Bell–Clauser–Horne–Shimony–Holt Behavior Under Quantum Loss and Decoherence

Abstract

We present a detailed analysis of the effect of quantum loss and decoherence in the Bell-CHSH scenario. Adopting a device-independent approach, we study the change in the bipartite conditional probability distribution, i.e., the behavior of the realized nonlocal box pair when the elements of the entangled qubit pair subjected to independent noisy quantum channels modeled by completely positive maps. As the verification of Bell inequalities is crucial in device-independent quantum cryptography, our considerations are instructive from the perspective of quantum realizations of nonlocal box pairs. We find that the impact of quantum channels cannot be described by an equivalent classical noise channel.

1. Introduction

The violation of Bell inequalities is one of the most intriguing features of quantum mechanics [1]. Besides its fundamental and philosophical relevance, it has practical importance too, especially in quantum cryptography. The celebrated Ekert91 protocol [2], which is the first and one of the most important quantum key distribution protocols based on quantum entanglement, is based on the verification of the violation of the Clauser–Horne–Shimony–Holt (CHSH) Bell inequality [3].

As the violation of a Bell inequality can be verified solely by the statistics of the measurement choices and outputs—that is, the behavior of the “nonlocal boxes”, i.e., the bipartite conditional probability distribution of the measurements in a Bell-type scenario—the Ekert91 protocol is also device-independent in that it self-certifies the presence of the required entangled resource. Indeed, device-independent quantum cryptography [4,5,6] is mainly based on self-certification based on the verification of Bell-type inequalities [7]. Without any assumptions on the particular quantum states and measurements used for the realization of the protocol, the security is based on the presence of nonlocal correlations in the behavior of nonlocal boxes. The presence of the required nonlocality is verified by testing Bell inequality violations in the behavior of the boxes in the course of the protocol.

The behavior of the boxes in Bell-type setups falls into three categories [1,8,9]. Local behaviors do not violate Bell’s inequalities; these can be reproduced using pre-shared randomness. In the linear space of the conditional probabilities, these form the local polytope, whose nontrivial facets correspond to Bell’s inequalities. Quantum behaviors are those that can be realized using a quantum setup, with an entangled resource and measurements. These form a broader closed convex set. Finally, no-signaling behaviors are even more general; their defining property is that they cannot be used for communication between the parties. They also form a polytope which contains the set of quantum behaviors. The nontrivial facets of the polytope are defined by the no-signaling conditions.

Whenever an entanglement-based or device-independent quantum cryptography protocol is realized, it will be subject to imperfections, such as noise, loss, and detector imperfections. Detection loopholes have always been the main issues that arise with practical Bell-inequality-violation testing. Meanwhile, detector inefficiencies can hide nonlocal correlations, turning the behavior into a local one [8,10]. Loss and decoherence, on the other hand, can decrease the entanglement of the entangled resource, which is required for quantum nonlocality.

As also summarized in the review by Brunner et al. [1], the tolerance of nonlocal correlations to the addition of various types of noise can be considered an operational measure of nonlocal correlation. This aspect has been extensively studied for different noise types, including white noise [11,12], local noise [13,14], and detection inefficiencies [15,16].

The quantum bit error rate (QBER), which is a key feature of quantum cryptographic protocols, depends on Bell inequality violation. The effect of the atmosphere, detector imperfections, and other relevant factors is extensively studied in the literature on optical satellite quantum cryptography [17,18,19], based on particular physical models [20].

From a theoretical point of view, the effect of losses and noise is modeled most generally by quantum channels, i.e., completely positive trace preserving (CPTP) maps. The effect of a local quantum channel on bipartite entanglement has been studied in detail before. Entanglement breaking channels are those CPTP maps which eliminate entanglement when applied to one part of an entangled pair; they have been characterized in a mathematically elegant fashion [21,22]. In the present work, we approach the question from the Bell inequalities’ point of view: we study the effect of the CPTP maps on nonlocal behaviors. The use of CPTP maps is not uncommon in the literature of device-independent quantum cryptography (see, e.g., [23]).

The particular effect of CPTP maps on nonclassical correlations, which is the subject of the present contribution, is less broadly discussed in the literature. Rastegin [24] considers information–theoretic Bell inequalities in a dephasing environment, whereas Ma et al. [25] study the behavior of qubit Bell inequality violations under qubit amplitude damping. These contributions focus on the Bell inequality violations, and not the detailed structural properties of the correlations. D. Park et al. [26] studied the Ekert protocol by calculating concurrence as the measure of the entanglement and Bell inequality violation. For the purpose of the investigation of the Ekert91 protocol, it is sufficient to consider Bell inequality violations. Their considerations could be repeated for the different channels and maps discussed here, when it comes to the analysis of that particular protocol. In the present contribution, however, we consider the structural changes in nonlocal behavior in more detail.

The scope of the present study is limited to the case of the CHSH scenario, involving a pair of quantum bits; a two-input–two-output nonlocal behavior; while this bears practical relevance for the Ekert91 protocol, our motivation is to gain a fundamental understanding of the effect of quantum channels to nonlocal behaviors. One is tempted to assume that all imperfections can be studied by adding local classical noise to the measurement outputs, as is the case when modeling detector inefficiencies. Our main observation will be that this is not always the case: quantum noise destroys nonlocal correlations differently from detector imperfections. Methodologically, our analysis adopts a picture based on the theory of classical discrete memoryless channels, providing an alternative view of nonlocal behaviors.

This paper is organized as follows: In Section 2, we introduce the concepts and methods used in the rest of this work and give more details on certain aspects that are already known from the literature. Section 3 summarizes the features and descriptions of the considered qubit channels. Section 4 contains the results of calculation of the nonlocal behaviors, assuming different imperfections and the analysis of the Bell-CHSH violation. Section 5 contains our main results: we analyze the extent to which the effect of the quantum channels is (in)equivalent to local classical noise. Finally, in Section 6, conclusions are drawn, and possible ways of continuing this research are outlined.

