferometers in the photon-number basis at arbitrary squeezing strength. Within this framework, we provide intuitive physical interpretations of several known quantum-interference effects and identify a new multi-photon interference phenomenon in a four-crystal geometry that could readily be observed in laboratories. Our work provides a compact analytic toolkit and explicit design rules for engineering multi-photon interference, with applications in quantum sensing, precision metrology, and quantum state generation.

Quantum asymmetry and coherence are genuinely quantum resources that are essential to realize quantum advantage in information technologies. However, all quantum processes are fundamentally constrained by quantum speed limits, which raises the question on the corresponding bounds on the rate of consumption of asymmetry and coherence. In the present work, we derive a formulation of the quantum speed limit for observables in terms of the trace-norm asymmetry of the time-dependent quantum state relative to the observable. This quantum speed limit can be directly observed in experiment through weak value measurement and provides a lower bound to the quantum Fisher information about the parameter conjugate to the observable. It can be further related to quantum coherence relative to the eigenbasis of the observable. We obtain a complementary relation for the speed of three mutually unbiased observables for a single qubit. As an application, we derive a notion of a quantum thermodynamic speed limit.

In Ref. [1], dynamical quantum phase transitions (DQPTs) – non-analyticities in the Loschmidt return rate at critical times – are investigated in the presence of noise for a two-band model. The authors report that DQPTs persist even after averaging over the noise and they use their results to derive dynamical phase diagrams. The protocol used approximates the noise-averaged mixed state, obtained using a master equation, by a pure state, characterized by its excitation probability. In this comment we rigorously show that: (1) This approximation is exponentially poor in the thermodynamic limit. (2) When using the correct metric, the Loschmidt echo of two density matrices in any two-dimensional Hilbert space can become zero if and only if both density matrices are pure, ruling out DQPTs for non-zero noise. (3) An a posteriori reinterpretation of the results as an interferometric protocol is possible but such a protocol is unsuitable to investigate the effects of noise on DQPTs because it is inherently blind to decoherence. We also investigate alternative natural ways to average over noise realizations and show that in all of them DQPTs are smoothed out.

We explore the non-reciprocal intracell hopping mediated non-Hermitian topological phases of an extended Su-Schrieffer-Heeger model hosting second-nearest-neighbour hopping. We microscopically analyze the phase boundaries using the non-Bloch momentum while the off-critical (critical) phases are directly associated with the gapped (gapless) nature of the non-Bloch bands that we derive from the characteristic equation using the gauge freedom. The non-Bloch momentum accurately reflects the bulk boundary correspondence (BBC) explaining the winding number profile under open boundary conditions. We examine the adiabatic dynamics to promote the concept of adiabatic charge transport in a non-Hermitian scenario justifying the BBC in spatio-temporal Bott index and non-Bloch Chern number. Once the non-Bloch bands experience no (a) gap-closing during the evolution of time, quantized flow of is preserved (broken). Our study systematically unifies the concept of non-Bloch bands for both static and driven situations.

We introduce a novel analytical approach for studying free-fermion (XX) chains with smoothly varying, site-dependent hoppings and magnetic fields. Building on a discrete WKB-like approximation applied directly to the recurrence relation for the single-particle eigenfunctions, we derive a closed-form expression for the local fermion density profile as a function of the Fermi energy, which is valid for arbitrary fillings, hopping amplitudes and magnetic fields. This formula reproduces the depletion and saturation effects observed in previous studies of inhomogeneous free-fermion chains, and provides a theoretical framework to understand entanglement entropy suppression in these models. We demonstrate the accuracy of our asymptotic formula in several chains with different hopping and magnetic field profiles. Our findings are thus the first step towards an analytical treatment of entanglement in free-fermion chains beyond the reach of conventional field-theoretic techniques.

