Robustness of quantum correlations in photonic systems under non-Markovian dephasing: theoretical analysis in the experimental range

Abstract

We investigate the preservation and dynamics of entanglement and Bell nonlocality in a two-photon system subject to local dephasing, particularly emphasizing environments exhibiting non-Markovian memory effects. By analyzing the interplay between entanglement, as measured by concurrence, and Bell inequality violation, we identify time regions where Bell inequality is not violated, even when concurrence remains high. This indicates that strongly entangled states do not always exhibit nonclassical correlations with certainty. Our study further explores the effects of refractive index variations, spectral width, and interaction symmetry, demonstrating how structured environments and controlled environmental interactions can mitigate decoherence and enhance the resilience of quantum correlations. We propose that these theoretical predictions can be tested using modifications of recent experimental optical setups. These findings provide valuable insights into the conditions under which quantum correlations remain irreproducible by classical local hidden-variable models, with direct implications for quantum communication, quantum information processing, and foundational studies in quantum mechanics. Furthermore, our findings highlight the crucial importance of quantum coherence in maintaining entanglement and nonlocality over extended time evolution. These results show remarkable alignment with the experimental observations, demonstrating high accuracy and consistency within the expected parameter range.

Introduction

Controlling entanglement dynamics and mitigating decoherence is crucial for quantum information processing. In photonic systems^1–4, decoherence from environmental interactions can lead to sudden entanglement death (ESD)^5,6, imposing constraints on quantum communication and computation. Unlike solid-state systems, photonic platforms offer advantages such as long coherence times and high-speed information transfer, making them ideal for quantum networking^7,8. However, photon loss, phase damping, and spectral diffusion challenge preserving quantum correlations. Entanglement and nonlocality are fundamental yet distinct quantum resources. While nonlocal states must be entangled, not all entangled states exhibit nonlocality. Extensive research has focused on their detection, quantification, and evolution in open quantum systems. A key challenge in quantum information processing is decoherence from environmental interactions, which degrades quantum resources and impacts protocol reliability. Understanding entanglement and nonlocality under noise is crucial for designing robust quantum technologies. This has driven research into sustaining quantum correlations, with non-Markovian environments^9–13 enabling entanglement and nonlocality revivals. Additionally, the quantum Zeno effect and structured environments support entanglement trapping, enhancing correlation resilience. To address these challenges, hybrid quantum systems leverage structured photonic environments, engineered reservoirs, and controlled spectral profiles to enhance entanglement resilience via non-Markovian memory effects. Quantum networking across fiber-optic and free-space channels is emerging as a scalable approach¹⁴. These advancements support key applications such as security-proof quantum key distribution^15,16, quantum teleportation¹⁷, and quantum communication complexity, all relying on quantum correlations and nonlocality¹⁸. Bell inequality violations remain essential in confirming correlation robustness, underscoring their importance in quantum information science¹⁹. A deeper understanding of Markovian and non-Markovian decoherence, especially in photonic systems^10,20, is vital for optimizing quantum protocols and ensuring the resilience of photonic quantum technologies.

Certain bipartite mixed entangled states exhibit correlations describable by a local hidden variable model, meaning they can be classically reproduced²¹. This implies that in practical scenarios involving mixed states, entanglement alone cannot guarantee fundamentally nonclassical correlations. To distinguish such cases, we introduce inherently nonlocal correlations (INCs) as quantum correlations that no classical local model can replicate. INCs are crucial in quantum mechanics, particularly in device-independent quantum cryptography, security-proof quantum key distribution^22,23, and quantum computing. A key indicator of INCs in bipartite systems is Bell inequality violation, confirmed through a Bell function^19,24. A central question is identifying entanglement-related quantum features essential for quantum computation and impossible to classically mimic. In two-qubit systems, the relationship between concurrence²⁵ and Bell inequality violations²⁴ has been extensively studied, characterizing the violation range for a given concurrence value²⁶. It has been shown that for every bipartite entangled state, a protocol using an auxiliary state and local filtering ensure the resulting state cannot be classically reproduced²⁷. However, not all entangled states are directly usable for tasks requiring INCs. Recent investigations have offered significant advancements in understanding the nature of quantum entanglement and the limitations of local hidden variable models. A notable study provides a rigorous reexamination of entanglement, differentiating between correlations that originate from quantum entanglement and those that can be attributed to local realistic models²⁸. Another explores hidden-variable models for multipartite entangled states and measurements, advancing the understanding of classical simulations of quantum correlations²⁹. These developments help delineate the boundaries between quantum and classical correlations, especially for mixed states.

