# Fast quantum state discrimination with nonlinear PTP channels

Michael R. Geller Center for Simulational Physics, University of Georgia,
Athens, Georgia 30602, USA

(November 10, 2021)

###### Abstract

We investigate models of nonlinear quantum computation based on deterministic
positive trace-preserving (PTP) channels and associated master equations. The
models are defined in any finite Hilbert space, but the main results are for
dimension $N\\!=\\!2$. For every normalizable linear or nonlinear positive map
$\phi$ on bounded linear operators $X$, there is an associated normalized PTP
channel $\phi(X)/{\rm tr}[\phi(X)]$. Normalized PTP channels include unitary
mean field theories, such as the Gross-Pitaevskii equation for interacting
bosons, as well as models of linear and nonlinear dissipation. They classify
into 4 types, yielding 3 distinct forms of nonlinearity whose computational
power we explore. In the qubit case these channels support Bloch ball torsion
and other distortions studied previously, where it has been shown that such
nonlinearity can be used to increase the separation between a pair of close
qubit states, resulting in an exponential speedup for state discrimination.
Building on this idea, we argue that this operation can be made robust to
noise by using dissipation to induce a bifurcation to a phase where a pair of
stable fixed points create an intrinisically fault-tolerant nonlinear state
discriminator.

Quantum nonlinearity, beyond the stochastic nonlinearity already provided by
projective measurement, might be a powerful computational resource Mielnik
(1980); Abrams and Lloyd (1998); Bechmann-Pasquinucci  _et al._ (1998);
Czachor (1998, ); Bacon (2004); Aaronson ; Brun  _et al._ (2009); Bennett  _et
al._ (2009); Kahou and Feder (2013); Meyer and Wong (2013, 2014); Childs and
Young (2016); Di Molfetta and Herzog . However there is no experimental
evidence for physics beyond standard linear quantum mechanics Bollinger  _et
al._ (1989); Chupp and Hoare (1990); Walsworth  _et al._ (1990); Majumder  _et
al._ (1990). It is challenging to even formulate a consistent, fundamentally
nonlinear quantum theory in accordance with general principles Kibble (1978);
Weinberg (1989, 1998); Gisin (1989, 1990); Polchinski (1991); Czachor (1996,
1995); Gisin and Rigo (1995); Czachor and Kuna (1998); Kuna  _et al._ ;
Mielnik (2001); Czachor and Doebner (2002); Kent (2005); Jordan (2006, ,
2010); Ho and Hsu ; Bassi and Hejazi (2015); Diósi ; Rembieliński and Caban
(2020a); Budiyono and Dipojono ; Kaplan and Rajendran . In this paper we
consider the application of effective quantum nonlinearity to information
processing, while at the same time accepting that quantum physics is
fundamentally linear. Effective means that it arises in some approximate
(e.g., low-energy) quantum description, is a consequence of constraints on a
linear system, or emerges in the limit of a large number of particles Alicki
and Messer (1983); Man’ko and Marmo (1992); Ciaglia  _et al._ (2019); Drossel
; Fernengel and Drossel (2020); Breuer and Petruccione (2002). An early
example from particle physics is the low-energy reduction of electroweak
theory to Fermi’s simpler 1933 model
$\mathscr{L}={\overline{\psi}}i{\not{\partial}}\psi-g\,({\overline{\psi}}\gamma_{a}\psi)^{2}\\!,$
containing a four-fermion interaction. Effective nonlinearity is also common
in condensed matter (Breuer and Petruccione Breuer and Petruccione (2002) give
an excellent introduction). It’s a natural byproduct of dimensional reduction,
where the dynamics of a complex quantum many-body system is described by a
model with fewer degrees of freedom. A familiar example is self-consistent
mean field theory: For $n$ quantum particles moving in $D$ dimensions, such an
effective model reduces a problem with $nD$ degrees of freedom to one
involving only $D$, at the expense of nonlinearity and errors (usually).
Examples include mean field models for superfluids, superconductors, and laser
fields. Beyond mean field theory, various forms of nonlinearity have been
proposed to describe friction and dissipation in quantum mechanics Alicki and
Messer (1983); Drossel ; Fernengel and Drossel (2020); Breuer and Petruccione
(2002); Gisin (1981); Gisin and Cibils (1992); Kowalski and Rembieliński
(2019); Kowalski (2020) and for open systems more generally Gisin (1981);
Alicki and Messer (1983); Drossel ; Gisin and Cibils (1992); Kowalski and
Rembieliński (2019); Kowalski (2020); Breuer and Petruccione (2002); Fonseca
Romero  _et al._ (2004); Fernengel and Drossel (2020); Rembieliński and Caban
(2020b).

It is not known whether effective nonlinearity can actually be used to enhance
quantum information processing. Perhaps any nonlinear advantage is an artifact
of approximations and could never be realized Meyer and Wong (2013, 2014);
Childs and Young (2016). Childs and Young Childs and Young (2016) used the
speedup predicted for optimal qubit state discrimination with Gross-Pitaevskii
nonlinearity to derive a complexity theoretic bound on the long-time accuracy
of the Gross-Pitaevskii equation itself (though weaker than a known bound
Childs and Young (2016)). Can a different nonlinearity be used, or can the
Gross-Pitaevskii nonlinearity be used in another way? Is speedup the only
possible advantage; what about noise resilience? The purpose of this paper is
to explore these questions by providing a framework for studying known
nonlinear channels from an information processing perspective, and for
proposing new ones that might be experimentally realizable in the near future.
The framework should be applicable to specially designed strongly correlated
quantum materials that are open and possibly operated under extreme conditions
(think non-Hermitian exciton-polariton condensates Gao  _et al._ (2015);
Kalozoumis and Petrosyan ). We aim to impose as little structure as possible
on the allowed evolution beyond preserving the Hermitian symmetry, positive
semi-definiteness, and trace of the density matrix $X$. Stochastic
nonlinearity Gisin (1989); Gisin and Cibils (1992); Gisin and Rigo (1995),
including projective measurement with post-selection Lloyd and Slotine (2000);
Kiss  _et al._ (2006, 2011); Muñoz-Arias  _et al._ (2020), is also a useful
resource, but will not be covered here. Therefore we restrict ourselves to
deterministic nonlinear positive trace-preserving (PTP) channels Sudarshan
_et al._ (1961). We do not attempt to derive an effective nonlinear theory
from an underlying physical model, but instead try to identify what types of
nonlinearity are desirable, and why. However we will only scratch the surface
of this interesting but formidable problem.

Interestingly, according to Abrams and Lloyd Abrams and Lloyd (1998), Aaronson
Aaronson , and Childs and Young Childs and Young (2016), nonlinearity is not
required in large quantities: Even one “nonlinear qubit” would provide a
computational benefit when coupled to a linear quantum computer. This is
because nonlinearity can be used to increase the trace distance between a pair
of qubit states Mielnik (1980); Abrams and Lloyd (1998); Bechmann-Pasquinucci
_et al._ (1998); Aaronson ; Childs and Young (2016), resulting in an
exponential speedup for unstructured search and hence for any problem in the
class NP. We mainly focus on this $N\\!=\\!2$ case. While these results are
intriguing, it must be emphasized that they apply in an idealized setting,
where model errors are neglected.

Building on a growing body of similarly motivated work Alicki and Messer
(1983); Man’ko and Marmo (1992); Ciaglia  _et al._ (2019); Drossel ; Fernengel
and Drossel (2020); Breuer and Petruccione (2002); Gisin (1981); Gisin and
Cibils (1992); Kowalski and Rembieliński (2019); Kowalski (2020); Fonseca
Romero  _et al._ (2004); Rembieliński and Caban (2020b), we study a family of
normalized PTP channels of the form $\phi(X)/{\rm tr}[\phi(X)]$, where
$\phi(X)$ is a positive linear or nonlinear map on bounded linear operators
$X$ satisfying ${\rm tr}[\phi(X)]\\!\neq\\!0$. Normalized PTP channels fall
into 4 classes, yielding 3 distinct forms of nonlinearity. As in the classical
setting, rich dynamical structures result from the interplay of nonlinearity
and dissipation, and we will see that PTP channels allow for greater control
over engineered linear dissipation than completely positive channels do. Our
main result is the identification of a nonlinear channel where the Bloch ball
separates into two basins of attraction, which can be used to implement fast
intrinsically fault-tolerant state discrimination. Although we do not address
the model error issue directly, we hope that the predicted phase will survive
in realistic models and be observable experimentally. Section I mainly covers
the definition of a PTP channel and can be skipped by most readers. Normalized
PTP channels are classified in Sec. II, and fault-tolerant nonlinear state
discrimination is explained in Sec. III. Section IV contains the conclusions.