2. Background, Notation

In this Section, we summarize the notation and the concepts used throughout the rest of the paper.

2.1. Discrete Memoryless Channels

A discrete memoryless channel without feedback, which will be referred to as a “channel” in what follows, is specified by the triple $(X,A,p(a\left|x\right))$, where X is a finite set of input symbols, $x\in X$, A is a finite set of output symbols, $a\in A$, and $P\left(a\right|x)$ is the conditional probability that the channel outputs, a, when the input is x. The channel is often considered to connect two parties; somewhat unusually, we shall call the sender Xenia, and the receiver Alice, in the present case.

Having an input probability distribution, $p_x^{\mathrm{in}}$, on Xenia’s side, the output probability distribution $p_a^{\mathrm{out}}$ at Alice’s side is calculated as follows:

$$p_a^{\mathrm{out}}=\sum _x{p}_x^{\mathrm{in}}P\left(a\right|x).$$

(1)

In information theory, it is common to define the channel transfer matrix

$$T_{x,a}=P\left(a\right|x).$$

(2)

Considering the input and output probability distributions as row vectors, ${\mathbf{p}}^{\mathrm{in}}$ and ${\mathbf{p}}^{\mathrm{out}}$, respectively, we have

$${\mathbf{p}}^{\mathrm{out}}={\mathbf{p}}^{\mathrm{in}}T.$$

(3)

For a matrix T to represent a channel, it should be a stochastic matrix: its elements have to be between 0 and 1, and the sum of each row should be 1.

Assume now that Alice sends her received symbols through another channel, $T^{\prime }$, to Ronnie. (Thus, $T^{\prime }$ represents a channel $(A,R,P(r\left|a\right))$ in the sense of (2).) It is straightforward to see that the matrix of the so-arising channel between Xenia and Ronnie will be $T{T}^{\prime }$. When composing channels, the subsequent channel’s matrix multiplies the preceding one from the right.

2.2. Bipartite Channels

In addition to the sender, Xenia, and the receiver, Alice, consider now another sender, Yvette, and another receiver, Bob. When all these parties are considered, the most general channel between the Xenia–Yvette pair and the Alice–Bob pair is a channel, $(X\times Y,A\times B,P(ab\left|xy\right))$, where the ordered pairs in the Cartesian product are denoted as strings, e.g., $(x,y)=xy$. The elements of the respective channel transfer matrix will be $T_{xy,ab}$; thus, they will be in the direct (tensor) product space of the spaces of the possible (Xenia to Alice) and (Yvette to Bob) matrices.

Indeed, if the channel $T^{(X\to A)}$ between Xenia and Alice is completely independent of the channel $T^{(Y\to B)}$ between Yvette and Bob, for the joint channel, we have

$$T_{xy,ab}^{(XY\to AB)}=T_{x,a}^{(X\to A)}{T}_{y,b}^{(Y\to B)}$$

(4)

because of the independence of the conditional probabilities; thus,

$$T^{(XY\to AB)}=T^{(X\to A)}\otimes T^{(Y\to B)}.$$

(5)

The channels can, however, be dependent. In this case, the channel matrix, $T_{xy,ab}^{(XY\to AB)}$, is not in the product form of Equation (4), but it is a generic stochastic matrix, which obeys the constraints

$$\begin{array}{c}\hfill \forall a,b,x,y0\le T_{xy,ab}^{(XY\to AB)}\le 1\\ \hfill \forall a,b\sum _{x,y}{T}_{xy,ab}^{(XY\to AB)}=1.\end{array}$$

(6)

2.3. Two-Party Bell Experiments as Channels

The bipartite joint probabilities, $P\left(ab\right|xy)$, are crucial objects for the study of quantum nonlocality and no-signaling theories. Consider a bipartite Bell-type experiment. Using the previous metaphor of actors, Xenia and Alice are now at the same location, whereas Yvette and Bob are together at another, separated place. So, the parties will be referred to as Alice and Bob. Alice and Bob share a pair of quantum systems in an entangled state.

Alice (and Xenia) can choose from a set of quantum measurements, X. Upon carrying out the selected measurement, x, they obtain a result $a\in A$. Similarly, Bob (and Yvette) choose a measurement, $y\in Y$, and obtain a result, $b\in B$. Given the set of measurements and the entangled state, according to the rules of quantum mechanics, the conditional probability distribution, $P\left(ab\right|xy)$, can be calculated, which is commonly termed as “behavior” in the present context. A behavior can exhibit so-called nonclassical correlations between Alice and Bob. The peculiarity of such correlations lies in the fact that Alice + Xenia are separated from Bob + Yvette, so their channels do not interact. Yet, there can be correlations that cannot be explained, even assuming the pre-shared randomness (i.e., the keys) between the two locations. The Bell inequalities are inequalities that can be expressed in terms of $P\left(ab\right|xy)$ and their violation indicates this unexplainability.

This scenario can be extended to study general behaviors, $P\left(ab\right|xy)$. The no-signaling ones, that is, those that do not facilitate communication between Alice+Xenia and Bob+Yvette, are of special interest. Formally, the no-signaling constraints in our notation can be written as

$$\begin{array}{cc}\hfill \sum _b{T}_{xy,ab}^{(XY\to AB)}& =\sum _b{T}_{x{y}^{\prime },ab}^{(XY\to AB)} \mathrm{for} \mathrm{all} x,a,y,y^{\prime }\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill \sum _a{T}_{xy,ab}^{(XY\to AB)}& =\sum _a{T}_{x^{\prime }y,ab}^{(XY\to AB)} \mathrm{for} \mathrm{all} y,b,x,x^{\prime }.\hfill \end{array}$$

(8)

These constraints imply that the local marginal probabilities of Alice can be defined unambiguously as

$$T_{x,a}^{(X\to A)}:=T_{xy,a}^{(XY\to AB)}=\sum _b{T}_{xy,ab}^{(XY\to AB)},$$

(9)

as these values are independent of Bob’s measurement setting, y (and the other way around). These form a polytope in the vector space of the behaviors, $T^{(XY\to AB)}$.