Quantum coherence forms a foundational element of quantum physics, capturing essential features ranging from superposition to quantum correlations³⁰. Its importance is evident in multiple quantum information and metrological tasks, where coherence offers a decisive advantage over classical methods³¹. More broadly, coherence serves as an indispensable resource in low-temperature thermodynamics^32,33, facilitates exciton and electron transport in biomolecular complexes^34–36, and underpins various processes at the nanoscale³⁷. Historically, quantum optics has provided a strong framework for studying coherence in specific scenarios^38–41. However, a more general information-theoretic approach has recently emerged, allowing for the quantification of coherence across arbitrary states^42,43. This approach defines a set of incoherent states along with “incoherent operations,” which are operations that preserve this set⁴². Analogous to the resource theory of entanglement, where local operations and classical communication maintain separable states⁴⁴, this formalism treats coherence as a quantifiable resource, linking it more directly to modern quantum information science.

Expanding on the foundational concepts discussed previously, this paper presents a detailed study of entanglement, Bell nonlocality, and coherence within a two-photon system, focusing specifically on the transition between Markovian and non-Markovian regimes. These transitions are especially significant in realistic environments, as decoherence and the resulting loss of quantum properties greatly affect nonlocal phenomena. Here, we model a two-photon setup subject to local dephasing and examine how such interactions govern the time evolution of quantum correlations. By pinpointing the conditions that give rise to these effects, we show that they are attainable with contemporary optical techniques. Crucially, we reveal that states displaying high entanglement (quantified through concurrence) do not always violate Bell inequality, underlining that strong entanglement need not consistently manifest nonclassical attributes. We further investigate situations where entanglement persists over prolonged intervals, contrasting these with intervals where INCs remain robust. Since each photon interacts with a memory-capable environment, revivals of entanglement, coherence, and Bell nonlocality are feasible when parameters are carefully chosen, suggesting a concrete strategy for addressing decoherence in open quantum systems. In addition, we provide a systematic analysis of how refractive index differences, spectral bandwidth, and symmetry or asymmetry in coupling determine the resilience of these quantum resources in the presence of noise. Finally, we propose that modest modifications to current experimental optical setups can validate these theoretical results, offering a practical avenue for verification and deeper insight into preserving entanglement, nonlocality, and coherence in photonic systems.

The paper is organized as follows. In Sect. Photon dynamics and interactions, we present the physical model and the quantum formalism that describe the two-photon system, focusing on its interaction with local dephasing. This section also explains the dependence of the system’s quantum properties on key parameters such as the dephasing rate, initial state, and the nature of symmetric and asymmetric interactions. Section Bell nonlocality and entanglement details the quantum quantifiers used in the analysis and provides an in-depth discussion of the results. In Sect. Conclusion, we summarize the key findings of this study, highlighting their importance in enhancing our understanding of open quantum systems and their implications for quantum optics and information science.

Photon dynamics and interactions

In this section, we investigate the non-Markovian framework, beginning with a thorough analysis of the one-photon model before extending to the two-photon scenario. This approach allows us to examine the manifestation of non-Markovian effects across varying photon quantities, promoting understanding applicable to more complex systems.

In the present study, we assume weak coupling between the system and the environment, where it describes a weak interaction that leads to pure dephasing without energy exchange–a common scenario in photonic systems exposed to dispersive media. This assumption is realistic because photonic systems, such as those involving photons in optical fibers or birefringent materials like quartz plates, typically experience weak interactions that cause dephasing rather than strong coupling effects like energy transfer. The manuscript supports this with experimental parameters, including a small refractive index difference and spectral width, which are consistent with weak coupling and align with referenced experimental studies (e.g., Ref.⁹).