##  I PTP channels

###  I.1 Notation

Let ${\cal H}=({\rm span}\\{|e_{i}\rangle\\}_{i=1}^{N},\langle x|y\rangle)$ be
the system Hilbert space with inner product $\langle
x|y\rangle=\sum_{i=1}^{N}x_{i}^{*}y_{i}$, complete orthonormal basis
$\\{|e_{i}\rangle\\}_{i=1}^{N}\ (\langle
e_{i}|e_{j}\rangle\\!=\\!\delta_{ij},\ \sum_{i=1}^{N}e_{ii}\\!=\\!I_{N},\
e_{ij}\\!:=\\!|e_{i}\rangle\langle e_{j}|),$ and norm
$\||x\rangle\|=\sqrt{\langle x|x\rangle}$. Here $x^{*}$ denotes complex
conjugation and $I_{N}$ is the $N\\!\times\\!N$ identity. Let $X:{\cal
H}\rightarrow{\cal H}$ be a linear operator on ${\cal H}$ (isomorphic to a
matrix $X\in{\mathbb{C}}^{N\\!\times\\!N}$) and let $X^{\dagger}$ be its
adjoint. The set of these bounded linear operators form a second complex
vector space $B({\cal H},{\mathbb{C}})$; this space is our main focus. Let
${\rm Her}({\cal H},{\mathbb{C}})=\\{X\in B({\cal
H},{\mathbb{C}}):X=X^{\dagger}\\}$ be the subset of self-adjoint observables,
and ${\rm Her}^{\geq 0}({\cal H},{\mathbb{C}})=\\{X\in{\rm Her}({\cal
H}):X\succeq 0\\}$ be the positive semidefinite (PSD) subset. Quantum states
live in the subset of ${\rm Her}^{\geq 0}({\cal H},{\mathbb{C}})$ with unit
trace: ${\rm Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})=\\{X\in{\rm Her}^{\geq
0}({\cal H},{\mathbb{C}}):{\rm tr}(X)=1\\}.$ In the qubit case the elements
$X=(I_{2}+{\bf r}\cdot\bm{\sigma})/2$ of ${\rm Her}^{\geq 0}_{1}({\cal
H},{\mathbb{C}})$ are mapped, using the basis of Pauli matrices
$(\sigma^{a})_{a=1,2,3}$, to real vectors ${\bf r}=(x,y,z)\in{\mathbb{R}}^{3}$
with $|{\bf r}|\leq 1$, the closed Bloch ball ${\mathbb{B}}_{1}[{\bf 0}]$.

###  I.2 Nonlinear positive maps

######  Definition 1 (Linear map).

Let $L:B({\cal H},{\mathbb{C}})\rightarrow B({\cal H},{\mathbb{C}})$ be a map
on bounded linear operators satisfying (i) $L(X+Y)=L(X)+L(Y)$, and (ii)
$L(\alpha X)=\alpha L(X)$, for every $X,Y\in B({\cal H},{\mathbb{C}})$ and
$\alpha\in{\mathbb{C}}$. Then $L$ is a linear map on $B({\cal
H},{\mathbb{C}})$.

######  Lemma 1.

Let $L:B({\cal H},{\mathbb{C}})\rightarrow B({\cal H},{\mathbb{C}})$ be a
linear map on finite-dimensional bounded linear operators. Then $L$ has a
representation

$\displaystyle X\mapsto L(X)=\sum_{\alpha=1}^{m}A_{\alpha}XB_{\alpha},\ \
A_{\alpha},B_{\alpha}\in{\mathbb{C}}^{N\\!\times\\!N},\ \ m\\!\leq\\!N^{2},\ \
N=\dim({\cal H}).$ (1)

Proof:  Every linear map is specified by its action on a complete matrix basis
$e_{ab}=|e_{a}\rangle\langle e_{b}|\in{\mathbb{C}}^{N\\!\times\\!N},\
(e_{ab})_{a^{\prime}b^{\prime}}=\delta_{aa^{\prime}}\delta_{bb^{\prime}}$:

$\displaystyle X\mapsto
L(X)=L\bigg{(}\sum_{a,b=1}^{N}X_{ab}\,e_{ab}\bigg{)}=\sum_{a,b=1}^{N}X_{ab}\,L(e_{ab}).$
(2)

The set $\\{L(e_{ab})\\}_{a,b=1}^{N}$ defines the map. Let $\alpha=(a,b)$ be a
composite index with $a,b\in\\{1,\dots,N\\}$, and rewrite (1) with $m=N^{2}$
as $L(X)=\sum_{a,b=1}^{N}A_{ab}XB_{ab}$ with
$A_{ab},B_{ab}\in{\mathbb{C}}^{N\\!\times\\!N}$. $A_{ab}$ and $B_{ab}$ can
always be chosen to implement the map (2): the choice $A_{ab}=e_{ab}$ leads to
$\langle e_{a}|L(X)|e_{d}\rangle=L(X)_{ad}=\sum_{b,c}X_{bc}\,(B_{ab})_{cd}$.
The same matrix element of (2) is $\sum_{b,c}X_{bc}\,L(e_{bc})_{ad}$, so the
choice $(B_{ab})_{cd}=L(e_{bc})_{ad}$ reduces this to (2) as required. $\Box$

######  Definition 2 (Hermitian map).

Let $T:B({\cal H},{\mathbb{C}})\rightarrow B({\cal H},{\mathbb{C}})$ be a map
on bounded linear operators satisfying $T(X)^{\dagger}=T(X^{\dagger})$ for
every $X\in B({\cal H},{\mathbb{C}})$. Then $T$ is a Hermitian map on $B({\cal
H},{\mathbb{C}})$.

Hermitian maps have the property $T({\rm Her}({\cal
H},{\mathbb{C}}))\subseteq{\rm Her}({\cal H},{\mathbb{C}})$, and they can be
nonlinear.

######  Definition 3 (Positive map).

Let $\phi:B({\cal H},{\mathbb{C}})\rightarrow B({\cal H},{\mathbb{C}})$ be a
Hermitian map on bounded linear operators satisfying $\phi(X\succeq 0)\succeq
0$ for all $X\in B({\cal H},{\mathbb{C}})$, where $\cdot\succeq 0$ means it’s
an element of the PSD subset ${\rm Her}^{\geq 0}({\cal H},{\mathbb{C}})$. Then
$\phi$ is a positive map on $B({\cal H},{\mathbb{C}})$.

We reserve the symbol $\phi$ for positive maps.

######  Proposition 1 (Linear positive map Kraus (1971); Choi (1975)).

Let $\phi:B({\cal H},{\mathbb{C}})\rightarrow B({\cal H},{\mathbb{C}})$ be a
positive linear map on finite-dimensional bounded linear operators. Then
$\phi$ has a representation111The existence of the representation (3) does not
actually require $\phi$ to be positive, only Hermitian.

$\displaystyle
X\mapsto\phi(X)=\sum_{\alpha=1}^{m}\lambda_{\alpha}A_{\alpha}XA_{\alpha}^{\dagger},\
\ A_{\alpha}\in{\mathbb{C}}^{N\\!\times\\!N},\ \ {\rm
tr}(A^{\dagger}_{\alpha}A_{\beta})=\delta_{\alpha\beta},\ \
\lambda_{\alpha}\in{\mathbb{R}},\ \ m\\!\leq\\!N^{2}.$ (3)

Proof:  Define a Choi operator $C:=\sum_{i,j=1}^{N}\phi(e_{ij})\otimes e_{ij}$
in an expanded Hilbert space ${\cal H}_{A}\\!\otimes\\!{\cal H}_{B}$
consisting of two copies of our system ${\cal H}$, with ${\dim(\cal
H}_{A,B})\\!=\\!N.$ The matrix elements of $C$ in the product basis
$\\{|e_{a}\rangle\otimes|e_{b}\rangle\\}_{a,b=1}^{N}$ are $(\langle
e_{a}|\otimes\langle
e_{b}|)\,C\,(|e_{a^{\prime}}\rangle\otimes|e_{b^{\prime}}\rangle)=\langle
e_{a}|\phi(e_{bb^{\prime}})|e_{a^{\prime}}\rangle=\phi(e_{bb^{\prime}})_{aa^{\prime}}$.
Using the Hermitian property $\phi(X)^{\dagger}=\phi(X^{\dagger})$ we see that
$C\in{\mathbb{C}}^{N^{2}\\!\times\\!N^{2}}$ is Hermitian: $(\langle
e_{a}|\otimes\langle
e_{b}|)\,C^{\dagger}\,(|e_{a^{\prime}}\rangle\otimes|e_{b^{\prime}}\rangle)=\big{(}\langle
e_{a^{\prime}}|\otimes\langle
e_{b^{\prime}}|\,C\,|e_{a}\rangle\otimes|e_{b}\rangle\big{)}^{*}=\phi(e_{b^{\prime}b})_{a^{\prime}a}^{*}=\phi(e_{bb^{\prime}})_{aa^{\prime}}=(\langle
e_{a}|\otimes\langle
e_{b}|)\,C\,(|e_{a^{\prime}}\rangle\otimes|e_{b^{\prime}}\rangle)$. It
therefore has a spectral decomposition
$C=\sum_{\alpha=1}^{N^{2}}\lambda_{\alpha}\,|V_{\alpha}\rangle\langle
V_{\alpha}|,\ \lambda_{\alpha}\in{\mathbb{R}},$ where the eigenvectors
$\\{|V_{\alpha}\rangle\\}_{\alpha=1}^{N^{2}}$ form a complete orthonormal
basis in ${\cal H}_{A}\\!\otimes\\!{\cal H}_{B}$. Any $|\psi\rangle\in{\cal
H}_{A}\\!\otimes\\!{\cal H}_{B}$ can be obtained by the application of an
operator $N^{\frac{1}{2}}\,A\otimes I_{N}$ to
$|\Psi\rangle:=N^{-\frac{1}{2}}\sum_{i=1}^{N}|e_{i}\rangle\otimes|e_{i}\rangle$,
with $A_{ij}=\langle e_{i}|A|e_{j}\rangle=\,(\langle e_{i}|\otimes\langle
e_{j}|)\,|\psi\rangle$. Then $|V_{\alpha}\rangle\langle
V_{\alpha}|=N\,(A_{\alpha}\otimes
I)|\Psi\rangle\langle\Psi|(A^{\dagger}_{\alpha}\otimes I)$ where $\langle
e_{i}|A_{\alpha}|e_{j}\rangle=(\langle e_{i}|\otimes\langle
e_{j}|)\,|V_{\alpha}\rangle$. Also note that
$|\Psi\rangle\langle\Psi|=N^{-1}\sum_{i,j=1}^{N}e_{ij}\otimes e_{ij}$, so

