Quantum Bell nonlocality as a form of entanglement

Bell nonlocality describes a manifestation of quantum mechanics that cannot be explained by any local hidden variable model. Its origin lies in the nature of quantum entanglement, although understanding the precise relationship between nonlocality and entanglement has been a notorious open problem. In this paper, we develop a dynamical framework in which quantum Bell nonlocality emerges as a special form of entanglement and both are unified as resources under local operations and classical communication (LOCC). Our framework is built on the notion of classical and quantum processes, which are defined as channels that map elements between specific intervals in space and time. Entanglement is identified as a process that cannot be generated by LOCC while Bell nonlocality is the subset of these processes that have an instantaneous input-to-output delay time. LOCC preprocessing is a natural set of free operations in this theory, thereby providing previous nonlocality activation results a clear resource-theoretic foundation. We provide a systematic method to quantify the Bell nonlocality of a bipartite quantum channel. It is shown that both the relative entropy and the max relative entropy of nonlocality are nonadditive for a family of bipartite classical channels. This family includes the channel obtained when using the singlet state to maximally violate the CHSH inequality. We also find that the regularized relative entropy of Bell nonlocality provides an upper bound on the asymptotic rate of converting (i.e., simulating) many copies of one classical instantaneous resource to another.

Figure 1

Local operations and classical communications (LOCC) describes a class of distributed quantum information processing protocols in which only classical information is exchanged between spatially separated parties. Each “round” of the protocol consists one party, say Alice, performing a local quantum operation $\left\{{\mathcal{A}}_{i_1}\right\}$ and sending classical data $i_1$ to Bob who conditions his next local action on these data [22, 23].

Figure 2

A state $\rho$ that cannot violate a Bell inequality can be transformed into a state that violates one by performing a pre-LOCC map, like that in Fig. 1. The key property we identify in this work is that after the LOCC map, what remains is a bipartite classical channel with essentially zero delay time between the inputs $(x_0,y_0)$ and outputs $(x_1,y_1)$. Dots with the same color correspond to the same time.

Figure 3

A quantum process $(\mathcal{N},\mathrm{\Delta }\mathbf{x},\mathrm{\Delta }t)$ takes an input $\rho$ at arbitrary space-time point $({\mathbf{x}}_{A_0},t_0)$ and generates an output $\mathcal{N}\left(\rho \right)$ at space-time point $({\mathbf{x}}_{A_1}={\mathbf{x}}_{A_0}+\mathrm{\Delta }\mathbf{x},t_1=t_0+\mathrm{\Delta }t)$. The input-to-output delay time of $\mathcal{N}$ in this process is $\mathrm{\Delta }t$.

Figure 4

The bipartite channel ${\mathcal{E}}_2\circ \mathcal{N}\circ {\mathcal{E}}_1$ can have differing delay times for the different subsystems if it is built by composing multiple processes (top). The result is a multiprocess (bottom). Dots with the same color correspond to the same time.

Figure 5

A quantum superprocess $(\mathrm{\Theta },\mathrm{\Delta }\mathbf{x}\to \mathrm{\Delta }{\mathbf{x}}^{\prime },\mathrm{\Delta }t\to \mathrm{\Delta }{t}^{\prime })$ converts process $(\mathcal{N},\mathrm{\Delta }\mathbf{x},\mathrm{\Delta }t)$ to process $({\mathcal{N}}^{\prime },\mathrm{\Delta }{\mathbf{x}}^{\prime },\mathrm{\Delta }{t}^{\prime })$. The shaded yellow area represents the action of the superprocess. Since ${\mathcal{E}}_{\mathrm{pre}}$ and ${\mathcal{E}}_{\mathrm{post}}$ might belong to multiprocess, the times $s_0$ and $s_1$ are not necessarily equal to $t_0$ or $t_1$. For example, in Fig. 6, we will see an example in which $t_0<t_0^{\prime }=s_0=s_1=t_1$.