This paper proposes an optimization of Quantum Key Distribution (QKD) Networks using Graph Neural Networks (GNN) framework. Today, the development of quantum computers threatens the security systems of classical cryptography. Moreover, as QKD networks are designed for protecting secret communication, they suffer from multiple operational difficulties: adaptive to dynamic conditions, optimization for multiple parameters and effective resource utilization. In order to overcome these obstacles, we propose a GNN-based framework which can model QKD networks as dynamic graphs and extracts exploitable characteristics from these networks' structure. The graph contains not only topological information but also specific characteristics associated with quantum communication (the number of edges between nodes, etc). Experimental results demonstrate that the GNN-optimized QKD network achieves a substantial increase in total key rate (from 27.1 Kbits/s to 470 Kbits/s), a reduced average QBER (from 6.6% to 6.0%), and maintains path integrity with a slight reduction in average transmission distance (from 7.13 km to 6.42 km). Furthermore, we analyze network performance across varying scales (10 to 250 nodes), showing improved link prediction accuracy and enhanced key generation rate in medium-sized networks. This work introduces a novel operation mode for QKD networks, shifting the paradigm of network optimization through adaptive and scalable quantum communication systems that enhance security and performance.

Classical particles in random potentials typically experience a percolation phase transition, being trapped in clusters of mean size $\chi$ that diverges algebraically at a percolation threshold. In contrast, quantum transport in random potentials is controlled by the Anderson localization length, which shows no distinct feature at this classical critical point. Here, we present a comprehensive theoretical analysis of the semi-classical crossover between these two regimes by studying particle propagation in a one-dimensional, red speckle potential, which hosts a percolation transition at its upper bound. As the system deviates from the classical limit, we find that the algebraic divergence of $\chi$ continuously connects to a smooth yet non-analytic increase of the localization length. We characterize this behavior both numerically and theoretically using a semi-classical approach. In this crossover regime, the correlated and non-Gaussian nature of the speckle potential becomes essential, causing the standard Dorokhov-Mello-Pereyra-Kumar (DPMK) description for uncorrelated disorder to break down. Instead, we predict the emergence of a bimodal transmission distribution, a behavior normally absent in one dimension, which we capture within our semi-classical analysis. Deep in the quantum regime, the DMPK framework is recovered and the universal features of Anderson localization reappear.

Simulated annealing provides a heuristic solution to combinatorial optimization problems. The cost function of a problem is mapped onto the energy function of a physical many-body system, and, by using thermal or quantum fluctuations, the system explores the state space to find the ground state, which corresponds to the optimal solution of the problem. Studies have highlighted both the similarities and differences between thermal and quantum fluctuations. Nevertheless, fundamental understanding of thermal and quantum annealing remains incomplete, making it unclear how quantum annealing outperforms thermal annealing in which problem instances. Here, we investigate the many-body dynamics of thermal and quantum annealing by examining all possible interaction networks of $\pm J$ Ising spin systems up to seven spins. Our comprehensive investigation reveals that differences between thermal and quantum annealing emerge for particular interaction networks, indicating that the structure of the energy landscape distinguishes the two dynamics. We identify the microscopic origin of these differences through probability fluxes in state space, finding that the two dynamics are broadly similar, but that quantum tunnelling produces qualitative differences. Our results provide insight into how thermal and quantum fluctuations navigate a system toward the ground state in simulated annealing, and are experimentally verifiable in atomic, molecular, and optical systems. Furthermore, these insights may improve mappings of optimization problems to Ising spin systems, yielding more accurate solutions in faster simulated annealing and thus benefiting real-world applications in industry. Our comprehensive survey of interaction networks and visualization of probability flux can help to understand, predict, and control quantum advantage in quantum annealing.

The $W$ state, a canonical representative of multipartite quantum entanglement, plays a crucial role in quantum information science due to its robust entanglement properties. Quantum uncertainty relations, on the other hand, are a fundamental cornerstone of quantum mechanics. This paper introduces a novel approach to Identifying tripartite $W$ states by leveraging tripartite quantum uncertainty relations. By employing a specific set of non-commuting observables, we formulate an uncertainty-based criterion for identifying $W$ states and rigorously demonstrate its generality in distinguishing them from other tripartite entangled states, such as the Greenberger-Horne-Zeilinger state. Our approach bypasses the need for complete quantum state tomography, as it requires only the verification of a set of uncertainty inequalities for efficient $W$-state identification. This work provides a new theoretical tool for identifying multipartite entangled states and underscores the significant role of quantum uncertainty relations in entanglement characterization.