$\displaystyle C$ $\displaystyle=$ $\displaystyle
N\sum_{\alpha=1}^{N^{2}}\lambda_{\alpha}\,(A_{\alpha}\otimes
I)|\Psi\rangle\langle\Psi|(A^{\dagger}_{\alpha}\otimes
I)=\sum_{i,j=1}^{N}\bigg{(}\sum_{\alpha=1}^{N^{2}}\lambda_{\alpha}A_{\alpha}e_{ij}A_{\alpha}^{\dagger}\bigg{)}\otimes
e_{ij}.$ (4)

Then
$\phi(e_{ij})=\sum_{\alpha=1}^{N^{2}}\lambda_{\alpha}A_{\alpha}e_{ij}A_{\alpha}^{\dagger}$,
and by linearity and completeness,
$\phi(X)=\sum_{\alpha=1}^{N^{2}}\lambda_{\alpha}A_{\alpha}XA_{\alpha}^{\dagger}$,
as required. Furthermore, because the $|V_{\alpha}\rangle$ are orthonormal,
$\langle V_{\alpha}|V_{\beta}\rangle={\rm
tr}(A^{\dagger}_{\alpha}A_{\beta})=\delta_{\alpha\beta}.\ \Box$

Having established a general representation for positive linear maps, we give
some nonlinear examples. The first is
$X\mapsto\phi(X)=(A^{\dagger}+CXB^{\dagger})(A+BXC^{\dagger})=[\phi(X^{\dagger})]^{\dagger},\
\ A,B,C\in{\mathbb{C}}^{N\\!\times\\!N}.$ On any PSD input,
$\phi(X)=|A+BXC^{\dagger}|^{2}\succeq 0$, so $\phi$ is positive. Two examples
of discrete nonlinear positive maps are $X\mapsto\phi_{\pm}(X)=({\rm
tr}(X)I_{N}\pm X)^{2}=[\phi_{\pm}(X^{\dagger})]^{\dagger},$ which reduce to
$|{\rm tr}(X)I_{N}\pm X|^{2}\succeq 0$ on PSD inputs. Another example is
$X\mapsto\phi(X)=|{\rm det}(X)|\cdot I_{N}=[\phi(X^{\dagger})]^{\dagger},$
which (after normalization) maps every input to the infinite-temperature
thermal state. Permanents and determinants have also been used to construct
nonlinear completely positive trace-preserving (CPTP) channels Ando and Choi
(1986).

######  Definition 4 (PTP channel Sudarshan  _et al._ (1961)).

Let $\Lambda:{\rm Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})\rightarrow{\rm
Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})$ be a positive map satisfying ${\rm
tr}(\Lambda(X))=1$ for all $X\in{\rm Her}^{\geq 0}_{1}({\cal
H},{\mathbb{C}})$. Then $\Lambda$ is a positive trace-preserving (PTP)
channel.

In this paper $\Lambda$ always refers to a PTP channel. The PTP channels
defined here are endomorphisms on the state space ${\rm Her}^{\geq
0}_{1}({\cal H},{\mathbb{C}})$, and are only required to act properly on
physical states. They may be composed of maps defined on operators outside of
${\rm Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})$ as well. In this case, the
behavior of the extended PTP channels on inputs outside of ${\rm Her}^{\geq
0}_{1}({\cal H},{\mathbb{C}})$ is not constrained by Definition 4.222In
particular, they might not be trace preserving in the usual sense of ${\rm
tr}(\Lambda(X))\\!=\\!{\rm tr}(X)$, because the trace preservation in
Definition 4 is really a trace fixing condition.

##  II PTP Models

In this paper we investigate models of nonlinear quantum computation based on
a category of PTP channels that take the form of a nonlinear positive channel
rescaled to conserve trace Kraus (1971); Kowalski and Rembieliński (2019);
Rembieliński and Caban (2020b); Kowalski (2020). Although normalized PTP
channels do not provide an exhaustive classification of nonlinear channels,
they are sufficient to illustrate some of the different types of effective
nonlinearity that are available or might become available in the near future.

######  Definition 5 (Normalized PTP channel Kraus (1971); Kowalski and
Rembieliński (2019); Rembieliński and Caban (2020b); Kowalski (2020)).

Let $\phi:{\rm Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})\rightarrow{\rm
Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})$ be a positive map satisfying ${\rm
tr}[\phi(X)]\neq 0$ for all $X\in{\rm Her}^{\geq 0}_{1}({\cal
H},{\mathbb{C}})$. Then the PTP map

$\displaystyle\Lambda_{\phi}:{\rm Her}^{\geq 0}_{1}({\cal
H},{\mathbb{C}})\rightarrow{\rm Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})\ \
{\rm given\ by}\ \ X\mapsto\Lambda_{\phi}(X)=\frac{\phi(X)}{{\rm
tr}[\phi(X)]}$ (5)

is a normalized PTP channel.

Normalized PTP channels are common because they can be constructed from any
positive map $\phi$ satisfying a normalizability condition ${\rm
tr}[\phi(X)]\neq 0$.333To be precise, normalized PTP channels are only defined
on a subset of physical states ${\rm Her}^{\geq 0}_{1}({\cal
H},{\mathbb{C}})$, because their action on inputs with ${\rm tr}[\phi(X)]=0$
is not specified. However this situation does not arise in the cases
considered here. From positivity, ${\rm tr}[\phi(X)]\geq 0$. Normalizability
requires the more restrictive condition that ${\rm tr}[\phi(X)]>0$. The
condition ${\rm tr}[\phi(X)]>0$ is important because it ensures the positivity
of $\Lambda_{\phi}$. Normalized PTP channels naturally fall into four classes,
according to whether $\phi$ is linear (or not) and trace-preserving (or not):

  1. II.1.

Linear positive $\phi$ and ${\rm tr}[\phi(X)]=1\ {\rm for\ all}\ X\in{\rm
Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})$. This is the class of general linear
positive channels, which includes linear CPTP channels.

  2. II.2.

Linear positive $\phi$ and ${\rm tr}[\phi(X)]\neq 1\ {\rm for\ some}\ X\in{\rm
Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})$. These are called nonlinear in
normalization only (NINO) channels.

  3. II.3.

Nonlinear positive $\phi$ and ${\rm tr}[\phi(X)]=1\ {\rm for\ all}\ X\in{\rm
Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})$. This class includes state-dependent
CPTP channels.

  4. II.4.

Nonlinear positive $\phi$ and ${\rm tr}[\phi(X)]\neq 1\ {\rm for\ some}\
X\in{\rm Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})$. These are the most general
channels considered here. They support rich dynamics similar to that of
classical nonlinear systems.

These classes are discussed below in Secs. II.1 \- II.4.