Figure 6

When the input to $\mathcal{N}$ does not depend on the input to ${\mathcal{E}}_{\text{pre}}$ it is possible to construct a superprocess $(\mathrm{\Theta },\mathrm{\Delta }\mathbf{x}\to \mathrm{\Delta }{\mathbf{x}}^{\prime },\mathrm{\Delta }t\to \mathrm{\Delta }{t}^{\prime })$ that decrease the input-to-output delay time, $\mathrm{\Delta }{t}^{\prime }<\mathrm{\Delta }t$. Observe in particular that we absorbed ${\mathcal{L}}^{A_0^{\prime }\to E}$ that appears in the expression (6) of ${\mathcal{E}}_{\text{pre}}$ into ${\mathcal{E}}_{\text{post}}$. In this case the delay time $\mathrm{\Delta }{t}^{\prime }$ depends only on the implementation of ${\mathcal{E}}_{\text{post}}$ and in general can be much smaller than $\mathrm{\Delta }t$. Dots with the same color correspond to the same time.

Figure 7

The most general superprocess that can convert a quantum process $({\mathcal{N}}^A,\mathrm{\Delta }\mathbf{x},\mathrm{\Delta }t)$ with $\mathrm{\Delta }t>0$ into an instantaneous quantum process. Irrespective of the time delays of $\mathcal{N}$ and ${\mathcal{F}}_1$, since ${\mathcal{F}}_2$ is instantaneous we have $t_1^{\prime }=t_0^{\prime }⩾t_1>t_0$. In particular, $\mathrm{\Delta }{t}^{\prime }=t_1^{\prime }-t_0^{\prime }=0$.

Figure 8

A bipartite superprocess $(\mathrm{\Theta },\mathrm{\Delta }t\to \mathrm{\Delta }{t}^{\prime })$ converts process $(\mathcal{N},\mathrm{\Delta }t)$ into process $\left(\mathrm{\Theta }\right[\mathcal{N}],\mathrm{\Delta }t)$, where $\mathrm{\Theta }$ is a superchannel that transforms $\mathcal{N}$ using pre- and postprocessing maps ${\mathcal{E}}_{\text{pre}}$ and ${\mathcal{E}}_{\mathrm{post}}$. The processes implementing ${\mathcal{E}}_{\text{pre}}$ and ${\mathcal{E}}_{\mathrm{post}}$ must themselves have delay times consistent with the overall change $\mathrm{\Delta }t\to \mathrm{\Delta }{t}^{\prime }$.

Figure 9

Implementation of an LOSE quantum process with an entangled state $\omega \in \mathfrak{D}\left(A_2{B}_2\right)$ and local superprocess $\mathrm{{\rm Y}}$ that takes the quantum state ${\omega }^{A_2{B}_2}$ as its input and then outputs the bipartite quantum channel ${\mathcal{N}}^{AB}≔\mathrm{{\rm Y}}\left[{\omega }^{A_2{B}_2}\right]$.

Figure 10

An LOSR superprocess with $\mathrm{\Delta }t=\mathrm{\Delta }{t}^{\prime }$.

Figure 11

An LOCC superprocess containing preprocessing LOCC simulating an instantaneous dynamical resource.

Figure 12

A comparison between the different classes of processes relevant to the study of quantum entanglement and Bell nonlocality. All processes belonging to the physically-realizable region outside LOCC possess entanglement. The instantaneous classical LOSE channels lying in the intersecting region are the Bell nonlocal processes.

Figure 13

The solid line is $2E_r\left({\mathcal{W}}_{\lambda }\right)$ whereas the dashed line is $D({\mathcal{W}}_{\lambda }^{\otimes 2}\parallel \overline{\mathcal{L}})$. We observe nonadditivity for $\lambda >0.74$. The $y$ axis is the relative entropy (units being defined as nonlocal bits) and the $x$ axis is channel parameter $\lambda$.