Entanglement of two-qubit pure or mixed states can be quantified by several entangled measures (Plenio and Virmani, 2007), and they are shown to be monotonic (Singh et al., 2020) with respect to each other. Here, we use concurrence as a computable measure of entanglement that is also shown to be an entanglement monotone for pure and mixed states for low-dimensional systems. Concurrence was introduced by Hill and Wootters (1997) for two-qubit pure states. Subsequently, Wootters (1998) developed a closed-form expression for its convex roof extension and provided a computable formula for the entanglement of formation in the two-qubit case (Wootters, 2001). For a general two-qubit state represented by a density matrix ρ, concurrence is given as follows: C ρ = max 0 , λ 1 − λ 2 − λ 3 − λ 4 , (1) where λ i ′ s are the square roots of the eigenvalues of the matrix ρ ρ ̃ in decreasing order. Here, ρ ̃ is obtained by applying the spin-flip matrix to ρ as follows: ρ ̃ = σ y ⊗ σ y ρ * σ y ⊗ σ y , (2) where the complex conjugation (ρ*) is determined in the computational basis {|00⟩, |01⟩, |10⟩, |11⟩}.

Quantum coherence (Streltsov et al., 2017), one of the basic features of quantum physics and a key resource in the field of quantum information processing, requires a quantitative measure for studying its dynamical evolution under different quantum maps. A broad range of coherence measures is utilized, often relying on the off-diagonal terms of the density matrix. The primary justification for employing these measures is typically based on physical intuition. Recently, the characterization and quantification of coherence have gained much attention, and a quantitative theory of coherence as a resource was developed (Baumgratz et al., 2014) using entanglement-based approaches (Vedral and Plenio, 1998; Brandão and Plenio, 2008; Brandão and Plenio, 2010). These articles suggested that any proper measure of coherence must satisfy several criteria, some of which focused on the properties of specific coherence measures based on entanglement (Napoli et al., 2016), operation (Yuan et al., 2015), or convex roof construction (D’Arrigo et al., 2013). In this work, we adopt two proper measures of coherence that are computable and monotonic to study the evolution of coherence in the presence of correlated noise channels. The first measure is the l ₁-norm, and the second is the relative entropy of coherence.

The l ₁-norm of coherence is based on the absolute sum of the off-diagonal elements of the density matrix and is defined as C l 1 ρ = min D ρ , δ = ∑ i ≠ j | ρ i , j | , (3) where D(ρ, δ) = ‖ρ − δ‖ denotes the distance of ρ to a set of incoherent states I(δ ∈ I), and ρ _i,j are the off-diagonal elements of ρ (Baumgratz et al., 2014). Note that the l ₁-norm of coherence depends on the basis in which the density matrix is expressed. In general, for a d-dimensional system, 0 ≤ C l 1 ( ρ ) ≤ d − 1 , and the maximum value is achieved for a maximally coherent state that is in a uniform superposition of the basis states: | ψ d 〉 ≔ 1 d ∑ i = 1 d | i 〉 . The relative entropy of coherence (C _R ) is defined as C R ρ = min S ρ ‖ δ = S ρ d i a g − S ρ , (4) where ρ _diag is the diagonal density matrix of ρ and S(ρ) indicates the Von Neumann entropy (Breuer and Petruccione, 2002; Baumgratz et al., 2014). For a given state ρ, the relative entropy is bounded as C _R ≤ S(ρ _diag ) ≤ log₂(d) and, again, the maximum value is achieved for the maximally coherent state.

Let us begin with a brief introduction of the initial state and the noise channels in terms of memory and memoryless channels (D’Arrigo et al., 2013; Paulson et al., 2022). We chose MEMS as the initial state for our study as this is the class of mixed states that are maximally entangled for a given purity. This state was introduced by Munro et al. (2001) and Munro and Nemoto (2001); its density matrix is given by ρ M = g γ 0 0 γ 2 0 1 − 2 g γ 0 0 0 0 0 0 γ 2 0 0 g γ , (5) where g γ = 1 3 , 0 ≤ γ < 2 3 , γ 2 , 2 3 ≤ γ ≤ 1 .

From here onward, we will denote g(γ) simply as g, and its dependence on γ is considered implied. The state (5) is entangled for all values of γ > 0. Here, we discuss two different cases of MEMS, that is., 0 ≤ γ < 2 3 and with 2 3 ≤ γ < 1 . For comparison, the Werner state (or W-state) (Werner, 1989) is another class of mixed entangled states that are a mixture of a maximally entangled state and the maximally mixed state: | W 〉 = ( 1 − γ 4 ) I 2 ⊗ I 2 + γ | ϕ + 〉 〈 ϕ + | . It is entangled for γ > 1/3 and maximally entangled for γ = 1. When considering a specific level of mixture, the maximally entangled mixed state is typically much more entangled than the Werner state with an equivalent degree of mixture.