###  II.1 Linear PTP and CPTP channels

A standard form for linear PTP channels is provided in Proposition 1. While
every linear positive map can be put in the form (3), not every map of the
form (3) is positive (because the $\lambda_{\alpha}$ can be negative). The
necessary conditions for (3) to represent a positive (P) or completely
positive (CP) map can be obtained as follows: Let
$S_{>}\\!=\\!\\{\alpha\\!:\\!\lambda_{\alpha}\\!>\\!0\\}\neq\emptyset$ and
$S_{<}\\!=\\!\\{\alpha\\!:\\!\lambda_{\alpha}\\!<\\!0\\}$ be the index sets of
positive and negative Choi eigenvalues, and decompose $\phi$ into
$\phi=\phi_{>}-\phi_{<}$, where $\phi_{>}(X)=\sum_{\alpha\in
S_{>}}\lambda_{\alpha}\,A_{\alpha}XA_{\alpha}^{\dagger}$ and
$\phi_{<}(X)=\sum_{\alpha\in
S_{<}}|\lambda_{\alpha}|\,A_{\alpha}XA_{\alpha}^{\dagger}$ are each manifestly
positive. Upon rescaling, each can be put into the form
$\Phi(X)=\sum_{\alpha}A_{\alpha}XA^{\dagger}_{\alpha},\
A_{\alpha}\in{\mathbb{C}}^{N\\!\times\\!N}$. Maps of this form also satisfy
the stronger condition of complete positivity, meaning that they remain
positive when combined with a second Hilbert space ${\cal H}_{B}$, of any
finite dimension, on which the identity acts: $[\Phi\otimes{\rm
id}_{B}](X\succeq 0)\succeq 0$. Here $X\in B({\cal H}_{A}\otimes{\cal
H}_{B},{\mathbb{C}})$. Each term in $\Phi$ clearly has the required property:
$\left\langle\psi\right|(A_{\alpha}\otimes I)X(A^{\dagger}_{\alpha}\otimes
I)\left|\psi\right\rangle=\left\langle\psi_{\alpha}\right|X\left|\psi_{\alpha}\right\rangle\geq
0$ for every $\left|\psi\right\rangle\in{\cal H}_{A}\\!\otimes\\!{\cal
H}_{B}$, where $\left|\psi_{\alpha}\right\rangle=(A^{\dagger}_{\alpha}\otimes
I)\left|\psi\right\rangle$. The condition for positivity is therefore
$\phi_{>}(X)\succeq\phi_{<}(X)\ {\rm for\ every}\ X\succeq 0$, whereas the
condition for complete positivity is $\phi_{<}(X)=0$. An important CP map is
the completely-positive trace-preserving (CPTP) channel, which has a
nonnegative Choi spectrum $\lambda_{\alpha}\geq 0$ and therefore an operator-
sum representation Kraus (1971); Choi (1975)

$\displaystyle
X\mapsto\Phi(X)=\sum_{\alpha=1}^{m}A_{\alpha}XA_{\alpha}^{\dagger},\ \
\sum_{\alpha=1}^{m}A_{\alpha}^{\dagger}A_{\alpha}=I_{N},\ \
A_{\alpha}\in{\mathbb{C}}^{N\\!\times\\!N},\ \ m\\!\leq\\!N^{2}.$ (6)

This follows from Proposition 1 (but with different $A^{\rm{}^{\prime}s}$ as
to enforce trace conservation and absorb a factor of
$\sqrt{\lambda_{\alpha}}$). The integer $m$ is the Choi rank. In this paper
$\Phi$ always refers to a linear CP (and usually TP) map.

There is a large body of work on linear P channels that are not CP, called
non-CP maps Pechukas (1994); Shaji and Sudarshan (2005); Carteret  _et al._
(2008); Dominy  _et al._ (2016). The distinction between CP and non-CP maps is
the possibility of non-CP maps generating negative states on entangled inputs,
requiring their restriction to a subset of the state space where this
unphysical output is avoided. Although non-CP operations are well known for
their use in entanglement detection Peres (1996), the question of whether
linear non-CP channels could provide a computational advantage over linear CP
channels appears to be largely unexplored. However, non-CP communication
channels in which the environment is measured (and the results used to correct
the channel) enable increased capacity Gregoratti and Werner (2004); Hayden
and King (2005); Winter . In the presence of nonlinearity, we also find that
the distinction between CP and non-CP is significant, due to the larger space
of nonunitary processes supported by non-CP channels.

###  II.2 NINO channels

######  Definition 6 (NINO channel).

Let $\phi:{\rm Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})\rightarrow{\rm
Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})$ be a positive linear map with ${\rm
tr}[\phi(X)]\neq 0$ for every $X\in{\rm Her}^{\geq 0}_{1}({\cal
H},{\mathbb{C}})$, and ${\rm tr}[\phi(X)]\neq 1$ for one or more $X\in{\rm
Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})$. Then the PTP map

$\displaystyle\Lambda_{\phi}:{\rm Her}^{\geq 0}_{1}({\cal
H},{\mathbb{C}})\rightarrow{\rm Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})\ \
{\rm given\ by}\ \ X\mapsto\Lambda_{\phi}(X)=\frac{\phi(X)}{{\rm
tr}[\phi(X)]}$ (7)

is called a nonlinear-in-normalization only (NINO) channel.

We stress that the positive map $\phi$ in Definition 6 is linear, but not
unitary $\phi_{U}(X)=UXU^{\dagger},\ U^{\dagger}U\\!=\\!I_{N}$ (because ${\rm
tr}[\phi_{U}(X)]={\rm tr}(X)=1$ for all $X$). NINO channels inherit, from
linear maps, the powerful ability to characterize them through their action on
a complete basis (because $\phi$ has this property). Next we obtain a general
representation for NINO channels.

######  Proposition 2 (NINO representation).

Let $\Lambda$ be a NINO map of the form (7) with $\phi$ linear. Then $\Lambda$
has a representation

$X\mapsto\Lambda(X)=\frac{\displaystyle\sum_{\alpha=1}^{m}\zeta_{\alpha}A_{\alpha}XA_{\alpha}^{\dagger}}{{\rm
tr}(FX)},\ \
F:=\sum_{\alpha=1}^{m}\zeta_{\alpha}A_{\alpha}^{\dagger}A_{\alpha}\neq I_{N},\
\ A_{\alpha}\in{\mathbb{C}}^{N\\!\times\\!N},\ \ \zeta_{\alpha}=\pm 1,\ \
m\\!\leq\\!N^{2}.\ \ \ \ $ (8)

Proof:  This follows from Proposition 1 after substituting
$\lambda_{\alpha}=\zeta_{\alpha}|\lambda_{\alpha}|,\ \zeta_{\alpha}={\rm
sign}(\lambda_{\alpha})$, and rescaling the $A^{\rm{}^{\prime}s}$. $\ \Box$

Our main objective is to study NINO channels from a dynamical perspective, via
master equations. Some of the analysis will apply to the other classes as
well. Let $X\in{\rm Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})$ be the state of
a physical system which evolves continuously in time according to

$\displaystyle X(t)\mapsto X(t+\Delta t)=\Lambda_{\Delta
t,t}\big{(}X(t)\big{)},\ \ \Delta t\geq 0,\ \ \Lambda_{0,t}={\rm id}\ {\rm
for\ all}\ t\in{\mathbb{R}}.$ (9)

Here $\Lambda_{\Delta t,t}$ is a two-parameter family of PTP channels
continuous in both $t$ and $\Delta t$. Because we want to study the simplest
instances that illustrate computational advantages, we make several additional
simplifying assumptions:

  1. (i)

Stationarity: $\Lambda_{\Delta t,t}=\Lambda_{\Delta t}$ for all
$t\in{\mathbb{R}}$.

  2. (ii)

Semigroup: $\Lambda_{s}\circ\Lambda_{t}=\Lambda_{s+t},\ \ 0\leq s\ll 1,\ \
0\leq t\ll 1.$

  3. (iii)

The nonlinearity can be turned off, recovering linear CPTP evolution.

  4. (iv)

$N=2$.

From (9) we have that $\Lambda_{\Delta t}$, defined in (i), is continuous and
satisfies $\Lambda_{0}={\rm id}$. Stationarity excludes time-dependent
Hamiltonians that arise when a physical system is driven with time-dependent
fields (precisely what we want to do to run a device). While it is possible to
formulate the problem with time-dependent generators and obtain some of the
results in terms of time-ordered exponentials, we will not cover that case
here. Instead we assume that the nonlinearity can be turned on and off, and
while in the off state the full toolkit of linear quantum information
processing can be applied. The additional assumptions (i) and (ii) are
sufficient to define, for any PTP channel, a Markovian master equation that
extends linear Markovian CPTP evolution by the Gorini-Kossakowski-Sudarshan-
Lindblad (GKSL) equation Gorini  _et al._ (1976); Lindblad (1976). Our
approach follows a large body of work on nonlinear master equations Alicki and
Messer (1983); Man’ko and Marmo (1992); Ciaglia  _et al._ (2019); Drossel ;
Fernengel and Drossel (2020); Breuer and Petruccione (2002); Gisin (1981);
Gisin and Cibils (1992); Kowalski and Rembieliński (2019); Kowalski (2020);
Fonseca Romero  _et al._ (2004); Rembieliński and Caban (2020b). Especially
relevant are the recent papers by Kowalski and Rembieliński Kowalski and
Rembieliński (2019), Fernengel and Drossel Fernengel and Drossel (2020), and
Rembieliński and Caban Rembieliński and Caban (2020b). The restriction to
qubits is not used in the derivation of the master equations, but is needed
for their subsequent analysis.