Quantum interference underpins many quantum information protocols but is typically studied in lossless Hermitian systems. Here, we reveal an exceptional point induced phase transition in two photon Hong Ou Mandel interference within a lossy coupled waveguide system. In the PT symmetric phase. interference is ultrasensitive to coupling strength, yielding sharp bunching antibunching switches. In the PT broken phase. it becomes robust oscillation free and propagation independent with coincidence probability stably tunable via coupling. These regimes enable enhanced quantum sensing and reliable two photon control for robust quantum information processing.

We explore several aspects of categorical symmetry-resolved entanglement entropy (SREE) directly within two-dimensional rational conformal field theory (RCFT) (without invoking any SymTFT construction arXiv:2409.02806). We derive a general formula applicable whenever the action of the relevant topological defect lines on the annulus Hilbert space is known. This framework accommodates weakly and strongly symmetric boundaries, cloaking states, and fusion rings with multiplicities. We verify the formula in a range of diagonal unitary and non-unitary examples, including theories with generalized Haagerup-Izumi modular data. Furthermore, we extend the analysis to non-diagonal RCFTs. The $\frac{1}{2}E_6$ example demonstrates that closed-channel modular data and NIM-rep multiplicities alone do not suffice to determine the defect action on the complete open-channel Hilbert space.

In this work, we investigate the emergence and control of genuine tripartite entanglement in a hybrid cavity quantum electrodynamics architecture consisting of two linearly coupled single mode resonators, one of which interacts coherently with a two level atom. An analytical framework is developed in a weak driving regime, where the system dynamically supports a delocalized hybrid excitation shared by the two photonic modes and the atomic degree of freedom. Tripartite concurrence fill has been used to characterize and identify parameter regimes of maximal multipartite quantum correlation that can be generated in this model. Additionally, we demonstrate how dissipative rates and detuning asymmetries govern the conversion of bipartite entanglement into a genuinely tripartite state, establishing a controllable transition from localized Jaynes Cummings correlations to delocalized photonic atomic entanglement networks. These findings outline a clear route to engineering steady state multipartite quantum resources in coupled cavity QED platforms, with direct relevance to quantum networking, distributed quantum information processing, and photonic state routing in scalable quantum architectures.

The development of quantum internet demands on-chip quantum processor nodes and interconnection between the nodes. Path-encoded photonic qubits are suitable for on-chip quantum information processors, while time-bin encoded ones are good at long-distance communication. It is necessary to develop an on-chip converter between the two encodings to satisfy the needs of the quantum internet. In this work, a quantum photonic circuit is proposed to convert time-bin-encoded photonic qubits to path-encoded ones via a thin-film lithium niobate high-speed optical switch and low-loss matched optical delay lines. The performance of the encoding converter is demonstrated by the experiment of time-bin to path encoding conversion on the fabricated sample chip. The converted path qubits have an average fidelity higher than 97%. The potential of the encoding converter on applications in quantum networks is demonstrated by the experiments of entanglement distribution and quantum key distribution. The results show that the on-chip encoding converter can serve as a foundational component in the future quantum internet, bridging the gap between quantum information transmission and on-chip processing based on photons.