In the beginning, we outline the procedure for deriving the dynamics of multiple independent bodies that interact locally with s, without limiting the nature of all noise when the dynamics of a single body are known. Subsequently, we will use this approach to explicitly examine the coherence and Bell non-locality of two qubits interacting locally with a non-Markovian.

We consider a qubit (S) and its associated environment () in the initial state , where (with ) represents the polarization state and denotes the time-evolved polarization state is given by:

1

where is the quantum map, and is the time evolution operator, and is the interaction Hamiltonian of the system.

To extend our discussion to two-qubit systems, we consider a composite system consisting of , where the subsystems and each interact with their respective s, and . For each qubit (denoted as S), the reduced density matrix evolves according to the following equation:

2

This is expressed in terms of the Kraus operators ^45–47.

Furthermore, the time evolution operator for the total system can be factored as . Consequently, the reduced density matrix for the two-qubit system is represented as follows:

3

On the other hand, the evolution of each qubit is described by:

4

where the matrix elements are:

5

For the two-qubit system, the dynamics are similarly given by:

6

This formulation illustrates the independent dynamics of the two qubits and their interactions with their respective environments.

Using the previous findings to examine the non-Markovian effects on quantum coherence and Bell nonlocality in a system of two independent qubits, each interacting exclusively and independently with its respective local environment. Specifically, our focus will be on analyzing the non-Markovian dynamics of an open quantum system in which the polarization state of a photon constitutes the primary system. At the same time, its frequency acts as the al degree of freedom. To simulate and study the behavior of the system, we utilize a carefully designed experimental setup comprising a rotatable Fabry–Pérot (FP) cavity, an interference filter, and a quartz plate (Fig. 1)^9,20,48–50. The (FP) cavity generates a frequency comb for the photon, which is subsequently refined by the interference filter to isolate two dominant peaks. The resulting frequency distribution, , characterizes the probability density of finding the photon in a mode with frequency . A two-peaked Gaussian profile mathematically represents this distribution⁵¹:

7

Here, and represent the central frequencies, defines the width, and governs the relative weighting of the peaks. The deviation angle of the FP cavity provides precise control over , enabling fine-tuning of the distribution.

Figure 1.

Schematic theoretical setup for studying quantum correlations in photonic systems. (a) Single-photon configuration including optical elements: HWP (half-wave plate) for polarization adjustment, QWP (quarter-wave plate) for linear-circular polarization conversion, IF (interference filter) for selecting wavelength ranges, QP (quartz plate) for introducing phase shifts, PBS (polarizing beam splitter) for separating or combining beams by polarization, FP cavity for isolating resonant frequencies, and SPD (single-photon detector) for high-precision photon detection. (b) Two-photon configuration extending the setup to investigate entanglement, Bell nonlocality, and coherence under non-Markovian dephasing.

In contrast, this configuration represents a structured setting frequently encountered in practical quantum optical setups, such as those utilizing cavities or interference-filtering methodologies. It is analytically manageable and offers versatile manipulation of environmental features like peak separation, width, and relative amplitudes through the parameter . Additionally, previous studies have notably employed these dual-peaked spectra to investigate non-Markovian dynamics in photonic systems^9,20,49.

Following the filtering process, the polarization state experiences non-Markovian dephasing due to its interaction with the frequency modes within the quartz plate. Hamiltonian mathematically characterizes this interaction citeLaine2013:

8

where and are the refractive indices for horizontal and vertical polarizations, respectively, inducing a differential phase shift based on frequency. This interaction results in non-Markovian dynamics for the photon’s polarization.

For an initial state , where (with ) represents the polarization state and is the al state, the polarization state with time-evolved is given by:

9

where is the quantum map, and is the time evolution operator.

The density matrix for the polarization is expressed as⁹:

10

Using the distribution from Eq. (7), the dephasing rate can be determined,

11

where . The formula for allows for the analysis of single-photon dynamics, as outlined in Eq. (10). This also leads to the derivation of two-photon dynamics by employing the reduced dynamics of a single photon. Using this framework, one can obtain insights into the quantum coherence and Bell nonlocality that can arise in the two-photon dynamics.