Before deriving a master equation for (8), we briefly consider the $m=1$ case
to illustrate one of the differences between NINO and linear CPTP channels.
Suppose $A=e^{tL}$ is a continuous time-dependent linear operator
infinitesimally generated by some $L\in B({\cal H})$. Decompose the generator
into Hermitian and anti-Hermitian contributions $L=L_{+}+L_{-}$, with
$L_{\pm}:=(L\pm L^{\dagger})/2$. Then $X(0)\mapsto
X(t)=\Lambda_{t}(X(0))=(e^{tL}\,X(0)\,e^{tL^{\dagger}})/[{\rm
tr}[e^{tL^{\dagger}}e^{tL}X(0)]]$ and

$\displaystyle\frac{dX}{dt}=[L_{-},X]+\\{L_{+},X\\}-2\,{\rm tr}(L_{+}X)\,X.$
(10)

Here $\\{\cdot,\cdot\\}$ is an anticommutator. Tracing gives $\frac{d}{dt}{\rm
tr}(X)\\!=\\!2\,{\rm tr}(L_{+}X)\\!-\\!2\,{\rm tr}(L_{+}X)\,{\rm
tr}(X)\\!=\\!0$ assuming ${\rm tr}(X)=1$, showing how the nonlinear term fixes
the normalization. The equation of motion (10) includes unitary evolution
generated by a Hamiltonian $H=iL_{-}$, together with linear dissipation and
amplification (if $L_{+}$ has positive eigenvalues). Or we can say that the
dynamics is generated by a non-Hermitian Hamiltonian444We distinguish between
infinitesimal generators $G$ and Hamiltonians $iG$. Thus, unitary evolution
results from Hermitian Hamiltonians but from anti-Hermitian generators.
$H_{\rm non}:=iL=H+iL_{+}$. NINO channels expand the utilization of linear
maps by conserving the trace nonlinearly.

Now we obtain a master equation for general $m$. Continuous one-parameter NINO
channels have the form

$\displaystyle\Lambda_{t}(X)=\frac{\displaystyle\sum_{\alpha=1}^{m}\zeta_{\alpha}\,A_{\alpha}(t)\,X\,A_{\alpha}^{\dagger}(t)}{{\rm
tr}(F_{t}X)},\ \
F_{t}:=\sum_{\alpha=1}^{m}\zeta_{\alpha}\,A_{\alpha}^{\dagger}(t)\,A_{\alpha}(t)\neq
I_{N},\ \ \zeta_{\alpha}=\pm 1,\ \ t\geq 0.$ (11)

The operators $A_{\alpha}(t)\in{\mathbb{C}}^{N\\!\times\\!N}$ are analytic
functions of time for $t\geq 0$, which can be classified into two types, jump
and nonjump, according to their $t\rightarrow 0$ behavior. We assume that
$k>0$ of the $m$ operators have nonzero limits

$\displaystyle\lim_{t\rightarrow 0}A_{\alpha}(t)=A_{\alpha}^{0}\neq 0,\ \
\alpha\in\\{1,\cdots,k\\}\ \ {\rm(nonjump)},$ (12)

while the others vanish

$\displaystyle\lim_{t\rightarrow 0}A_{\alpha}(t)=0,\ \
\alpha\in\\{k\\!+\\!1,\cdots,m\\}\ \ {\rm(jump)}.$ (13)

The jump/nonjump terminology comes from the unraveled stochastic picture,
where one or more rare but disruptive “jump” operations are applied randomly
to a simulated open system, on top of a smooth background of unitary plus
nonunitary evolution, with the nonunitary component fixed to conserve trace.
There is no restriction on the number of jump operators; however no more than
$m^{2}$ are required. The only restriction on the number of nonjump operators
is that $k>0$: At least one is required to obtain the desired $t\rightarrow 0$
limit. Note that if $F_{t}=I_{N}$, the nonlinear generator disappears and we
recover linear GKSL evolution Gorini  _et al._ (1976); Lindblad (1976).

The constant matrices $A_{\alpha}^{0}\in{\mathbb{C}}^{N\\!\times\\!N}$ in (12)
are not arbitrary; the condition $\lim_{t\rightarrow 0}\Lambda_{t}(X)=X$ for
all $X\in{\rm Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})$ requires
$A_{\alpha}^{0}=z_{\alpha}\,I_{N},\ \ \alpha\in\\{1,\cdots,k\\},$ where the
$z_{\alpha}\in{\mathbb{C}}$ satisfy a “normalization” condition
$\sum_{\alpha=1}^{k}\zeta_{\alpha}|z_{\alpha}|^{2}=1$ (recall that
$\zeta_{\alpha}=\pm 1$). To satisfy the semigroup property it is sufficient to
let

$\displaystyle A_{\alpha}(t)=\begin{cases}z_{\alpha}\,e^{tL_{\alpha}}\ \ {\rm
for}\ \alpha\in\\{1,\cdots,k\\},\\\ B_{\alpha}\sqrt{t}\ \ \ {\rm for}\
\alpha\in\\{k\\!+\\!1,\cdots,m\\},\end{cases}$ (14)

where $L_{\alpha},B_{\alpha}\in B({\cal H},{\mathbb{C}})$. To see why, note
that in the short-time limit

$\displaystyle\sum_{\alpha=1}^{m}\zeta_{\alpha}\,A_{\alpha}(t)XA_{\alpha}^{\dagger}(t)=X+t\bigg{[}\sum_{\alpha=1}^{k}\zeta_{\alpha}|z_{\alpha}|^{2}\bigg{(}\\!L_{\alpha}X+XL_{\alpha}^{\dagger}\\!\bigg{)}+\sum_{\alpha>k}^{m}\zeta_{\alpha}B_{\alpha}XB_{\alpha}^{\dagger}\bigg{]}+O(t^{2})$
(15)

and

$\displaystyle F_{t}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{m}\zeta_{\alpha}\,A_{\alpha}^{\dagger}(t)A_{\alpha}(t),$
(16) $\displaystyle=$ $\displaystyle
I_{N}+t\bigg{[}\sum_{\alpha=1}^{k}\zeta_{\alpha}|z_{\alpha}|^{2}\big{(}\\!L_{\alpha}+L_{\alpha}^{\dagger}\\!\big{)}+\sum_{\alpha>k}^{m}\zeta_{\alpha}B_{\alpha}^{\dagger}B_{\alpha}\bigg{]}+O(t^{2}),$
(17) $\displaystyle=$ $\displaystyle I_{N}+t\,\frac{dF_{0}}{dt}+O(t^{2}).$
(18)

Then, if ${\rm tr}(X)=1$,

$\displaystyle\Lambda_{t}(X)$ $\displaystyle=$ $\displaystyle
X+t\bigg{[}\sum_{\alpha=1}^{k}\zeta_{\alpha}|z_{\alpha}|^{2}\bigg{(}\\!L_{\alpha}X+XL_{\alpha}^{\dagger}\\!\bigg{)}+\sum_{\alpha>k}^{m}\zeta_{\alpha}B_{\alpha}XB_{\alpha}^{\dagger}-{\rm
tr}\bigg{(}\\!X\frac{dF_{0}}{dt}\bigg{)}X\bigg{]}+O(t^{2})\ \ \ \ \ $ (19)

and

$\displaystyle\Lambda_{s}(\Lambda_{t}(X))=\Lambda_{s+t}(X)+O(s^{2})+O(st)+O(t^{2})\
\ {\rm for\ every\ }X\in{\rm Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}}),$ (20)

as required for short times. The NINO evolution equation follows from (19):

$\displaystyle\frac{dX}{dt}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{k}\zeta_{\alpha}|z_{\alpha}|^{2}\bigg{(}\\!L_{\alpha}X+XL_{\alpha}^{\dagger}\\!\bigg{)}+\sum_{\alpha>k}^{m}\zeta_{\alpha}B_{\alpha}XB_{\alpha}^{\dagger}-X\,{\rm
tr}\bigg{(}\\!\\!X\frac{dF_{0}}{dt}\\!\bigg{)}$ (21) $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{k}\zeta_{\alpha}|z_{\alpha}|^{2}\bigg{(}\\![L_{\alpha-},X]+\\{L_{\alpha+},X\\}\\!\bigg{)}+\sum_{\alpha>k}^{m}\zeta_{\alpha}B_{\alpha}XB_{\alpha}^{\dagger}-X\,{\rm
tr}\bigg{(}\\!\\!X\frac{dF_{0}}{dt}\\!\bigg{)},$

where, in the second line, each linear operator $L_{\alpha}$ has been
decomposed into Hermitian and anti-Hermitian parts according to
$L_{\alpha}=L_{\alpha+}+L_{\alpha-}$, with $L_{\alpha\pm}:=(L_{\alpha}\pm
L_{\alpha}^{\dagger})/2$. The anti-Hermitian
$\\{L_{\alpha-}\\}_{\alpha=1}^{k}$ each generate a unitary time evolution with
Hamiltonian $H_{\alpha}=iL_{\alpha-}=H_{\alpha}^{\dagger}$, whereas the
$\\{L_{\alpha+}\\}_{\alpha=1}^{k}$ and $\\{B_{\alpha}\\}_{\alpha>k}^{m}$
generate nonunitary time evolution. The brackets in (21) are commutators and
anticommutators $\\{A,B\\}=AB+BA$. Note that the parameters $z_{\alpha}$ can
be absorbed into rescaled generators
${\tilde{L}}_{\alpha}=|z_{\alpha}|^{2}\,L_{\alpha}$ with no essential change.