A quantum channel can be broadly classified into two categories: a memoryless channel and a memory channel. In the case of a memoryless channel, the environmental correlation time is smaller than the time between successive uses of the channel over two qubits. Conversely, in the case of a memory channel, the environmental correlation time is greater than the time between two consecutive uses of the channel. Mathematically, a quantum channel can be defined as a completely positive trace-preserving (CPTP) map between input and output states (density matrices), and the action of the channel (ϵ) on a quantum state ρ can be expressed as follows: ϵ ρ = ∑ i E i ρ E i † , (6) where E _i are the Kraus operators of the channel satisfying the completeness condition ∑ i E i E i † = 1 , and E i = P i 1 … i N A i . Here, P i 1 … i N is the joint probability distribution for a random sequence of operations applied to N qubits passing through quantum channels. Kraus operators are obtained by tracing the environmental degrees of freedom from the global unitary operation between the system and the environment. When P i 1 … . i N = P i 1 P i 2 … P i N , these operations are said to be independent, and the corresponding channel is called memoryless. However, for memory channels, P i 1 … i N = P i 1 P i 2 | i 1 … P i N | i N − 1 are correlated, and P i N | i N − 1 = ( 1 − μ ) P i N + μ δ i N , i N − 1 is the conditional probability with the memory parameter of the channel 0 ≤ μ ≤ 1 (Guo et al., 2017; Awasthi et al., 2022). The evolution of an initial state ρ under two successive uses of a memory channel can be written as follows (Yeo and Skeen, 2003; Sharma and Gerdt, 2020): ϵ ρ = 1 − μ ∑ i , j E i , j u ρ E i , j u † + μ ∑ k E k , k c ρ E k , k c † . (7)

It can be seen from the aforementioned expression that the action of the correlated noise is specified by the Kraus operator E k , k c with probability μ, and the action of the uncorrelated noise is specified by the Kraus operator E i , j u with probability (1 − μ).

ADC can be conceptualized as a simplified representation of the decay of the energy of an excited state in a two-level atom over time due to the emission of a photon through spontaneous decay. The ADC Kraus operators (Preskill, 1998) for a single qubit are given as E 0 u = 1 − p 0 0 1 , E 1 u = 0 0 p 0 , (8) where p = 1 − exp(−λt) is the probability of decay (0 ≤ p ≤ 1) of the qubit from the excited state to the ground state, which we call the noise parameter for ADC, and λ is the decay rate of the excited state. The Kraus operators governing the evolution of a two-qubit quantum system are given as E i j u = E i u ⊗ E j u , i , j = 0,1 . (9)

In contrast, the Kraus operators for the correlated part of the ADC are given as E 00 c = 1 − p 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , E 11 c = 0 0 0 0 0 0 0 0 0 0 0 0 p 0 0 0 , (10) where the value of p ranges between 0 and 1 (Yeo and Skeen, 2003). Substituting Eqs 5, 8–10 into Eq. 7, the density matrix of MEMS after evolution through correlated ADC is given as ρ M A = g 1 − μ 1 − p 2 + g μ 1 − p 0 0 1 2 γ 1 − μ 1 − p + 1 2 γ μ 1 − p 0 1 − 2 g μ + 1 − μ 1 − 2 g 1 − p + g p 1 − p 0 0 0 0 g 1 − μ 1 − p p 0 1 2 γ 1 − μ 1 − p + 1 2 γ μ 1 − p 0 0 1 − μ g p 2 + 1 − 2 g p + g + μ g p + g . (11)

3D plots of concurrence (C) vs. ADC parameter (p) and memory parameter (μ) for MEMS with γ = 0.5 and 0.9 are shown in Figures 1, 3, respectively. Two-dimensional projections of these plots in the (C − p) plane for different memory parameters are shown in Figures 2, 4.

According to Eq. 5, we have bifurcated the study of MEMS into two ranges of the g(γ) parameter: (i) when 0 ≤ γ < 2 3 and (ii) 2 3 ≤ γ < 1 . It is evident from Figures 1, 3 that MEMS undergoes ESD for both choices of γ parameter (γ = 0.5 & 0.9) in the presence of pure memoryless ADC (μ = 0). It can be easily verified from Eq. 5 that the higher the value of γ, the higher the entanglement in MEMS. Figures 1–4 show that as the value of the memory parameter (μ) increases from zero, MEMS contains a higher amount of entanglement for the same value of ADC parameter (p). As the value of μ increases, there is a delay in the sudden death of entanglement (Ficek and Tanaś, 2008). Moreover, ESD can be completely avoided for sufficiently high memory parameters for a given γ as MEMS undergoes asymptotic decay of entanglement (ADE). Further increase in μ increases the amount of entanglement for a given p.