Note that, from the classical channel’s point of view, in the Bell scenario, there are two sender–receiver pairs Xenia–Alice and Yvette–Bob. The members of each of these pairs are, however, located at the same place. So, the correlations are not between the “sender” and the “receiver” in the channel sense, but between the two parts of the channel. Yet, in some cases, it is useful to view them from the channel’s perspective, writing the behavior $P\left(ab\right|xy)$ in the form of a channel matrix, $T^{(XY\to AB)}$.

As an example, consider the celebrated Clauser–Horne–Shimony–Holt (CHSH) experiment, which will be the subject of the present study. Alice and Bob share a pair of quantum bits in a maximally entangled state; let it be the Bell state:

$$|{\mathrm{\Phi }}^{(+)}\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(|00\rangle +|11\rangle \right),$$

(10)

where we have fixed a product basis, $|i,j\rangle ,(i,j)\in {\{0,1\}}^2$, on the space of the two qubits; the first index is for Alice’s side.

Alice’s measurement bases are:

$$\begin{array}{c}\hfill {|a=0\rangle }_{x=0}=|0\rangle ,{|a=1\rangle }_{x=0}=|1\rangle ,\\ \hfill {|a=0\rangle }_{x=1}={\displaystyle \frac{1}{\sqrt{2}}}\left(|0\rangle +|1\rangle \right),{|a=1\rangle }_{x=1}={\displaystyle \frac{1}{\sqrt{2}}}\left(|0\rangle -|1\rangle \right),\end{array}$$

(11)

whereas Bob’s ones are:

$$\begin{array}{c}\hfill {|b=0\rangle }_{y=0}=cos\vartheta |0\rangle +sin\vartheta |1\rangle ,{|b=1\rangle }_{y=0}=-sin\vartheta |0\rangle +cos\vartheta |1\rangle ,\\ \hfill {|b=0\rangle }_{y=1}=cos\vartheta |0\rangle -sin\vartheta |1\rangle ,{|b=1\rangle }_{y=1}=sin\vartheta |0\rangle +cos\vartheta |1\rangle ,\end{array}$$

(12)

where $\vartheta =\pi /8$. To calculate the conditional probabilities $P\left(ab\right|xy)=T_{xy,ab}^{(XY\to AB)}$, it is convenient to introduce the density operator of the entangled resource,

$${\varrho }^{\left(|{\mathrm{\Phi }}^{(+)}\rangle \right)}=|{\mathrm{\Phi }}^{(+)}\rangle \langle {\mathrm{\Phi }}^{(+)}|,$$

(13)

and then, we have

$$T_{xy,ab}^{(XY\to AB)}={\langle a|}_x{\langle b|}_y{\varrho }^{\left(|{\mathrm{\Phi }}^{(+)}\rangle \right)}{|a\rangle }_x{|b\rangle }_y.$$

(14)

Carrying out the calculations, this leads to the following matrix:

$$T^{(XY\to AB)}=\left(\begin{array}{cccc}\alpha & \beta & \beta & \alpha \\ \alpha & \beta & \beta & \alpha \\ \alpha & \beta & \beta & \alpha \\ \beta & \alpha & \alpha & \beta \end{array}\right),$$

(15)

where $\alpha ={\displaystyle \frac{2+\sqrt{2}}{8}}$ and $\beta ={\displaystyle \frac{2-\sqrt{2}}{8}}$.

Concerning nonlocality, the famous Clauser–Horne–Shimony–Holt inequality will read

$$\begin{array}{c}\hfill T_{00,00}^{(XY\to AB)}+T_{00,11}^{(XY\to AB)}-T_{00,01}^{(XY\to AB)}-T_{00,10}^{(XY\to AB)}\\ \hfill +T_{01,00}^{(XY\to AB)}+T_{01,11}^{(XY\to AB)}-T_{01,01}^{(XY\to AB)}-T_{01,10}^{(XY\to AB)}\\ \hfill +T_{10,00}^{(XY\to AB)}+T_{10,11}^{(XY\to AB)}-T_{10,01}^{(XY\to AB)}-T_{10,10}^{(XY\to AB)}\\ \hfill -T_{11,00}^{(XY\to AB)}-T_{11,11}^{(XY\to AB)}+T_{11,01}^{(XY\to AB)}+T_{11,10}^{(XY\to AB)}\le 2.\end{array}$$

(16)

For our choice of the basis, the left-hand side will be approx. $4{cos}^2(\pi /8)=2+\sqrt{2}\approx 3.41$, which violates the inequality. This is the highest violation which is possible with a quantum setup.

2.4. Detector Imperfections

One of the most frequently studied issues with Bell-type experiments is that of detector imperfections. An imperfect detector on Alice’s side can be modeled as follows: suppose Alice has the accurate result of the measurement, but she could only communicate with Ronnie through a noisy channel, $T^{(A\to R)}$. Similarly, on Bob’s side, the access to the measurement results of Bob is only available for Sam through $T^{(B\to S)}$. As the detectors are at different locations, they are independent; thus, we have

$$T^{(XY\to RS)}=T^{(XY\to AB)}{T}^{(AB\to RS)}=T^{(XY\to RS)}\left(T^{(A\to R)}\otimes T^{(B\to S)}\right),$$

(17)

and the signatures of nonlocality, such as the no-signaling conditions in Equation (7) or the violations of the Bell inequalities, are to be evaluated on the so-arising behavior, $P\left(rs\right|xz)$.

The observable behavior, $P\left(rs\right|xz)$, will depend on the parameters of $T^{(AB\to RS)}$, such as detector efficiencies. Ref. [8] presents calculations that show which the critical values are, for which the no-signaling polytope of the given setup is shrunk by $T^{(AB\to RS)}$ to the local polytope—the set of behaviors not violating any Bell-type inequalities. This implies that no device-independent cryptographic protocol can be implemented with detectors worse than the threshold. This is, however, a rough picture which yields too strict a threshold. A more accurate approach is to consider the convex set of quantum correlations. The latter can be characterized by a hierarchy of semidefinite programs and it is contained inside the no-signaling polytope. The authors of [10] consider the effect of detector imperfections on this convex set.