A principal difference between Markovian NINO and Markovian CPTP evolution is
in the form of the nonjump operators. In a CPTP channel, $\zeta_{\alpha}=1$
and trace is conserved through the requirement
$\sum_{\alpha=1}^{k}|z_{\alpha}|^{2}L_{\alpha+}=-\frac{1}{2}\sum_{\alpha>k}^{m}B_{\alpha}^{\dagger}B_{\alpha}$,
which sets $dF_{0}/dt=0$ in (21). In this case total Hermitian generator
$\sum_{\alpha=1}^{k}|z_{\alpha}|^{2}L_{\alpha+}$ is always negative
semidefinite, leading to nonexpansive evolution and usually to a single stable
fixed point. However in a NINO channel the $L_{\alpha+}$ are free parameters,
and they can have positive eigenvalues. An example of this distinction occurs
when $m\\!=\\!1$: Rank 1 CPTP channels are unitary and nondissipative, whereas
$m\\!=\\!1$ NINO channels already support dissipation and amplification [see
(10)]. Therefore we can think of the NINO master equation (21) as a
generalization of the linear GKSL equation Gorini  _et al._ (1976); Lindblad
(1976) to support linear evolution by one or more non-Hermitian Hamiltonians.

###  II.3 State-dependent CPTP channels

Next we discuss the class of normalized PTP channels (5) with nonlinear
positive $\phi$ and ${\rm tr}[\phi(X)]=1\ {\rm for\ all}\ X\in{\rm Her}^{\geq
0}_{1}({\cal H},{\mathbb{C}})$. These include the important subset of
parametrically nonlinear CPTP channels, which we call state-dependent CPTP
channels.

######  Definition 7 (State-dependent CPTP).

Let $X\in B({\cal H},{\mathbb{C}})$ and
$A_{\alpha}(X)\in{\mathbb{C}}^{N\\!\times\\!N}$ be a set of $X$-dependent
matrices satisfying

  1. 1.

$A_{\alpha}(X^{\dagger})=A_{\alpha}(X),$

  2. 2.

${\displaystyle\sum_{\alpha=1}^{m}A_{\alpha}(X)^{\dagger}A_{\alpha}(X)=I_{N}}$,

for all $X\in B({\cal H},{\mathbb{C}})$ and any finite $m$. Then

$\displaystyle
X\mapsto\Lambda(X)=\sum_{\alpha=1}^{m}A_{\alpha}(X)\,X\,A_{\alpha}(X)^{\dagger}$
(22)

is a state-dependent CPTP channel.

Channels in this class have been investigated by many authors Gisin (1981);
Abrams and Lloyd (1998); Kahou and Feder (2013); Meyer and Wong (2013, 2014);
Czachor and Kuna (1998); Childs and Young (2016); Fernengel and Drossel
(2020). The associated master equation is the state-dependent GKSL equation
Gorini  _et al._ (1976); Lindblad (1976). Many early proposals for nonlinear
extensions of quantum mechanics, including the Weinberg model Weinberg (1998),
and unitary models based on a nonlinear Schrodinger equation, are in this
class. A rank 1 example is

$\displaystyle X\mapsto\Lambda(X)=U(X)\,X\,U(X)^{\dagger},\ \ U(X):=e^{i\,{\rm
tr}(AX)\,B}=U(X^{\dagger}),\ \ A,B\in{\rm Her}({\cal H},{\mathbb{C}}).$ (23)

This map applies a generator $B$ scaled by the mean $\langle A\rangle={\rm
tr}(AX)$ of observable $A$. A generalization of (23) to multiple nonlinear
generators is $U(X)=e^{i\sum_{\alpha}{\rm tr}(A_{\alpha}X)\,B_{\alpha}}$,
which includes arbitrary state-dependent Hamiltonians and unitary mean field
theories, including the Gross-Pitaevskii equation for interacting bosons.

For a qubit in the Pauli basis, $X=(I_{2}+{\bf r}\cdot\bm{\sigma})/2\in{\rm
Her}^{\geq 0}_{1}({\cal H},{\mathbb{C}})$, ${\bf r}={\rm
tr}(X{\bm{\sigma}})\in{\mathbb{B}}_{1}[{\bf 0}]$, any Markovian PTP master
equation can be put in the form555Let
$dX/dt\\!=\\!Y(X)\\!=\\!\xi^{a}(X)\sigma^{a}=\xi^{a}({\bf r})\sigma^{a}$ where
each $\xi^{a}:{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}$ is a continuous
function of ${\bf r}$, which can be decomposed as $\xi^{a}({\bf
r})=2C^{\alpha}+2G^{ab}({\bf r})r^{b}=2C^{\alpha}+2L^{ab}r^{b}+2N^{ab}({\bf
r})r^{b}$, where $2C^{\alpha}$ captures any ${\bf r}$-independent part of
$\xi^{a}({\bf r})$.

$\displaystyle\frac{dX}{dt}=\frac{\sigma^{a}}{2}\bigg{(}\frac{dr^{a}}{dt}\bigg{)},\
\ \frac{dr^{a}}{dt}={\rm
tr}\bigg{(}\\!\frac{dX}{dt}\sigma^{a}\\!\bigg{)}\\!=G^{ab}({\bf
r})\,r^{b}+C^{a},\ \ G({\bf r})\in{\mathbb{R}}^{3\times 3},\ \
C^{a}\in{\mathbb{R}}^{3},$ (24)

where we sum over repeated indices $a,b\in(1,2,3)$. Here $G({\bf r})$ is a
state-dependent generator, which can be decomposed into linear and nonlinear
parts: $G^{ab}({\bf r})=L^{ab}+N^{ab}({\bf r})$. If $N^{ab}({\bf r})\\!=\\!0$,
(24) describes a general affine transformation on $X$ and ${\bf r}$ (strictly
linear if $C^{a}\\!=\\!0$). Every $G({\bf r})$ can be decomposed into
symmetric and antisymmetric components $G=G_{+}+G_{-}$, with $G_{\pm}:=(G\pm
G^{\top})/2$, which have distinct actions on the Bloch vector length:
$\frac{d}{dt}|{\bf r}|^{2}=2G^{ab}({\bf r})\,r^{a}r^{b}=2G^{ab}_{+}({\bf
r})\,r^{a}r^{b}$. Antisymmetric components $G_{-}$ conserve Bloch vector
length; they result from (possibly state-dependent) “unitary” transformations
$X\mapsto U(X)\,X\,U(X)^{\dagger}$. Linear antisymmetric generators correspond
to rigid rotations of the Bloch ball and result from strictly linear unitary
transformations on $X$. General symmetric generators $G_{+}({\bf r})$ can
amplify some qubit states, increasing their Bloch vector, while decreasing
others. Linear symmetric generators $G_{+}$ resulting from entropy-increasing
CPTP channels have nonpositive $G_{+}$ [see discussion following (21)] and
cannot increase $|{\bf r}|$. A process that increases (decreases) $|{\bf r}|$
is called amplifying (dissiptive). Note that channel characterization via
changes in $|{\bf r}|$ is not specifically sensitive to nonlinearity. Next we
consider a geometric characterization of the evolution: The divergence of the
qubit velocity field is

$\displaystyle\nabla\cdot(d{\bf r}/dt)={\rm tr}[G_{+}({\bf
r})]+r^{b}\partial_{a}G^{ab}({\bf r}),$ (25)

which has contributions from both symmetric and nonlinear generators, and can
take either sign. By contrast, the divergence is nonpositive in linear CPTP
channels (because $G_{+}\preceq 0$). The vorticity

$\displaystyle{\bm{\omega}}=\nabla\times(d{\bf r}/dt),\ \
\omega^{a}=\varepsilon^{abc}\partial_{b}[G^{cd}({\bf
r})r^{d}]=\varepsilon^{abc}G^{cb}({\bf
r})+\varepsilon^{abc}[\partial_{b}G^{cd}({\bf r})]\,r^{d}$ (26)

also has linear and nonlinear contributions ($\varepsilon$ is the Levi-Civita
symbol). The $\varepsilon^{abc}G^{cb}({\bf r})$ term will contribute if
$G({\bf r})\in{\mathbb{R}}^{3\times 3}$ has an antisymmetric ($|{\bf
r}|$-conserving) part. The divergence and vorticity characterize the velocity
field, but don’t fully expose the computational benefits of nonlinearity. This
is because the velocity field describes how single states $X_{\alpha}$ follow
their streamlines, but does not directly convey the relative motion between
evolving states. For a more sensitive characterization we want to consider how
pairs of states $(X_{\alpha},X_{\beta})$ transform under the channel. To
further motivate this, consider a common setting for quantum algorithms, where
a subroutine accepts as input a sequence of quantum states
$(X_{1},X_{2},X_{3},\cdots)$, then applies the same channel $\Lambda$ to each
in order to learn something about those states or compute some function of
those states. For example, we might know that the states can only take values
from a given set $\\{Y_{1},Y_{2},\cdots\\}$, and we want to identify which.
Previous authors Mielnik (1980); Abrams and Lloyd (1998); Bechmann-Pasquinucci
_et al._ (1998); Aaronson ; Childs and Young (2016) have noted the intriguing
computational power afforded by the ability to increase the distinguishability
between a pair of potential inputs $X_{\alpha}$ and $X_{\beta}$, i.e., to
increase their trace distance $\|X_{\alpha}-X_{\beta}\|_{1}$, which is
prohibited in linear CPTP channels. Let’s examine this for a qubit in the
Pauli basis: The differential of $\|X\|_{p}:=[{\rm
tr}(|X|^{p})]^{\frac{1}{p}}$ for any square matrix $X$ is
$d\|X\|_{p}=\|X\|_{p}^{1-p}\ {\rm tr}(|X|^{p-1}d|X|).$ Now let
$X=X_{\alpha}-X_{\beta}=\frac{1}{2}({\bf r}_{\alpha}-{\bf
r}_{\beta})\cdot{\bm{\sigma}}$ be the difference between a pair of qubit
states with Bloch vectors ${\bf r}_{\alpha,\beta}\in{\mathbb{B}}_{1}[{\bf 0}]$
and separation $\|X_{\alpha}-X_{\beta}\|_{p}=2^{\frac{1}{p}-1}\,|{\bf
r}_{\alpha}-{\bf r}_{\beta}|$ measured in Schatten norm. Then

