Nonlinear evolution and signaling

TL;DR

This paper introduces a convex quasi-linearity condition for deterministic nonlinear quantum evolutions, ensuring they do not enable arbitrarily fast signaling, and provides an explicit example of such an evolution.

Contribution

It proposes a new condition called convex quasi-linearity for nonlinear quantum evolutions and demonstrates its implications for signaling constraints.

Findings

Nonlinear evolutions satisfying convex quasi-linearity do not permit arbitrary fast signaling.

An explicit example of a nonlinear qubit evolution satisfying the quasi-linearity condition.

The condition helps distinguish physically plausible nonlinear quantum evolutions.

Abstract

We propose a condition, called convex quasi-linearity, for deterministic nonlinear quantum evolutions. Evolutions satisfying this condition do not allow for arbitrary fast signaling, therefore, they cannot be ruled out by a standard argument. We also give an explicit example of a nonlinear qubit evolution satisfying quasi-linearity.

Equations

$${\mathbf{\zeta }}_{\phi }=(\cos \phi ,\sin \phi ,0).$$

$${\rho }_{\phi }={\pi }_{\phi }{\rho }_{\mathsf{Bell}}{\pi }_{\phi }+(I-{\pi }_{\phi }){\rho }_{\mathsf{Bell}}(I-{\pi }_{\phi })\equiv \frac{1}{2}{\rho }_1+\frac{1}{2}{\rho }_2.$$

$${\rho }_B=\frac{1}{2}{\rho }_{B_{\phi }^1}+\frac{1}{2}{\rho }_{B_{\phi }^2}=\frac{1}{2}I,$$

$${\rho }_{B_{\phi }^1}$$

$${\rho }_{B_{\phi }^2}$$

$$\lambda {\rho }_a+(1-\lambda ){\rho }_b\equiv \rho ,$$

$$\Phi [\rho ]=\Phi [\lambda {\rho }_a+(1-\lambda ){\rho }_b]=\bar{\lambda }\Phi [{\rho }_a]+(1-\bar{\lambda })\Phi [{\rho }_b].$$

$$\Phi [\rho ]$$

$$=\lambda \frac{\mathrm{Tr}(\Pi {\rho }_a)}{\mathrm{Tr}(\Pi \rho )}\Phi [{\rho }_a]+(1-\lambda )\frac{\mathrm{Tr}(\Pi {\rho }_b)}{\mathrm{Tr}(\Pi \rho )}\Phi [{\rho }_b].$$

$$1-\lambda \frac{\mathrm{Tr}(\Pi {\rho }_a)}{\mathrm{Tr}(\Pi \rho )}=(1-\lambda )\frac{\mathrm{Tr}(\Pi {\rho }_b)}{\mathrm{Tr}(\Pi \rho )}.$$

$$\bar{\lambda }=\lambda \frac{\mathrm{Tr}(\Pi {\rho }_a)}{\mathrm{Tr}(\Pi \rho )}$$

$$\lambda {\rho }_{a0}+(1-\lambda ){\rho }_{b0}={\rho }_0$$

$$\begin{gathered} {\Phi }_t[\lambda {\rho }_{a0}+(1-\lambda ){\rho }_{b0}] \\ =\lambda (t){\Phi }_t[{\rho }_{a0}]+(1-\lambda (t)){\Phi }_t[{\rho }_{b0}], \end{gathered}$$

$${\text{v}}^{\prime }=\frac{\text{v}+\text{e}[\sinh \eta +(\cosh \eta -1)(\text{e}\cdot \text{v})]}{\cosh \eta +(\text{e}\cdot \text{v})\sinh \eta },$$

$$\text{n}(t)=\frac{\mathbf{\xi }+\text{e}[\sinh (gt)+(\cosh (gt)-1)(\text{e}\cdot \mathbf{\xi })]}{\cosh (gt)+(\text{e}\cdot \mathbf{\xi })\sinh (gt)},$$