PDC is a unital channel that describes the loss of quantum information without the loss of energy. The single-qubit Kraus operators for a memoryless PDC are given as P i σ i , where i = {0, 3}, P ₀ = 1 − p/2, P ₃ = p/2, and 0 ≤ p ≤ 1. For two successive uses of channels, no correlation is generated for memoryless channels. The uncorrelated or memoryless Kraus operators for two qubits (Datta et al., 2018; D’Arrigo et al., 2007) are given as E i , j u = P i P j σ i ⊗ σ j , i , j = 1,3 . (12)

In the presence of memory, two uses of channels generate some correlation, and the correlated Kraus operators (Datta et al., 2018; D’Arrigo et al., 2007) for two qubits are given as E k , k c = P K σ k ⊗ σ k , k = 0,3 , (13) where P ₀ = 1 − p/2, P ₃ = p/2. Substituting Eqs 5, 12, 13 into Eq. 7, the final density matrix of the MEMS can be expressed as ρ M P = g 0 0 1 2 γ − γ μ − 1 p − 2 p 0 1 − 2 g 0 0 0 0 0 0 1 2 γ − γ μ − 1 p − 2 p 0 0 g . (14)

3D plots of concurrence (C) vs. PDC parameter (p) and memory parameter (μ) for γ = 0.5 and 0.9 are shown in Figures 5, 7. Two-dimensional projections of these plots in the (C − p) plane for different memory parameters are shown in Figures 6, 8.

It is evident from Figures 5, 7 that MEMS undergoes ADE for both choices of γ parameter (γ = 0.5 & 0.9) in the presence of a pure memoryless ADC (μ = 0). As soon as the value of μ becomes non-zero, MEMS retains a non-zero entanglement even at p → 1. Figures 5–8 show that as the value of the memory parameter (μ) increases from zero, MEMS contains a higher amount of entanglement for a non-zero p-value. For a perfectly correlated channel (μ = 1), the entanglement of MEMS (for arbitrary γ values) freezes at the same value as the initial state for all values of p.

In Awasthi and Joshi (2023), we studied the effect of memory and memoryless channels on entanglement for different values of noise and MEMS parameters in the context of ADCs and PDCs. In previous work, a delay in the sudden death of entanglement in two different cases was observed. Motivated by our previous work, we explored our results for other channels on entanglement and quantum coherence. We again bifurcated our calculations into two cases based on the γ parameter of MEMS. Here, our main focus is on two noise channels, that is, depolarizing and bit-flip channels, and how they compare with the ADC and PDC channels with memory.

A depolarizing channel describes how the density matrix is dynamically replaced by the state I ₂/2, where I ₂ denotes the identity matrix. The Kraus operators for a single qubit (Macchiavello et al., 2004) are given as A i = P i σ i , (15) where P ₀ = 1 − p, P ₁ = P ₂ = P ₃ = p/3, and p = 1 − e ^−λt . The Kraus operators of the depolarizing channel without memory, where no correlation is generated by successive uses of channels, can be written as E i , j u = P i P j σ i ⊗ σ j . (16)

The case of two successive uses of a channel with partial memory that generates some correlation is given as E k , k c = P k σ k ⊗ σ k , (17) where i, j, k = {0, 1, 2, 3}.

Substituting Eqs 5, 16, 17 into Eq. 7, the final density matrix of MEMS can be expressed as ρ M D = g μ − 1 9 μ − 1 g 3 − 4 p 2 + 2 p 3 − 2 p 0 0 1 18 γ 9 − 8 μ − 1 p 2 p − 3 0 1 9 2 p − 3 g 8 μ − 1 p + 6 − 2 μ − 1 p − 3 0 0 0 0 2 9 p 3 − 6 g μ − 2 μ − 1 g 3 − 4 p + p 0 1 18 γ 9 − 8 μ − 1 p 2 p − 3 0 0 g μ − 1 9 μ − 1 g 3 − 4 p 2 + 2 p 3 − 2 p . (18)

3D plots of concurrence (C) vs. depolarizing channel parameter (p) and memory parameter (μ) for γ = 0.5 & 0.9 are shown in Figures 9, 11. Two-dimensional projections of these plots in the (C − p) plane for different memory parameters are shown in Figures 10, 12.

It is evident from Figures 9, 11 that MEMS undergoes ESD for both choices of γ parameter (γ = 0.5 & 0.9) in the presence of a pure memoryless depolarizing channel (μ = 0). In Figures 9–12, one can see that as μ increases from zero, MEMS contains a higher amount of entanglement for the same value of p. Furthermore, as the value of μ increases, there is a delay in the sudden death of entanglement. Moreover, ESD can be completely avoided for sufficiently high memory parameters for a given γ as MEMS undergoes ADE. Further increase in μ increases the amount of entanglement for a given p.