Neutral-atom quantum computers encode qubits in individually trapped atoms arranged in optical lattices. Achieving defect-free atom configurations is essential for high-fidelity quantum gates and scalable error correction, yet stochastic loading and atom loss during rearrangement hinder reliable large-scale assembly. This work presents ATLAS, an open-source atom transport algorithm that efficiently converts a randomly loaded $W \times W$ lattice into a defect-free $L \times L$ subarray while accounting for realistic physical constraints, including finite acceleration, transfer time, and per-move loss probability. In the planning phase, optimal batches of parallel moves are computed on a lossless virtual array; during execution, these moves are replayed under probabilistic atom loss to maximize the expected number of retained atoms. Monte Carlo simulations across lattice sizes $W=10$–$100$, loading probabilities $p_{\mathrm{occ}}=0.5$–$0.9$, and loss rates $p_{\mathrm{loss}}=0$–$0.05$ demonstrate fill rates above $99\%$ within six iterations and over $90\%$ atom retention at low loss. The algorithm achieves sublinear move scaling ($\propto M^{0.55}$) and linear growth of required initial size with target dimension, outperforming prior methods in robustness and scalability – offering a practical path toward larger neutral-atom quantum arrays.

We study the optimal rates of emulation (also called interconversion) between quantum channels, in three regimes: when the encoder and the decoder are unassisted, when they share classical randomness, and when they share an unlimited amount of entanglement. When the source and the target channels are idempotent, we completely characterise these three regimes, in terms of the shape vectors of the two channels, which describe the structure of their ranges. In the unassisted and shared randomness regimes, we give a single-letter expression for the emulation capacity, show that it coincides with the zero-error emulation capacity, and prove a finite-blocklength converse which implies a strong converse. In these two regimes, shared randomness does not increase the emulation capacity, and the emulation of idempotent channels is not reversible. Moreover, we provide an efficient algorithm to approximate this capacity to arbitrary precision. When an unlimited amount of shared entanglement is allowed, we determine the minimal emulation error exactly at every finite blocklength. It is governed by a single quantity, the $\ell_2$-norm of the shape vector, so that the emulation becomes reversible, and exact emulation is achievable with finitely many maximally entangled states. Altogether, these results draw a comprehensive picture of the task of emulating idempotent channels with each other.

Quantum circuits will experience failures that lead to computational errors. We introduce Medusa, an automated compilation method for lowering a circuit's failure rate. Medusa uses flags to predict the absence of high-weight errors. Our method can numerically upper bound the failure rate of a circuit in the presence of flags, and fine tune the fault-tolerance of the flags in order to reach this bound. We assume the flags can have an increased fault-tolerance as a result of applying surface QECs to the gates interacting with them. We use circuit level depolarizing noise to evaluate the effectiveness of these flags in revealing the absence of the high-weight stabilizers. Medusa reduces the cost of quantum-error-correction (QEC) because the underlying circuit has a lower failure rate. We benchmark our approach using structured quantum circuits representative of ripple-carry adders. In particular, our flag scheme demonstrates that for adder-like circuits, the failure rate of large-scale implementations can be lowered to fit the failure rates of smaller-scale circuits. We show numerically that a slight improvement in the local fault-tolerance of the flag-qubits can lead to a reduction in the overall failure rate of the entire quantum circuit.

Optical cavities provide a powerful means to engineer light-matter hybrid states by coupling confined electromagnetic fields with matter excitations. Achieving in situ control of the coupling strength is essential for investigating how such hybridization evolves with the coupling strength. In this work, we use a symmetry-changing structural phase transition in lead halide perovskites to reversibly tune the phonon-photon coupling strength, leveraging the fact that their phonon frequencies and oscillator strengths are dictated by lattice symmetry. Terahertz time-domain spectroscopy of MAPbI3 embedded in nanoslot cavities reveals three polariton branches above the critical temperature Tc = 162.5 K, and the emergence of an additional branch below Tc, activated by a new phonon mode in the low-temperature phase. The full dispersion is accurately reproduced using a multimode Hopfield model, confirming that all normalized coupling strengths remain in the ultrastrong coupling regime. These results demonstrate symmetry-controlled tuning of ultrastrong coupling via phonon engineering in optical cavities.

Quantum batteries are quantum mechanical systems able to store and release energy in a controlled fashion. Among them, a special role is played by quantum structures defined as networks of two-level systems. In this context, it has recently been shown that the energy stored in free fermion quantum batteries is sensitive to the quantum phase diagram of the battery itself. This sensitivity is relevant for stabilizing the stored energy and designing optimal charging protocols. In this article, we explore universal charging behaviors of free fermion quantum batteries across quantum phase transitions. We first analyze a Dirac cone-like model to extract general features. Then, we verify our findings by means of two relevant lattice models, namely the Ising chain in a transverse field and the Haldane model.