In the present paper, our intention is to consider the effect of noisy quantum channels acting on quantum systems in a quantum Bell experiment in an analogous fashion. On the one hand, we calculate threshold values of quantum channel parameters for the violation of Bell-CHSH inequalities. More importantly, we analyze whether the presence of quantum loss and noise can be treated in a similar fashion as detector inefficiencies are treated using Equation (17).

3. Qubit Quantum Channels

Assume now that the two quantum bits do not arrive with Alice and Bob in the maximally entangled state, ${\varrho }^{\left(|{\mathrm{\Phi }}^{(+)}\rangle \right)}$, but they undergo some unwanted change on their way. For instance, optical qubits are subjected to loss or decoherence when traveling through an optical fiber. In quantum mechanics, such loss is modeled by completely positive trace preserving (CPTP) maps, i.e., quantum channels. After a brief summary of the description of the CPTP maps we will be using, we turn our attention to our main point: their effect on the behavior of the CHSH experiment.

We use the Choi–Jamiołkowski representation of CPTP maps of quantum channels to describe (quantum) loss and imperfections. A detailed introduction can be found, e.g., in [27]. Here, we just recapitulate the way of calculating the output density matrix from the input. Let ${\mathcal{H}}_{\mathrm{in}}$ and ${\mathcal{H}}_{\mathrm{out}}$ be the complex, finite-dimensional Hilbert spaces of the input and the output of the quantum channel; these are two dimensional in the case of qubits. Let us fix a basis in both spaces. A CPTP map $\mathcal{E}$ can be described by its Choi matrix: a Hermitian positive semidefinite matrix, $X_{\mathcal{E}}$, representing a map ${\mathcal{H}}_{\mathrm{in}}\otimes {\mathcal{H}}_{\mathrm{out}}\mapsto {\mathcal{H}}_{\mathrm{in}}\otimes {\mathcal{H}}_{\mathrm{out}}$ on the product of the two fixed bases. Then, the output of the quantum channel when its input is $\varrho$ is calculated as

$$\mathcal{E}\left(\varrho \right)={\mathrm{Tr}}_{\mathrm{in}}\left(\left({\varrho }^{\top }\otimes \mathbb{I}\right)X_{\mathcal{E}}\right).$$

(18)

Any positive semidefinite Hermitian matrix with the property

$${\mathrm{Tr}}_{\mathrm{out}}{X}_{\mathcal{E}}={\mathbb{I}}_{\mathrm{in}}$$

(19)

represents a completely positive map; thus, our analysis could be carried out for generic channels using a parametrization of these. This would, however, require to use a large number of parameters; hence, we will consider two particular types of qubit channels (as our investigation will be restricted to pairs of quantum bits). Each have a parameter $p\in [0,1]$; for $p=0$, they coincide with the identity channel, whereas for $p=1$, they change state the most. The considered channel types are presented here.

3.1. Phase-Damping Channel

This is also called a dephasing channel, and acts as follows:

$$\mathcal{E}\left(\left(\begin{array}{cc}{\varrho }_{0,0}& {\varrho }_{0,1}\\ {\varrho }_{1,0}& {\varrho }_{1,1}\end{array}\right)\right)=\left(\begin{array}{cc}{\varrho }_{0,0}& (1-p){\varrho }_{0,1}\\ (1-p){\varrho }_{1,0}& {\varrho }_{1,1}\end{array}\right).$$

(20)

This channel depletes the coherences (off-diagonal elements) of the density matrix in the given basis for $p=1$, keeping the diagonals only. Hence, it is a unital channel: it keeps the completely mixed state $(1/2)\mathbb{I}$ invariant. The Choi matrix of the phase-damping channel reads

$$X_{\mathrm{PD}}=\left(\begin{array}{cccc}1& 0& 0& 1-p\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1-p& 0& 0& 1\end{array}\right).$$

(21)

3.2. Amplitude-Damping Channel

This acts as

$$\mathcal{E}\left(\left(\begin{array}{cc}{\varrho }_{0,0}& {\varrho }_{0,1}\\ {\varrho }_{1,0}& {\varrho }_{1,1}\end{array}\right)\right)=\left(\begin{array}{cc}{\varrho }_{0,0}+p{\varrho }_{1,1}& \sqrt{1-p}{\varrho }_{0,1}\\ \sqrt{1-p}{\varrho }_{1,0}& (1-p){\varrho }_{1,1}\end{array}\right).$$

(22)

It is often used to model exponential decay. It is not unital, as it modifies the diagonal of the matrix too. The Choi representation of the amplitude-damping channel reads

$$X_{\mathrm{AD}}=\left(\begin{array}{cccc}1& 0& 0& \sqrt{1-p}\\ 0& 0& 0& 0\\ 0& 0& p& 0\\ \sqrt{1-p}& 0& 0& 1-p\end{array}\right).$$

(23)

4. CHSH Experiment with Quantum Imperfections

Here, we come to the main part of our investigation. Consider the aforementioned CHSH setup in which Alice and Bob choose between two measurements each, and they perform these measurements on a pair of quantum bits. Instead of having a perfect entangled resource, we assume that each of the qubits goes through a quantum channel, which can model, e.g., the noisy transmission of the quantum bit from the source of entangled quantum bits to the parties.

We assume the channel acting on Alice’s qubit and the channel acting on Bob’s qubit to be independent. That is, we neglect potential correlated imperfections at the locus of the generation of the qubit pair. It is plausible to assume that the loss and imperfections during the travel of each qubit to the given party, or its storage at the parties, can be described with a pair of independent channels.