$$\rho(t)=\tfrac{1}{2}\big{(}I+\textbf{n}(t)\cdot\boldsymbol{\sigma}\big{)}.$$

$$\rho (t)={\Phi }_t[{\rho }_0],$$

$$\dot{\textbf{n}}=g\big{(}\textbf{e}-\textbf{n}(\textbf{e}\cdot\textbf{n})\big{)}$$

$$\rho (t)=\lambda (t){\rho }_a(t)+(1-\lambda (t)){\rho }_b(t),$$

$$\rho_{a}(t)=\tfrac{1}{2}\big{(}I+\textbf{n}_{a}(t)\cdot\boldsymbol{\sigma}\big{)},\quad\rho_{b}(t)=\tfrac{1}{2}\big{(}I+\textbf{n}_{b}(t)\cdot\boldsymbol{\sigma}\big{)}$$

$$\lambda (t)=\frac{1+(\text{e}\cdot {\mathbf{\xi }}_a)\tanh (gt)}{1+(\text{e}\cdot \mathbf{\xi })\tanh (gt)}\lambda .$$

$$\rho_{B}(t)=\tfrac{1}{2}\big{(}I+(\textbf{e}\cdot\boldsymbol{\sigma})\tanh(gt)\big{)},$$

$$\rho_{B_{\varphi}^{1}}(t)=\tfrac{1}{2}\bigg{(}I\\ +\frac{-\boldsymbol{\zeta}_{\varphi}+\big{[}\sinh(gt)-(\textbf{e}\cdot\boldsymbol{\zeta}_{\varphi})\big{(}\cosh(gt)-1\big{)}\big{]}\textbf{e}}{\cosh(gt)-(\textbf{e}\cdot\boldsymbol{\zeta}_{\varphi})\sinh(gt)}\cdot\boldsymbol{\sigma}\bigg{)},$$

$$\rho_{B_{\varphi}^{2}}(t)=\tfrac{1}{2}\bigg{(}I\\ +\frac{\boldsymbol{\zeta}_{\varphi}+\big{[}\sinh(gt)+(\textbf{e}\cdot\boldsymbol{\zeta}_{\varphi})\big{(}\cosh(gt)-1\big{)}\big{]}\textbf{e}}{\cosh(gt)+(\textbf{e}\cdot\boldsymbol{\zeta}_{\varphi})\sinh(gt)}\cdot\boldsymbol{\sigma}\bigg{)}.$$

$$\lambda(t)=\tfrac{1}{2}\big{(}1-(\textbf{e}\cdot\boldsymbol{\zeta}_{\varphi})\tanh(gt)\big{)}.$$

$$\lambda(t)\rho_{B_{\varphi}^{1}}(t)+(1-\lambda(t))\rho_{B_{\varphi}^{2}}(t)\\ =\tfrac{1}{2}\big{(}I+(\textbf{e}\cdot\boldsymbol{\sigma})\tanh(gt)\big{)}=\rho_{B}(t).$$

$$\dot{\textbf{n}}=g\big{(}\textbf{e}\times\textbf{n}(\textbf{e}\cdot\textbf{n})\big{)}.$$

$$\textbf{n}(t)=\boldsymbol{\xi}\cos\theta(t)+(\textbf{e}\times\boldsymbol{\xi})\sin\theta(t)\\ +\textbf{e}(\textbf{e}\cdot\boldsymbol{\xi})\big{(}1-\cos\theta(t)\big{)},$$