Using the reduced density matrix derived for the degree of freedom of polarization, as given in Eq. (10)⁵², we extend the construction to obtain the full reduced density matrix, , for the two-qubit system comprising nonidentical qubits. The basis considered for this system is defined as^46,47:

We use the previous procedure, to obtain the full reduced density matrix, , at any given time t. Using the procedure described previously, we obtain the complete reduced density matrix, , at any given time t.

12

Since , where is a Hermitian operator, Eq. (12) can be used to determine the time evolution of the two-qubit density matrix for any given initial state.

With this foundation, we are now prepared to investigate the impact of non-Markovian dynamics in a bipartite system, with particular attention to its insightful and advantageous aspects.

Bell nonlocality and entanglement

To quantify entanglement in a two-qubit state, we use concurrence, a well-established measure determining whether a quantum state is entangled or separable. For a general X-structured density matrix, the concurrence is defined as^25,53–59:

13

where

14

This formulation ensures that the concurrence is nonzero if and only if the state is entangled, and it vanishes for separable states. It thus provides a clear quantitative measure of how entanglement evolves. In particular, for Bell-like states, which form a subset of X states, the system starts in a maximally entangled pure state:

15

where . These Bell-like states are instrumental in many quantum information protocols due to their maximal entanglement. Under the evolution induced by Eq. (12), we get

16

17

With this formalism, we can analyze the correlation dynamics by evaluating how the polarization systems evolve, considering their identical and distinct dephasing rates.

Bell nonlocality, another fundamental feature of quantum mechanics, provides a distinct way to probe quantum correlations that defy the classical explanation. The Bell-CHSH inequality is frequently employed to quantify the extent of nonlocality in quantum systems. A violation of the B-C-H-S-H inequality characterizes nonlocality, and this violation can be expressed in terms of the following:

18

where and . The terms , , and are related to the elements of the density matrix, defined as:

19

In this study, our objective is to identify regions where the Bell inequality remains below the classical bound , despite the presence of entanglement, thus exploring the circumstances under which quantum correlations fail to exhibit non-locality. The expressions illustrate how the off-diagonal elements of the density matrix directly influence the nonlocal properties of the system. It is significant to note that is generally larger than , ensuring that the Bell-CHSH inequality reflects the maximum violation of the system at . This framework is particularly applied to two-qubit X states, where the structure of the density matrix simplifies the calculations, offering a clearer path to understanding non-locality in such systems.

The characteristics of quantum coherence in a system are intricately related to the off-diagonal elements of its density matrix. The ability to quantify coherence allows for a deeper understanding of its role and the challenges that decoherence poses in open systems. One well-established method of quantifying coherence is based on relative entropy, which quantifies it as the “distance” between a quantum state and its closest incoherent state. The relative entropy of coherence is expressed as:

20

where denotes the von Neumann entropy, and is the diagonalized version of , representing the incoherent state. This measure provides a deeper understanding of the intrinsic information content carried by quantum coherence. The relative entropy is particularly valuable in quantum information theory, as it serves as a reliable indicator of the degree of “quantum-ness” of a system. In particular, the relative entropy is a monotonic measure, which means that coherence cannot increase under incoherent operations.