Independent channel pairs can be described with the direct product of their Choi matrices, as described in [27] in more detail. Thus, the scenario we consider is the following. We assume that the source generates the entangled state ${\mathrm{\Phi }}^{(+)}$ of Equation (10). Then, the qubits are passed to Alice and Bob, going through the independent quantum channels ${\mathcal{E}}_{\mathrm{Alice}}$ and ${\mathcal{E}}_{\mathrm{Bob}}$. Then, Alice and Bob carry out the measurements like in the original CHSH scenario. The respective behavior is calculated as in the ideal case using Equation (14), but instead of ${\varrho }^{\left(|{\mathrm{\Phi }}^{(+)}\rangle \right)}$, we use the density matrix

$${\varrho }^{\mathrm{ent}}=\left({\mathcal{E}}_{\mathrm{Alice}}\otimes {\mathcal{E}}_{\mathrm{Bob}}\right)\left({\varrho }^{\left(|{\mathrm{\Phi }}^{(+)}\rangle \right)}\right),$$

(24)

where the action of ${\mathcal{E}}_{\mathrm{Alice}}\otimes {\mathcal{E}}_{\mathrm{Bob}}$ is calculated using Equation (18), using the product of the respective qubit Choi matrices.

Let us now perform the calculations for the selected particular channel combinations. For the rest of the paper, for the sake of notational convenience, we introduce the following notations:

$$\begin{array}{c}\hfill x=\sqrt{(1-p_1)(1-p_2)},\end{array}$$

(25)

$$\begin{array}{c}\hfill c=cos(\pi /8)=\sqrt{2+\sqrt{2}}/2,\end{array}$$

(26)

$$\begin{array}{c}\hfill s=sin(\pi /8)=\sqrt{2-\sqrt{2}}/2,\end{array}$$

(27)

$$\begin{array}{c}\hfill cs=\sqrt{2}/4,\end{array}$$

(28)

thus, $c^2-s^2=2cs$. In the case of x, the parameters $p_1$ and $p_2$ stand for the parameter of the channel on Alice’s and Bob’s sides, respectively, regardless of whether the channel is amplitude- or phase-damping. As we will be in the need of them later, we will specify the right eigenvalues of the channel matrices $T^{(XY\to RS)}$, too. It is important to note that all the $T^{(XY\to RS)}$ matrices describe a no-signaling behavior: they obey the no-signaling conditions in Equation (7) with the respective inputs and outputs.

4.1. A Pair of Phase-Damping Channels

Let us consider a pair of independent phase-damping channels with parameters $p_1$ and $p_2$, respectively, acting on Alice’s and Bob’s sides of the entangled pair. To obtain the density matrix of the entangled resource

$${\varrho }^{\left(|{\mathrm{\Phi }}^{(+)}\rangle \right)}={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}1& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 1\end{array}\right)$$

(29)

after the action of the channel pair, we evaluate Equation (24), with the use of the Choi matrix in Equation (21), resulting in

$${\varrho }^{\mathrm{ent}, \mathrm{PD},\mathrm{PD}}={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}1& 0& 0& (1-p_1)(1-p_2)\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ (1-p_1)(1-p_2)& 0& 0& 1\end{array}\right).$$

(30)

Using this density matrix like in Equation (14), we obtain the respective nonideal behavior, which reads

$$T^{(XY\to RS),\mathrm{PD}, \mathrm{PD}}={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}{c}^2& s^2& s^2& c^2\\ c^2& s^2& s^2& c^2\\ 1/2+cs{x}^2& 1/2-cs{x}^2& 1/2-cs{x}^2& 1/2+cs{x}^2\\ 1/2-cs{x}^2& 1/2+cs{x}^2& 1/2+cs{x}^2& 1/2-cs{x}^2\end{array}\right).$$

(31)

Its eigenvalues and their multiplicities are:

$$\left(\left(-{\displaystyle \frac{\sqrt{2}(-p_2+p_1(p_2-1)+1)}{4}},1,0\right),\left(1,1,2\right)\right).$$

(32)

The left-hand side of the CHSH inequality in Equation (16) is plotted in the left part of Figure 1.

4.2. A Pair of Amplitude-Damping Channels

Next, let us discuss the case of a pair of amplitude dampings with parameters $p_1$ and $p_2$ on each side, respectively. We use the same consideration as in the previous Section, now with the Choi matrix of Equation (23). This results in the following nonideal entangled resource:

$${\varrho }^{\mathrm{ent}, \mathrm{AD},\mathrm{AD}}={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}1+p_1{p}_2& 0& 0& \sqrt{(1-p_1)(1-p_2)}\\ 0& p_1(1-p_2)& 0& 0\\ 0& 0& (1-p_1)p_2& 0\\ \sqrt{(1-p_1)(1-p_2)}& 0& 0& (1-p_1)(1-p_2)\end{array}\right).$$

(33)

The respective behavior will be

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & T^{(XY\to RS),\mathrm{AD},\mathrm{AD}}=\hfill \\ \hfill & {\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}{c}^2+2cs{p}_1{p}_2+s^2{p}_1& s^2-2cs{p}_1{p}_2+c^2{p}_1& c^2-c^2{p}_1-2cs{x}^2& s^2-s^2{p}_1+2cs{x}^2\\ c^2+2cs{p}_1{p}_2+s^2{p}_1& s^2-2cs{p}_1{p}_2+c^2{p}_1& c^2-c^2{p}_1-2cs{x}^2& s^2-s^2{p}_1+2cs{x}^2\\ 1/2+cs(p_2+x)& 1/2-cs(p_2+x)& 1/2+cs(p_2-x)& 1/2-cs(p_2-x)\\ 1/2+cs(p_2-x)& 1/2-cs(p_2-x)& 1/2+cs(p_2+x)& 1/2-cs(p_2+x)\end{array}\right).\hfill \end{array}\end{array}$$

(34)

The eigenvalues and their multiplicities are

$$\left(\left({\displaystyle \frac{p_1}{2}},-{\displaystyle \frac{\sqrt{2(1-p_1)(1-p_2)}}{4}},1,0\right),\left(1,1,1,1\right)\right).$$

(35)

The left-hand side of the CHSH inequality in Equation (16) is plotted in the right part of Figure 1.