Full text

Introduction.—From almost of century, quantum mechanics (QM), in his version based on the Hilbert space, was formulated as a linear theory preserving the superposition principle for pure states. However, many authors, for different reasons, undertaken attempts to generalize this theory by including nonlinear operations, too (see, e,g, Białynicki-Birula and Mycielski (1976); Weinberg (1989a, b); Doebner and Goldin (1992, 1996) and references therein). In fact a part of the QM formalism related to the measurement description uses nonlinear (stochastic) operations such as the selective projection postulate. Most of proposed attempts lie in replacement of the linear time evolution of quantum systems by a nonlinear one (see, e.g., Białynicki-Birula and Mycielski (1976); Weinberg (1989b)). It is widely believed that such deterministic nonlinear generalizations of the Schrödinger equation allow for signaling, i.e., allows one to send signals over arbitrarily large distances in a finite time (see, e.g., Gisin and Rigo (1995)). Arguments supporting this claim (in the context of the Weinberg model Weinberg (1989b)) were clearly given by Gisin in Ref. Gisin (1990), compare also Polchinski (1991); Czachor (1991). Gisin’s arguments are based on the observation that deterministic nonlinear time evolution destroys equivalence of quantum ensembles defining the same mixed state of the considered system. As a consequence, it creates a possibility of an instantaneous communication for space-like separated observers with the help of systems of entangled particles. Evidently such a possibility is in an apparent conflict with the special relativity.

Let us mention here that some authors gave arguments that deterministic nonlinear dynamics in special circumstances does not allow for signaling Polchinski (1991); Czachor (1998); Czachor and Doebner (2002); Kent (2005); Ferrero et al. (2004); Jordan (2009); Helou and Chen (2017). However, these arguments seems to be insufficient (compare Mielnik (2001); Jordan (2010)). For example, Czachor and Doebner Czachor and Doebner (2002) approach implies the modification of the state reduction postulate while Helou and Chen Helou and Chen (2017) postulate the extension of the Born rule. Our goal is different: we are looking for such an extension of deterministic linear dynamics that does not allow for faster than light signaling and at the same time does not require anything to be changed in the remaining part of quantum formalism.

Notice also that there exist nonlinear stochastic evolution equations free of the problems with signaling Gisin and Rigo (1995). Such models were proposed in various contexts, one of the most important are collapse models (see Ref. Ghirardi et al. (1986); Bassi et al. (2013) and references therein).

In this Letter we propose a new condition for a deterministic nonlinear quantum evolutions—quasi-linearity [Eq. (7)]. This condition guarantees that evolution preserves the equivalence of quantum ensembles. Consequently, the Gisin argument Gisin (1990) does not work in this case.

The Gisin’s argument.—Following Gisin Gisin (1990), let us assume that two distant observers, say A and B, want to establish instantaneous communication. In the half of the distance between them there is a source emitting pairs of spin-1/2 particles. Initial spin state of the particle is the Bell state. Particles move along the $z$-axis, one towards A and the second one towards B. The observer A performs polarization procedure of the flux of particles directed to him. To this end A measures without selection the projector $\pi_{\varphi}=\tfrac{1}{2}(I+\boldsymbol{\zeta}_{\varphi}\cdot\boldsymbol{\sigma})\otimes I$ with a possibility of changing the polarization angle $\varphi$. Here, the polarization vector

[TABLE]

As effect, the Bell state converts into the mixture

[TABLE]

Consequently, the reduced density matrix $\rho_{B}=\operatorname{Tr}_{A}(\rho_{\varphi})$ accessible to the observer B, has the form of the appropriate ensemble

[TABLE]

with

[TABLE]

The ensemble accessible to the observer B is equivalent to the fully depolarized state. Thus, any quantum-mechanical measurement of the observer B is incapable to distinguish between different ensembles of this form for different values of the polarization angle $\varphi$. Consequently, arbitrary fast communication is excluded by the QM rules.

However, according to the Gisin’s Gedanken prescription, a selective measurement by the observer B can be prefaced by a nonlinear evolution (like the Weinberg nonlinear evolution Weinberg (1989b)) applied to the reduced density $\rho_{B}$ and to the related ensembles. Gisin shows that the Weinberg nonlinear time evolution does not respect equivalence of ensembles so the evolved ensemble is polarization dependent ($\boldsymbol{\zeta}_{\varphi}$ dependent). Consequently, changes of polarizations made by the observer A can be registered by the observer B, i.e., instantaneous communication is possible. Thus, to avoid contradiction with relativity, nonlinear evolution must be ruled out of the QM formalism.

We show that this argument does not work for nonlinear evolutions satisfying the quasi-linearity condition.