$\displaystyle\frac{d}{dt}\|X_{\alpha}-X_{\beta}\|_{p}=2^{\frac{1}{p}-1}\,\frac{{\bf
r}_{\alpha}-{\bf r}_{\beta}}{|{\bf r}_{\alpha}-{\bf
r}_{\beta}|}\cdot\bigg{(}\frac{d{\bf r}_{\alpha}}{dt}-\frac{d{\bf
r}_{\beta}}{dt}\bigg{)}\leq 2^{\frac{1}{p}-1}\bigg{|}\frac{d{\bf
r}_{\alpha}}{dt}-\frac{d{\bf r}_{\beta}}{dt}\bigg{|}$ (27)

characterizes the expansivity of the channel:
$\frac{d}{dt}\|X_{\alpha}-X_{\beta}\|_{p}<0$ means that the channel is
strictly contractive on the pair, $\frac{d}{dt}\|X_{\alpha}-X_{\beta}\|_{p}=0$
means it’s distance preserving on the pair, and
$\frac{d}{dt}\|X_{\alpha}-X_{\beta}\|_{p}>0$ means it’s expansive. Expansivity
allows for the distance between two nearby states $(X_{\alpha},X_{\beta})$ to
increase. In the notation of (24) the rate of change of state separation is

$\displaystyle\frac{d}{dt}\|X_{\alpha}-X_{\beta}\|_{p}=2^{\frac{1}{p}-1}\,\frac{r^{a}_{\alpha}-r^{a}_{\beta}}{|{\bf
r}_{\alpha}-{\bf r}_{\beta}|}\bigg{[}G^{ab}({\bf
r}_{\alpha})\,r_{\alpha}^{b}-G^{ab}({\bf r}_{\beta})\,r_{\beta}^{b}\bigg{]}.$
(28)

Expanding about the midpoint ${\bf R}=({\bf r}_{\alpha}+{\bf r}_{\beta})/2$
gives, to second order in $|{\bf r}_{\alpha}-{\bf r}_{\beta}|$,

$\displaystyle\frac{d}{dt}\|X_{\alpha}-X_{\beta}\|_{p}=2^{\frac{1}{p}-1}\,\frac{[G^{ab}({\bf
R})\\!+\\!K^{ab}({\bf
R})](r^{a}_{\alpha}\\!-\\!r^{a}_{\beta})(r^{b}_{\alpha}\\!-\\!r^{b}_{\beta})}{|{\bf
r}_{\alpha}-{\bf r}_{\beta}|},\ \ K^{ab}({\bf
R}):=R^{c}\,\partial_{b}G^{ac}({\bf R}).\ \ \ \ \ \ $ (29)

We note the two distinct sources of expansivity in (29): Any antisymmetric
part of $G({\bf R})$, if present, doesn’t contribute to the expansivity, but
positive eigenvalues in the symmetric part do. This is an alternative
expression of the same results we found above for $d|{\bf r}|^{2}/dt$, and for
the ${\rm tr}[G_{+}({\bf r})]$ term in the divergence. The second term
contributes to expansivity if the symmetric part of the matrix
$K\in{\mathbb{R}}^{3\times 3}$ has positive eigenvalues.

In the remainder of this section we apply this geometric characterization to a
state-dependent CPTP channel with torsion Abrams and Lloyd (1998); Childs and
Young (2016),

$\displaystyle\frac{dr^{a}}{dt}=G^{ab}\\!({\bf r})\,r^{b}\\!,\ \ G^{ab}({\bf
r})=gzJ_{z}^{ab},\ \ J_{z}=\begin{pmatrix}0&-1&0\\\ 1&0&0\\\ 0&0&0\\\
\end{pmatrix}\\!,\ \ g\in{\mathbb{R}}.$ (30)

Here $J_{z}$ is an SO(3) generator. $G({\bf r})\in{\mathbb{R}}^{3\times 3}$ is
antisymmetric and hence $|{\bf r}|$-preserving. $G({\bf r})$ generates $z$
rotations with a rate that increases linearly with Bloch coordinate $z$,
changing direction for $z<0$, a type of twist. The divergence (25) vanishes
everywhere and the flow is incompressible. The vorticity (26) is
${\bm{\omega}}=(-x,-y,2z)g$. The $z$ component $\omega^{3}$ describes rigid
body rotation within each plane of constant $z$, with a $z$-dependent
frequency, while $\omega^{1}$ and $\omega^{2}$ reflect the associated shear.
We can use (29) to discover expansive trajectories: In the torsion model (30),
the matrix $K^{ab}({\bf R})$ defined in (29) is

$\displaystyle K=\frac{g}{2}\begin{pmatrix}0&0&-(y_{\alpha}+y_{\beta})\\\
0&0&(x_{\alpha}+x_{\beta})\\\ 0&0&0\\\ \end{pmatrix},\ \
K_{+}=\frac{g}{4}\begin{pmatrix}0&0&-(y_{\alpha}+y_{\beta})\\\
0&0&(x_{\alpha}+x_{\beta})\\\
-(y_{\alpha}+y_{\beta})&(x_{\alpha}+x_{\beta})&0\\\ \end{pmatrix}.$ (31)

$K_{+}$ has eigenvalues 0 and
$\pm(|g|/4)\sqrt{(x_{\alpha}+x_{\beta})^{2}+(y_{\alpha}+y_{\beta})^{2}}$. For
a pair of nearby states $X_{\alpha},X_{\beta}$, their difference ${\bf
r}_{\alpha}-{\bf r}_{\beta}$ is a short vector located at midpoint position
${\bf R}=({\bf r}_{\alpha}+{\bf r}_{\beta})/2$. Expansive trajectories occur
when $K^{ab}({\bf
R})(r^{a}_{\alpha}-r^{a}_{\beta})(r^{b}_{\alpha}-r^{b}_{\beta})=\partial_{b}G^{ac}({\bf
R})R^{c}(r^{a}_{\alpha}-r^{a}_{\beta})(r^{b}_{\alpha}-r^{b}_{\beta})$ is
positive. In the torsion model this condition simplifies to

$\displaystyle
g\big{[}R^{x}(y_{\alpha}-y_{\beta})-R^{y}(x_{\alpha}-x_{\beta})\big{]}(z_{\alpha}-z_{\beta})>0.$
(32)

Let ${\bf r}_{\alpha}=(\frac{1}{2},\frac{\eta_{y}}{2},\frac{\eta_{z}}{2})$ and
${\bf r}_{\beta}=(\frac{1}{2},-\frac{\eta_{y}}{2},-\frac{\eta_{z}}{2})$ be a
pair of states with midpoint position ${\bf R}=(\frac{1}{2},0,0)$ along the
positive $x$ axis. The states are separated by $\eta_{y}$ in the $y$ direction
and $\eta_{z}$ in the $z$ direction. For nonzero $\eta_{y}$ and $\eta_{z}$,
the two states move in opposite directions and separate at a rate
$\frac{d}{dt}\|X_{\alpha}-X_{\beta}\|_{p}=2^{\frac{1}{p}-2}g(\eta_{y}\eta_{z}/\\!\sqrt{\eta_{y}^{2}+\eta_{z}^{2}})$.