4.3. Different Channels on Each Side

Now, consider the case in which, on Alice’s side, the qubit is affected by an amplitude-damping channel, while on Bob’s side, the qubit is affected by a phase-damping channel, or vice versa. Note that, because of the basis dependence of amplitude damping, the two cases are not equivalent.

Let $p_1$ and $p_2$ be the parameters of the channels on Alice’s and Bob’s side, respectively, regardless of whether it is the phase- or amplitude-damping channel. The nonideal entangled resources will be

$${\varrho }^{\mathrm{ent}, \mathrm{AD},\mathrm{PD}}={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}1& 0& 0& \sqrt{1-p_2}\sqrt{(1-p_1)(1-p_2)}\\ 0& p_1& 0& 0\\ 0& 0& 0& 0\\ \sqrt{1-p_2}\sqrt{(1-p_1)(1-p_2)}& 0& 0& 1-p_1\end{array}\right),$$

(36)

and

$${\varrho }^{\mathrm{ent}, \mathrm{PD},\mathrm{AD}}={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}1& 0& 0& \sqrt{1-p_1}\sqrt{(1-p_1)(1-p_2)}\\ 0& 0& 0& 0\\ 0& 0& p_2& 0\\ \sqrt{1-p_1}\sqrt{(1-p_1)(1-p_2)}& 0& 0& 1-p_2\end{array}\right)$$

(37)

resulting in the following behaviors:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & T^{(XY\to RS),\mathrm{AD}, \mathrm{PD}}=\hfill \\ \hfill & {\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}{c}^2+s^2{p}_1& s^2+c^2{p}_1& s^2-s^2{p}_1& c^2-c^2{p}_1\\ c^2+s^2{p}_1& s^2+c^2{p}_1& s^2-s^2{p}_1& c^2-c^2{p}_1\\ 1/2+cs\sqrt{1-p_2}x& 1/2-cs\sqrt{1-p_2}x& 1/2-cs\sqrt{1-p_2}x& 1/2+cs\sqrt{1-p_2}x\\ 1/2-cs\sqrt{1-p_2}x& 1/2+cs\sqrt{1-p_2}x& 1/2+cs\sqrt{1-p_2}x& 1/2-cs\sqrt{1-p_2}x\end{array}\right),\hfill \end{array}\end{array}$$

(38)

and

$$T^{(XY\to RS),\mathrm{PD}, \mathrm{AD}}={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}{S}_{11}& S_{12}& S_{13}& S_{14}\\ S_{21}& S_{22}& S_{23}& S_{24}\\ S_{31}& S_{32}& S_{33}& S_{34}\\ S_{41}& S_{42}& S_{43}& S_{44}\end{array}\right)$$

(39)

where

$$\begin{array}{l}{S}_{11}=c^2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{S}_{21}=c^2\hfill \end{array}$$

(40)

$$\begin{array}{l}{S}_{12}=s^2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{S}_{22}=s^2\hfill \end{array}$$

(41)

$$\begin{array}{l}{S}_{13}=s^2+2cs{p}_2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{S}_{23}=s^2+2cs{p}_2\end{array}$$

(42)

$$\begin{array}{l}{S}_{14}=c^2-2cs{p}_2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{S}_{24}=c^2-2cs{p}_2\end{array}$$

(43)

$$\begin{array}{ll}{S}_{31}=1/2+cs(p_2+\sqrt{1-p_1}x)& \hspace{1em}\hspace{1em}\hspace{1em}{S}_{41}=1/2+cs(p_2-\sqrt{1-p_1}x)\end{array}$$

(44)

$$\begin{array}{ll}{S}_{32}=1/2-cs(p_2+\sqrt{1-p_1}x)& \hspace{1em}\hspace{1em}\hspace{1em}{S}_{42}=1/2-cs(p_2-\sqrt{1-p_1}x)\end{array}$$

(45)

$$\begin{array}{ll}{S}_{33}=1/2+cs(p_2-\sqrt{1-p_1}x)& \hspace{1em}\hspace{1em}\hspace{1em}{S}_{43}=1/2+cs(p_2+\sqrt{1-p_1}x)\end{array}$$

(46)

$$\begin{array}{ll}{S}_{34}=1/2-cs(p_2-\sqrt{1-p_1}x)& \hspace{1em}\hspace{1em}\hspace{1em}{S}_{44}=1/2-cs(p_2+\sqrt{1-p_1}x).\end{array}$$

(47)

The eigenvalues and their multiplicities of $T^{(XY\to RS),\mathrm{AD}, \mathrm{PD}}$ are:

$$\left(\left({\displaystyle \frac{\sqrt{1-p_1}(p_2-1)}{2\sqrt{2}}},{\displaystyle \frac{p_1}{2}},1,0\right),(1,1,1,1)\right),$$

(48)

while those of $T^{(XY\to RS),\mathrm{PD}, \mathrm{AD}}$ read:

$$\left(\left({\displaystyle \frac{(p_1-1)\sqrt{1-p_2}}{2\sqrt{2}}},1,0\right),(1,1,2)\right).$$

(49)

The left-hand side of the CHSH inequality in Equation (16) for these cases is plotted in Figure 2.

5. Decomposability of Behaviors

The question naturally arises whether the impact of the quantum channels, as modeled by the CPTP maps, can be replaced on the behavior’s level by the composition of the ideal behavior with an effective noise channel. In other words, we ask whether the behavior $T^{(XY\to RS)}$ obtained in the CHSH experiment—performing the original ideal measurement after the entangled state—was subjected to the effect of a pair of quantum noise channels, and whether it can be written as a composition of the ideal $T^{(XY\to AB)}$ given in Equation (15) and an effective classical noise channel $T^{(AB\to RS)}$ as follows:

$$T^{(XY\to RS)}=T^{(XY\to AB)}{T}^{(AB\to RS)}.$$

(50)

This seems plausible, and because of the CPTP maps are independent, one may even expect $T^{(AB\to RS)}$ to be in a product form, as in the case of modeling detector inefficiencies in Equation (17). Somewhat surprisingly, at least in case of the studied CPTP map pairs, it is not the case.