Quantum computing faces a key challenge: balancing the need for low circuit depth (crucial for fault tolerance) with the high accuracy required for complex computations like quantum chemistry and error correction, which typically require deeper circuits. We overcome this trade-off by introducing a novel reinforcement learning approach featuring engineered reward functions, called QASER, that take into account seemingly contradictory optimization goals. This reward enables the compilation of circuits with lower depth and higher accuracy, significantly outperforming state-of-the-art techniques. Benchmarks on quantum chemistry state preparation circuits demonstrate stable compilations. We achieve up to 50% improved accuracy, while reducing 2-qubit gate counts and depths by 20%. This advancement enables more efficient and reliable quantum compilation.

Optomechanics with levitated nanoparticles is a promising way to combine very different types of quantum non-Gaussian aspects induced by continuous dynamics in a nonlinear or time-varying potential with the ones coming from discrete quantum elements in dynamics or measurement. First, it is necessary to prepare quantum non-Gaussian states using both methods. The nonlinear and time-varying potentials have been widely analyzed for this purpose. However, feasible preparation of provably quantum non-Gaussian states in a single mechanical mode using discrete photon detection has not been proposed yet for optical levitation. We explore pulsed optomechanical interactions combined with non-linear photon detection techniques to approach mechanical Fock states and confirm their quantum non-Gaussianity. We also predict the conditions under which the optomechanical interaction can induce multiple-phonon addition processes, which are relevant for $n$-phonon quantum non-Gaussianity. The practical applicability of quantum non-Gaussian states for sensing phase-randomized displacements is shown. Besides such applications, generating quantum non-Gaussian states of levitated nanoparticles can help to study fundamental questions of quantum thermodynamics, and macroscopic quantum effects.

For pure states, the quantum Berry curvature was well studied. However, the quantum curvature for mixed states has received less attention. From the concept of symmetric logarithmic derivative, we introduce a mixed-state quantum curvature and find that it plays a key role in the field of multi-parameter precision estimations. Through spectral decomposition, we derive the mixed-state Berry curvature for both the full-rank and non-full-rank density matrices. As an example, we obtain the exact expression of the Berry curvature for an arbitrary qubit state.

Quantum neural networks (QNNs) are an analog of classical neural networks in the world of quantum computing, which are represented by a unitary matrix with trainable parameters. Inspired by the universal approximation property of classical neural networks, ensuring that every continuous function can be arbitrarily well approximated uniformly on a compact set of a Euclidean space, some recent works have established analogous results for QNNs, ranging from single-qubit to multi-qubit QNNs, and even hybrid classical-quantum models. In this paper, we study the approximation capabilities of QNNs for periodic functions with respect to the supremum norm. We use the Jackson inequality to approximate a given function by implementing its approximating trigonometric polynomial via a suitable QNN. In particular, we see that by restricting to the class of periodic functions, one can achieve a quadratic reduction of the number of parameters, producing better approximation results than in the literature. Moreover, the smoother the function, the fewer parameters are needed to construct a QNN to approximate the function.

We demonstrate channel-selective frequency up-conversion from telecom wavelengths around 1540 nm for optical fiber communication to visible wavelengths around 780 nm, based on second-order optical nonlinearity in a cavity of the converted modes. In our experiment, we selectively convert a light from any frequency mode within frequency-multiplexed telecom signals to a desired output mode, determined by the cavity resonances. Based on the experimental results of the frequency up-conversion, we derive the signal-to-noise ratio of the process at the single-photon level, and discuss its applicability to channel-selective quantum frequency conversion (CS-QFC) in the context of frequency-multiplexed quantum networks. Finally, we describe specific use cases of the CS-QFC, which show its utility as a reconfigurable switching element in frequency-multiplexed networks, particularly for selectively performing Bell-state measurements between two photons originating from different frequencies.