The convex quasi-linear map.—Let us consider ensembles of the form

[TABLE]

where $0\leq\lambda\leq 1$ and $\rho_{a}$, $\rho_{b}$ belong to the convex set $S$ of density matrices. A trace preserving map $\Phi$ in $S$ is the convex quasi-linear if for each non-negative $\lambda\leq 1$ and for arbitrary density operators $\rho_{a}$, $\rho_{b}$ there exists such $\bar{\lambda}$, $0\leq\bar{\lambda}\leq 1$ that it holds

[TABLE]

This class of quantum operations contains linear maps too. The essence of this map is that it transforms convex combinations of density operators into convex combinations of their images so it preserves the convex structure of $S$. Moreover, it preserves equivalence of ensembles related to each fixed density matrix.

Usually, it is assumed that deterministic quantum evolution preserves mixtures; it is realized by requirement of linearity of $\Phi$ Gisin (1989); Jordan (1991); Ferrero et al. (2004); Jordan (2006); Bassi and Hejazi (2015). However, in our opinion this assumption is too restrictive and can be generalized to the condition (7). In fact, standard quantum-mechanical selective measurements belong to the class of quasi-linear transformations. Indeed, let us define the trace-preserving nonlinear map corresponding to a selective measurement: $\Phi(\rho)=\frac{\Pi\rho\Pi}{\operatorname{Tr}(\Pi\rho\Pi)}$, where $\Pi$ is a projector. Applying $\Phi$ to the density matrix (6) we get

[TABLE]

Now, Eq. (8) has the form (7) provided that

[TABLE]

But using (6) one can easily verify that (9) really holds. Therefore, Eq. (8) can be cast in the form (7) with

[TABLE]

and $0\leq\bar{\lambda}\leq 1$. The convex quasi-linearity of some classes of quantum operations was noted by Kraus in Kraus (1983) (although Kraus didn’t use this name). In fact, the applicability of the selective measurements to the quantum mechanics is strongly related to the quasi-linearity property of this stochastic operation. Indeed, it implies that the set of ensembles representing a fixed density matrix $\rho$ is mapped on the set of ensembles representing $\Phi(\rho)$.

The nonlinear evolution satisfying the quasi-linearity condition.—In view of the above discussion it seems that there are no objections to consider deterministic convex quasi-linear evolutions. Such an evolution map $\rho(t)=\Phi_{t}[\rho_{0}]$, with the initial condition $\rho(0)=\rho_{0}$ should form a semigroup satisfying the relation (7) for each value of the time parameter $t$; namely if

[TABLE]

then

[TABLE]

with the conditions $\lambda(0)=\lambda$ and $0\leq\lambda(t)\leq 1$. To show that the set of deterministic convex quasi-linear evolutions is non-empty we will construct a simple toy model of a qubit evolution satisfying (12).

To define the evolution of a qubit we should determine nonlinear evolution of its Bloch vector. In construction of our model we use the well known transformation rule for a three-velocity v under Lorentz boosts in a given direction, say e ($|\textbf{e}|=1$):

[TABLE]

where $\eta$ is the rapidity (we work in natural units with $\hbar=c=1$). The above transformation (13) is one of standard examples of nonlinear transformations appearing in physics.

Let us notice two obvious facts: (i) Boosts in a given direction form a one-parameter subgroup of the Lorentz group, (ii) Length of a three-velocity is always less than or equal to 1.

Observation (i) allows us to treat the transformation rule (13) as an equation defining time evolution of a three vector. Namely, we can rewrite Eq. (13) as

[TABLE]

where the constant $g$ has been introduced from dimensional reasons and $\textbf{n}(0)\equiv\boldsymbol{\xi}$ and $\boldsymbol{\xi}^{2}\leq 1$. Now, point (i) implies that if we write $\textbf{n}(t)=f_{t}(\boldsymbol{\xi})$ then $f_{t_{2}}\circ f_{t_{1}}=f_{t_{1}+t_{2}}$, therefore, (14) is a nonlinear time evolution of $\textbf{n}(t)$.