Mathematically, we are searching for $T^{(AB\to RS)}$, obeying Equation (50), while the other two matrices are fixed. In addition, $T^{(AB\to RS)}$ has to be a stochastic matrix as it represents a channel. Let us postpone, for the time being, the check of the stochasticity and investigate other necessary conditions for any $T^{(AB\to RS)}$ obeying Equation (50) to exist. The problem of finding $T^{(AB\to RS)}$ is a matrix inversion. However, $T^{(XY\to AB)}$ is singular, and so is $T^{(XY\to RS)}$, at least for the studied CPTP map pairs. Hence, a necessary condition for the desired $T^{(AB\to RS)}$ to exist is that the left null space of the “ideal” channel, $T^{(XY\to AB)}$, is contained in the left null space of the “observed” channel, $T^{(XY\to RS)}$.

The eigenvalues of $T^{(XY\to AB)}$ are $1,\beta -\alpha ,0,0$ and the corresponding right eigenvectors read:

$$\left(\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right),\left(\begin{array}{c}-\alpha \\ -\alpha \\ -\alpha \\ 2\alpha +\beta \end{array}\right),\left(\begin{array}{c}1\\ 0\\ 0\\ -1\end{array}\right),\left(\begin{array}{c}0\\ 1\\ -1\\ 0\end{array}\right),$$

(51)

where the parameters $\alpha =c^2/2$ and $\beta =s^2/2$ are the same as in Equation (15). The left null space is orthogonal to those right eigenvectors that correspond to nonzero eigenvalues. Hence, this is a rank-2 matrix.

Let us now briefly consider the marginals defined in Equation (9); these are the behaviors $T^{(X\to R)}$ and $T^{(Y\to S)}$, observable on each individual side. Note that these could be obtained by applying the quantum channel on the given side to the respective part of the entangled resource, and then the measurement results can be calculated locally. In the case of the Bell-CHSH situation, the entangled resource is a Bell state; thus, the subsystems are in a completely mixed state. As these local behaviors depend on the channel on the given side, they can be calculated for the two channel types, regardless of the channel on the other side. The phase-damping channel is unital, i.e., it does not alter the completely mixed state. Thus, it does not affect the local behavior; it remains

$$T^{\mathrm{PD}, \mathrm{A}}=T^{\mathrm{PD}, \mathrm{B}}={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right).$$

(52)

In the case of amplitude damping with parameter p, we obtain

$$T^{\mathrm{AD}, \mathrm{A}}={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1+p& 1-p\\ 1& 1\end{array}\right),T^{\mathrm{AD}, \mathrm{B}}={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1+2csp& 1-2csp\\ 1+2csp& 1-2csp\end{array}\right)$$

(53)

on Alice’s and Bob’s sides, respectively. (Recall that the measurement bases differ on the two sides.) It is interesting to note that, on Bob’s side, we can decompose the local behavior $T^{\mathrm{AD}, \mathrm{B}}$ into the product of the ideal local behavior and a noise channel:

$${\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right){\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1+2csp& 1-2csp\\ 1+2csp& 1-2csp\end{array}\right)={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1+2csp& 1-2csp\\ 1+2csp& 1-2csp\end{array}\right).$$

(54)

This is, however, not possible on Alice’s side; this is because, while the ideal behavior is rank-1, the noisy behavior is a full-rank matrix.

Returning now to the question of joint behaviors, consider first the case when the local marginals are such that the output distribution is independent from the condition, i.e., the input. This is the case for the ideal behavior in which case Alice and Bob both observe a uniform distribution of their outputs locally, regardless of the input. In such case, there is only a single free parameter for the output distribution; therefore, the corresponding row must stay inside a two-dimensional subspace. Hence, all rows of the channel matrix must come from the same two dimensions and thus the rank of the matrix must be 2 (or 1). Apart from the ideal case, this holds whenever there is no amplitude-damping channel on Alice’s side. If there is one, it results in a rank-3 $T^{(XY\to RS)}$ (in the nontrivial cases). Therefore, it cannot be decomposed into the product of the ideal (rank-2) behavior and any classical (even correlated) noise channel.

Let us now investigate the remaining cases when there is no amplitude damping on Alice’s side. Here, the rank stays, at most, at 2, so this does not exclude the decomposition. For the decomposition to exist, the left null spaces of the matrices $T^{(XY\to AB)}$ and $T^{(XY\to RS)}$ have to coincide (this is necessary but not sufficient condition though). The left null spaces are determined by the two right eigenvectors with nonzero eigenvalue. One of the right eigenvectors coincides for $T^{(XY\to AB)}$ and $T^{(XY\to RS)}$ as they are stochastic matrices. Thus, for the null spaces to coincide, either $T^{(XY\to RS)}$ could have rank 1, which describes complete information erasure channels. This does not hold in our examples apart from some trivial extreme cases. For the nontrivial, rank-2 case, comparing its eigenvectors with the ones in Equation (51) of the ideal matrix, the second eigenvector of $T^{(XY\to RS)}$ with nonzero eigenvalue must have the form

$$\mathbf{v}=\left(\begin{array}{c}\xi \\ \xi \\ \xi \\ \eta \end{array}\right),$$

(55)

where the ratio of $\xi$ and $\eta$ is irrelevant; so, we have a single free parameter. In the behavior matrix, every row corresponds to a specific choice of $\xi$ and $\eta$. This has to be compared with the eigenvectors of the channel matrices of Equations (31), (34), (38), and (39). We have verified numerically that, for the case of the pair of phase-damping channels, no right eigenvector $\mathbf{v}$ of the form in Equation (55) exists.

For the case of phase damping on Alice’s side and amplitude damping on Bob’s, the verification of the existence of the eigenvector in the form of Equation (55) is a more complicated question. To find $T^{(AB\to RS),\mathrm{PD}, \mathrm{AD}}$, first, we conduct a numerical investigation: we plot the difference between the second and third coordinates of the respective eigenvector—this should be zero for the vector to obey the desired form in Equation (55); the result is displayed in Figure 3.