Figure 2 presents the time evolution of entanglement, Bell nonlocality, and quantum coherence under different Markovian and non-Markovian conditions, illustrating how quantum resources degrade and revive depending on system-environment interactions. The figure comprises two subplots corresponding to symmetric () and asymmetric () system-environment interactions, with distinct line styles representing various degrees of non-Markovianity. The system parameters, including refractive index difference , spectral width THz, and central photon frequencies PHz ( nm) and PHz ( nm), define the spectral characteristics governing the decoherence and memory effects in the photon system. The red, blue, and black curves represent entanglement, Bell nonlocality, and coherence measures, respectively, with different line styles corresponding to various values of and , which determine the degree of non-Markovianity in each environment. A fundamental distinction in quantum physics is that while nonlocal states must necessarily be entangled, an entangled state does not always exhibit nonlocality. The violation of the C-H-S-H Bell inequality, where , serves as a definitive indicator of INCs, which cannot be reproduced by any classical local model. Although such violations always occur for pure entangled states, they are not necessarily guaranteed for mixed states, making it essential to identify regions where the Bell inequality remains below the classical bound despite the presence of entanglement. The figure depicts the regions where as functions of time and the degree of entanglement. When both environments are fully Markovian (), where no memory effects are present, all three measures exhibit monotonic decay where system correlations are negligible and information is permanently lost. Furthermore, the entanglement undergoes sudden death after a finite time, while Bell’s nonlocality vanishes earlier, confirming that entanglement alone is not a sufficient condition for nonlocality. As non-Markovianity increases with both environments (), quantum measures display revival oscillations due to periodic information backflow from the environment. It is observed that reaches the classical threshold value slightly before the first occurrence of the disappearance of the entanglement as time progresses. In the strongly non-Markovian regime (), all three measures exhibit significant revival oscillations, where the figure illustrates the regions of revivals of Bell inequality violations and entanglement. These regions correspond to a return, after finite intervals during which and , of INCs and entanglement, respectively. Introducing non-Markovian effects in one environment (), the system exhibits partial memory effects, leading to asymmetric decay patterns of quantumness measures and decreasing the time intervals of partial revivals. Furthermore, comparing the symmetric and asymmetric cases, it is noted that asymmetric system-environment interactions exacerbate decoherence, leading to an accelerated degradation of quantum resources during the dynamics. These results highlight the crucial role of non-Markovian memory effects in extending quantum properties in photonic systems, the importance of symmetry in shaping quantum dynamics, and the ability to tune quantum resources through the independent control of and .

Figure 2.

Time evolution of quantum measures as a function of time t for different angle values under both Markovian and non-Markovian dynamics. Subfigures (a) and (b) represent symmetric () and asymmetric (), respectively. The system parameters are set as , THz, PHz (approximately 704.5 nm), and PHz (approximately 700.3 nm)⁹. The color code in the plots is such that: the black, blue, and red curves correspond to coherence, nonlocality, and entanglement measures, respectively. The color coding follows: solid lines represent in (a) and in (b); large-dashed lines correspond to in (a) and in (b); dot-dashed lines indicate in (a) and in (b).

Figure 3 illustrates the dynamics of entanglement, Bell nonlocality, and quantum coherence under different spectral width values () in both Markovian and non-Markovian regimes, revealing how these quantum resources evolve in the two-photon system. Crucially, variations in the spectral width modulate the rate of decoherence. In fact, broader spectra () lead to rapid deterioration of all three quantmness measures, while narrower ones () slow down decoherence, enhancing the lifetimes of quantum resources. In the Markovian case, a larger spectral width ( THz) accelerates decoherence, leading to earlier entanglement sudden death and faster disappearance of Bell nonlocality, whereas a smaller spectral width ( THz) allows coherence and quantum correlations to persist longer. In the non-Markovian regime, memory effects enable reviving of coherence and quantum correlations; however, increasing the spectral width weakens these revivals by reducing the information backflow from the environment. These findings emphasize that optimizing the spectral width is crucial for enhancing the resilience of quantum coherence and preserving entanglement and nonlocality, making it a key parameter in decoherence suppression and quantum resource management in noisy quantum systems.

Figure 3.

Time evolution of quantum measures as a function of time t for different spectral width values under both Markovian and non-Markovian dynamics. Subfigures (a) and (b) correspond to Markovian and non-Markovian dynamics, respectively. The system parameters are set as , PHz, and PHz. The color code in the plots is such that the black, blue, and red curves correspond to coherence, nonlocality, and entanglement measures, respectively. The color coding follows: solid lines represent ; large-dashed lines correspond to ; dot-dashed lines correspond to .