Next, point (ii) gives us the possibility of identifying the vector $\textbf{n}(t)$ with the Bloch vector defining a qubit density matrix $\rho(t)$:

[TABLE]

It means that the evolution $f_{t}$ corresponds to a nonlinear evolution of a qubit density matrix:

[TABLE]

where \rho_{0}=\tfrac{1}{2}\big{(}I+\boldsymbol{\xi}\cdot\boldsymbol{\sigma}\big{)}. Point (ii) implies that the condition $\textbf{n}(t)^{2}\leq 1$ is preserved for all $t$. Because the magnitude of a unit vector does not change under the evolution (14) the subset of pure states is invariant under the evolution $\Phi_{t}$.

Moreover, we can easily check that $\textbf{n}(t)\to\textbf{e}$ in the limit $t\to\infty$. Therefore, under the evolution (14) mixed states evolve into pure states. In such cases the von Neumann entropy of the state (15) decreases during the evolution. This observation, together with the second law of thermodynamics, suggest that a carrier physical system (e.g. a spin one-half particle or a two level atom) of the qubit evolving according to Eq. (14) cannot be isolated. Instead, it should be treated as an open quantum system interacting with an environment. Of course, it does not exclude the use of the evolution (14) in the considered Gisin Gedanken experiment.

Let us notice that the Bloch vector (14) is a solution of the following nonlinear differential equation:

[TABLE]

under the initial condition $\textbf{n}(0)=\boldsymbol{\xi}$.

Now, using the evolution (14) we can show that if the Eq. (11) holds then

[TABLE]

where

[TABLE]

and both Bloch vectors $\textbf{n}_{a}$ and $\textbf{n}_{b}$ evolve according to the nonlinear law (14) under the replacement $\boldsymbol{\xi}\to\boldsymbol{\xi}_{a}$ or $\boldsymbol{\xi}_{b}$, respectively. Using Eqs. (19,18,14) we can find the coefficient $\lambda(t)$. It is given by

[TABLE]

Therefore, we can conclude that each ensemble (11) equivalent to $\rho_{0}$ evolves under prescription (14) into ensemble (18) equivalent to $\rho(t)$. We see that the coefficient $\lambda(t)$ explicitly depends on time. However, in view of our previous remark that qubit evolving according to Eq. (14) cannot be treated as an isolated system, the dependence of $\lambda$ on time is rather expected.

Returning to the Gisin Gedanken experiment, the ensemble $\rho_{B}=\operatorname{Tr}_{A}(\rho_{\varphi})$, accessible to the observer B [Eq. (3)], evolves under (14) as follows

[TABLE]

and

[TABLE]

[TABLE]

where $\boldsymbol{\zeta}_{\varphi}$ is given in Eq. (1). So using (20) we find that in this case

[TABLE]

Therefore, finally, it really holds that:

[TABLE]

Thus, observer B cannot register any change of the polarization by the observer A, exactly as in the standard case.

Conclusions.—We have shown that time evolutions satisfying the quasi-linearity property (12) are admissible in the convex set of density operators even if they are nonlinear. As an example we discussed nonlinear time evolution of a qubit explicitly satisfying this property and we applied it to the famous Gedanken nonlocal correlation experiment by Gisin Gisin (1990). We concluded that this evolution does not allow for arbitrary fast signaling. Therefore, such an evolution is not in contradiction with the special relativity at this level. It remains an open question how big is the class of convex quasi-linear evolutions, work on this subject is in progress Rembieliński and Caban .

We can notice that the differential equation for the Bloch vector (17) resembles the equation for the Bloch vector in the simplified Weinberg model, discussed in Gisin (1990), adapted to the qubit case:

[TABLE]

Solution of the last equation reads

[TABLE]

where $\theta(t)=gt(\textbf{e}\cdot\boldsymbol{\xi})$ and $\textbf{n}(0)=\boldsymbol{\xi}$. However, it can be explicitly shown that in general solutions (27) cannot satisfy the quasi-linearity property (18). Consequently, evolution in the Weinberg model allows for arbitrary fast signaling.