Based on this numerical evaluation, our conjecture is that, to have a nontrivial solution, ${(1-p_1)}^2=1-p_2$ should hold for the channel parameters. Furthermore, indeed, if this condition is satisfied, the observable behavior takes the form

$$T^{(XY\to RS),\mathrm{PD}, \mathrm{AD}}={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}{c}^2& s^2& s^2+2cs{p}_2& c^2-2cs{p}_2\\ c^2& s^2& s^2+2cs{p}_2& c^2-2cs{p}_2\\ c^2& s^2& s^2+2cs{p}_2& c^2-2cs{p}_2\\ s^2+2cs{p}_2& c^2-2cs{p}_2& c^2& s^2\end{array}\right).$$

(56)

As the first three rows are apparently equal, we assume a particular form of the pseudo inverse:

$$\begin{array}{l}{\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}{c}^2& s^2& s^2& c^2\\ c^2& s^2& s^2& c^2\\ c^2& s^2& s^2& c^2\\ c^2& s^2& s^2& c^2\end{array}\right)\left(\begin{array}{cccc}{\alpha }_1& {\alpha }_2& {\alpha }_3& {\alpha }_4\\ {\beta }_1& {\beta }_2& {\beta }_2& {\beta }_4\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)=\\ {\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}{c}^2& s^2& s^2+2cs{p}_2& c^2-2cs{p}_2\\ c^2& s^2& s^2+2cs{p}_2& c^2-2cs{p}_2\\ c^2& s^2& s^2+2cs{p}_2& c^2-2cs{p}_2\\ s^2+2cs{p}_2& c^2-2cs{p}_2& c^2& s^2\end{array}\right).\end{array}$$

(57)

Using standard linear algebra, this equation can be solved; thus, the pseudo inverse reads:

$$\left(\begin{array}{cccc}{\alpha }_1& {\alpha }_2& {\alpha }_3& {\alpha }_4\\ {\beta }_1& {\beta }_2& {\beta }_2& {\beta }_4\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)=\left(\begin{array}{cccc}1-s^2{p}_2& s^2{p}_2& c^2{p}_2& 1-c^2{p}_2\\ c^2{p}_2& 1-c^2{p}_2& 1-s^2{p}_2& s^2{p}_2\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$

(58)

This is not a stochastic matrix yet. However, as in $T^{(XY\to AB)}$, the first column is identical to the last and the second is identical to the third; there is a freedom in interchanging and rearranging the elements within the first and last rows, as well as within the second and third rows. Since $c^2{p}_2\le 1$ and $s^2{p}_2\le 1$, each component of the matrix will be non-negative. We can obtain the desired pseudo inverse matrix, e.g.,:

$$T^{(AB\to RS),\mathrm{PD}, \mathrm{AD}}=\left(\begin{array}{cccc}1-s^2{p}_2& s^2{p}_2& 0& 0\\ c^2{p}_2& 1-c^2{p}_2& 0& 0\\ 0& 0& 1-s^2{p}_2& s^2{p}_2\\ 0& 0& c^2{p}_2& 1-c^2{p}_2\end{array}\right).$$

(59)

Thus, in the PD-AD case, the behavior is decomposable under nontrivial conditions, and the decomposition is not trivial.

In summary, albeit it cannot be excluded in general, in the specified cases, the nonideal behavior cannot be obtained as a decomposition of the form of Equation (50). The impact of quantum channels cannot be taken into account with an equivalent effective classical noise channel.

6. Conclusions and Outlook

We have studied the nonlocal behavior of the CHSH-Bell experiment under the presence of quantum noise affecting the elements of the entangled resource separately. To gain insight into how such imperfections influence the system, we adopted a picture of classical memoryless channels. Our main contribution is this picture of the imperfect Bell experiment, and the study of the decomposability of the so-arising classical channel.

Although we examined only a few basic examples, these demonstrate that the manner in which quantum nonlocality is disrupted by quantum channels differs from how detector imperfections affect such scenarios. Specifically, the modified nonlocal behavior is not equivalent to the behavior obtainable by sending ideal measurement results through a classical channel that models local noise. In the case of unital channels, the change in the local behavior can at least be viewed as a composition of ideal behavior and classical noise; however, this approach proves to be impossible for amplitude damping. This implies that, when studying the effects of imperfections—such as, e.g., on a device-independent quantum key distribution scheme—considering only the impact of local classical noise (e.g., detector inefficiencies) does not provide a complete picture. Quantum imperfections can cause changes that are not equivalent to detector imperfections; this has to be taken into account when analyzing entanglement-based protocols.

Even though our contribution is mainly theoretical and aims to build deeper understanding of the nonlocal behavior in a Bell-CHSH experiment based on the picture of a classical channel between the input pair and the output pair, the introduced technique can be useful in experimental or theoretical scenarios involving two-qubit Bell experiments. One could gain detailed insights into changes in behavior due to the imperfections by carrying out the consideration described here, for the case of the actual CPTP maps describing a practical scenario. This can be determined, e.g., via quantum channel tomography [28], which can be addressed with machine learning approaches too [29]. In this way, the change in the behavior can be analyzed, and the decomposability to the ideal behavior and classical noise channel can be addressed. The additional effect of detector efficiencies can also be considered.

Our consideration can be generalized to arbitrary pairs of qubit channels. For instance, the example in Ref. [25] considers a pair of identical amplitude-damping channels acting on a singlet state; this could be another example to study as the amplitude-damping process modifies the singlet state in a way different from the entangled resource considered by us. Further, by considering the Choi representation of the most general channels, we can see that, through the linear space of semidefinite matrices—with a trace condition and the property of positive semidefiniteness—a fully general description can be obtained. Relaxing the condition of positive semidefiniteness would introduce nonphysical channels, but the calculation remains feasible. The large number of parameters, however, would make the study less transparent. It would therefore be interesting to identify more useful general conditions on decomposability, and this could potentially applicable to many-input–many-output or multiparty scenarios.