In Fig. 4, the effect of the parameter on entanglement, Bell nonlocality, and quantum coherence is explored within both Markovian and non-Markovian frameworks. The figure underscores how the refractive index difference between horizontal and vertical polarizations critically shapes both the decay rate and any possible resurgence of quantum resources. The results indicate that smaller refractive index differences () slow the decay of quantumness measures, resulting in a longer persistence of coherence, entanglement, and nonlocality, while also allowing more pronounced and extended revival under non-Markovian conditions. Interestingly, a smaller leads to a less pronounced phase shift between the polarization components, thereby slowing the loss of off-diagonal density matrix elements. In contrast, larger refractive index differences () accelerate dephasing, leading to the degradation of quantumness measures and suppressing revival effects in the non-Markovian regime. Intermediate values () produce moderate behaviors, balancing slower decay rates with weaker revival oscillations. The long protection of quantum resources with small thus highlights how the refractive index not only pertains to optical considerations but also plays a crucial role in defining the timescales for quantum resources loss and recovery, influencing the robustness of quantum information in open systems. Therefore, tuning the refractive index difference can serve as an effective tool to control the longevity and revival characteristics of quantum resources during dynamics.

Figure 4.

Time evolution of quantum measures as a function of time t for various values of refractive index differences under both Markovian and non-Markovian dynamics. Subfigures (a) and (b) correspond to Markovian and non-Markovian dynamics, respectively. The system parameters are set as THz, PHz, and PHz. The black, blue, and red curves correspond to coherence, nonlocality, and entanglement measures, respectively. The color coding follows: solid lines represent ; large-dashed lines correspond to ; dot-dashed lines correspond to .

Finally, in Fig. 5 we present a detailed comparison of how varying the initial state parameter affects the evolution of entanglement, Bell nonlocality, and coherence under Markovian and non-Markovian dephasing. Physically, sets the balance between the (or ) and (or ) components of the initial Bell-like state. Near or , the system starts with relatively weak correlations, leading to rapid decay of quantum resources and scant revival behavior even when non-Markovian memory effects are present. In contrast, states initialized around are almost maximally entangled and display more pronounced resilience: their coherence and Bell inequality violations last longer, and they can partially recover after initial decay in non-Markovian environments. Consequently, tuning provides a direct means to control how robustly and persistently quantum correlations manifest during evolution, revealing a clear strategy for protocols based on optimizing quantum resources in open photonic systems.

Figure 5.

Time evolution of quantum measures as a function of time t and the initial state parameter under both Markovian and non-Markovian dynamics. Subfigures (a), (c), and (e) correspond to the Markovian regime with , where the system undergoes memoryless decoherence. Subfigures (b), (d), and (f) correspond to the non-Markovian regime with , where all memory effects lead to information backflow and quantum correlation revivals. The system parameters are set as , THz, PHz (approximately 704.5 nm), and PHz (approximately 700.3 nm).

Conclusion

In summary, our theoretical investigation demonstrates that a two-photon system subjected to local dephasing can maintain high levels of entanglement, Bell nonlocality, and coherence for extended periods, provided that the environment is suitably structured and exhibits non-Markovian memory effects. We have shown that strong entanglement does not necessarily guarantee Bell violation, highlighting the need to distinguish genuinely nonlocal correlations from those describable by classical models. By exploring variations in experimentally relevant physical parameters—particularly spectral width, differences in refractive index, interaction symmetry, and initial state configuration—we have identified accessible strategies to mitigate decoherence. These include narrowing the spectral bandwidth to slow dephasing, engineering smaller refractive index contrasts to reduce phase mismatches, and preparing initial states near maximal entanglement to better leverage environmental memory effects. Crucially, the values chosen for these parameters fall within typical experimental ranges, enabling direct testing and validation using contemporary photonic setups. Our findings thus provide practical design guidelines for preserving quantum resources, with potential applications in long-distance quantum communication, high-fidelity quantum computation, and foundational quantum optics. Moreover, the proposed modifications to existing optical experiments offer an experimentally accessible pathway to verify and expand on these theoretical predictions, ultimately strengthening our understanding of open quantum systems and their technological potential. As a future study, we plan to investigate N-photon systems to characterize multipartite entanglement and nonlocality under non-Markovian dephasing, enhancing our understanding of preserving quantum resources in complex open quantum systems.

Author contributions

K.B. and S.B wrote the manuscript. All authors reviewed the manuscript.

Data availability

No datasets were generated or analysed during the current study.

Declarations

Conflict of interest

The authors declare that they have no conflict of interest.