###  II.4 General normalized PTP channels

Next we discuss channels with nonlinear positive $\phi$ and ${\rm
tr}[\phi(X)]\neq 1\ {\rm for\ some}\ X\in{\rm Her}^{\geq 0}_{1}({\cal
H},{\mathbb{C}})$, the most general PTP channels considered here. This class
combines the nonunitary features of the NINO channels with the nonlinearity of
state-dependent CPTP channels. Suppose we want to add linear
dissipation/amplification to the torsion model (30) by adding a linear part
$G$ to the generator. What are the allowed values of $G$? To answer this
question, we use generators from the NINO master equation (21), namely
$dX/dt=[L_{-},X]+\\{L_{+},X\\}+\zeta_{2}BXB^{\dagger}$, where we have included
one jump operator $B$ and one nonjump operator $L$.666Here we assume that
$\zeta_{1}\\!=\\!1$; the normalization condition on the $z_{\alpha}$ is then
satisfied with $z_{1}=1$. In the Pauli basis the first term in $dX/dt$ leads
to $dr^{a}/dt=G^{ab}r^{b}$, with $G^{ab}={\rm
tr}(\sigma^{a}L_{-}\sigma^{b}-\sigma^{b}L_{-}\sigma^{a})/2$, resulting in an
antisymmetric contribution to $G$. To see its connection with unitary
dynamics, expand $L_{-}=-L_{-}^{\dagger}$ in the Pauli basis as
$L_{-}=i(\xi_{0}I+\xi_{a}\sigma^{a})$, where
$\xi_{0},\dots,\xi_{3}\in{\mathbb{R}}$ are real coordinates for $L_{-}$. In
this basis $G^{ab}=2\varepsilon^{abc}\xi_{c}=-i\,{\rm
tr}(L_{-}\sigma^{c})\,\varepsilon^{abc}$, a real but otherwise arbitrary
linear combination of SO(3) generators. Any linear antisymmetric $G$ can be
implemented by controlling these generators. Similarly, the $\\{L_{+},X\\}$
term leads to a real symmetric $G^{ab}={\rm
tr}(\sigma^{a}L_{+}\sigma^{b}+\sigma^{b}L_{+}\sigma^{a})/2$ plus an
inhomogeneous part $C^{a}={\rm tr}(\sigma^{a}L_{+})$. Expanding
$L_{+}=L_{+}^{\dagger}$ in the Pauli basis as
$L_{+}=\xi_{0}I+\xi_{a}\sigma^{a}$, where
$\xi_{0},\dots,\xi_{3}\in{\mathbb{R}}$ are again real, leads to a diagonal
matrix $G=2\xi_{0}I_{3}={\rm tr}(L_{+})I_{3}$. And the
$\zeta_{2}\,BXB^{\dagger}$ term leads to $G^{ab}=\zeta_{2}\,{\rm
tr}(\sigma^{a}B\sigma^{b}B^{\dagger})/2$ and $C^{a}=\zeta_{2}\,{\rm
tr}(\sigma^{a}BB^{\dagger})/2$ in the Pauli basis. Expanding
$B\in{\mathbb{C}}^{2\\!\times\\!2}$ as $B=\xi_{0}I+\xi_{a}\sigma^{a}$, with
$\xi_{0},\dots,\xi_{3}\in{\mathbb{C}}$ complex coordinates for $B$, we have

$\displaystyle
G^{ab}=\zeta_{2}\,\big{(}|\xi_{0}|^{2}-|\xi_{1}|^{2}-|\xi_{2}|^{2}-|\xi_{3}|^{2}\big{)}\delta^{ab}+2\zeta_{2}\,{\rm
Im}(\xi^{*}_{0}\xi_{c})\varepsilon^{abc}+2\zeta_{2}\,{\rm
Re}(\xi^{*}_{a}\xi_{b}).$ (33)

The first term is diagonal. The second term is antisymmetric (both $L_{-}$ and
$B$ contribute to unitary evolution if this term is nonzero). The third term
is symmetric. Let $\xi_{0}=0$; then

$\displaystyle
G=-\zeta_{2}\,(|\xi_{1}|^{2}+|\xi_{2}|^{2}+|\xi_{3}|^{2})I+2\zeta_{2}\,{\rm
Re}\\!\left[\\!\begin{pmatrix}\xi^{*}_{1}\\\ \xi^{*}_{2}\\\
\xi^{*}_{3}\end{pmatrix}\\!\\!\otimes\\!\\!\begin{pmatrix}\xi_{1}&\xi_{2}&\xi_{3}\end{pmatrix}\\!\right],$
(34)

where $I$ is the identity. Consider now a pair of jump operators with the same
$\zeta_{2}$ and coordinates $\xi=(1,1,0)$ and $(0,0,1)$:

$\displaystyle G_{(1,1,0)}=\zeta_{2}\begin{pmatrix}0&2&0\\\ 2&0&0\\\
0&0&-2\end{pmatrix},\ \ G_{(0,0,1)}=\zeta_{2}\begin{pmatrix}-1&0&0\\\
0&-1&0\\\ 0&0&1\end{pmatrix}.$ (35)

Combining them gives

$\displaystyle G_{(1,1,0)}+G_{(0,0,1)}=\zeta_{2}(2\lambda_{1}-I),\ \
\lambda_{1}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\ 0&0&0\end{pmatrix}\\!,$ (36)

where $\lambda_{1}$ is a Gell-Mann matrix. Similarly,
$G_{(1,0,1)}+G_{(0,1,0)}=\zeta_{2}(2\lambda_{4}-I)$ and
$G_{(0,1,1)}+G_{(1,0,0)}=\zeta_{2}(2\lambda_{6}-I)$, where $\lambda_{4}$ and
$\lambda_{6}$ are Gell-Mann matrices. By combining Hamiltonian control with
jump operator engineering, a set of linear generators $G$ can be implemented.

##  III Fault-tolerant nonlinear state discrimination

In the remainder of the paper we consider an extension of the qubit torsion
channel (30) that includes linear dissipation and amplification. Using the
techniques of Sec II.4, jump operators are chosen such that

$\displaystyle\frac{dX}{dt}=\frac{\sigma^{a}}{2}\bigg{(}\frac{dr^{a}}{dt}\bigg{)},\
\ \frac{dr^{a}}{dt}={\rm
tr}\bigg{(}\\!\frac{dX}{dt}\sigma^{a}\\!\bigg{)}\\!=G^{ab}({\bf
r})\,r^{b}=(m\,\lambda_{4}-\gamma I+gzJ_{z})^{ab}r^{b}\\!,$ (37)

where $I$ is the $3\\!\times\\!3$ identity,

$\lambda_{4}=\begin{pmatrix}0&0&1\\\ 0&0&0\\\ 1&0&0\\\ \end{pmatrix}\\!,\ \
{\rm and}\ \ J_{z}=\begin{pmatrix}0&-1&0\\\ 1&0&0\\\ 0&0&0\\\
\end{pmatrix}\\!.$ (38)

Here $\lambda_{4}$ is an SU(3) generator, $J_{z}$ is an SO(3) generator, and
we sum over repeated indices $a,b\in(1,2,3)$. The dimensionless model
parameters $m$, $\gamma$, and $g$ are real variables of either sign. The
fixed-point equations are

$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle mz-\gamma
x-gyz=0,$ (39) $\displaystyle\frac{dy}{dt}$ $\displaystyle=$
$\displaystyle-\gamma y+gxz=0,$ (40) $\displaystyle\frac{dz}{dt}$
$\displaystyle=$ $\displaystyle mx-\gamma z=0.$ (41)

The origin is always a fixed point, ${\bf r}^{\rm fp}_{0}=(0,0,0),$ although
not always stable. Assuming ${\bf r}\neq(0,0,0)$, $\gamma\neq 0$, and
eliminating $z$, the fixed point equations are

$\displaystyle m^{2}-\gamma^{2}=gmy,$ (42) $\displaystyle\gamma^{2}y=gmx^{2}.$
(43)

If $g=0$, any fixed points must be confined to the $y=0$ plane. There are no
additional fixed points unless $\gamma=\pm m$, in which case there is a set of
fixed points ${\bf r}^{{\rm fp},g=0}_{z=\pm x}$ on the line ${\bf r}_{z=\pm
x}=\\{(x,0,\pm x):x\in{\mathbb{R}}\\}$. The settings $\gamma=\pm m$ are
singular lines in the parameter space of the model. Manipulating these
singularities in the presence of nonlinearity is the key to engineering useful
information processing.

When $g>0$, any fixed points must be confined to the plane
$my=(m^{2}-\gamma^{2})/g$. However (43) requires $y$ to have the same sign as
that of $m$. Therefore $my>0$, which is only possible when $m^{2}>\gamma^{2}$.
Therefore, when $m^{2}<\gamma^{2}$, the only fixed point is ${\bf r}^{\rm
fp}_{0}$, and this fixed point is stable for all $m^{2}<\gamma^{2}$. If
instead the condition $m^{2}>\gamma^{2}$ is satisfied, and $g>0$, there is a
pair of stable fixed points at

$\displaystyle{\bf r}^{\rm
fp}_{\pm}=\bigg{(}\\!\pm\frac{|\gamma|}{g}\sqrt{\delta},\ \frac{m}{g}\delta,\
\pm\,{\rm sign}(\gamma)\frac{m}{g}\sqrt{\delta}\bigg{)},\ \
\delta:=\frac{m^{2}-\gamma^{2}}{m^{2}}\in(0,\infty].$ (44)

For these fixed points to be contained within the Bloch ball requires
$|g|>g_{\rm min}$, where $g_{\rm
min}=\sqrt{(\gamma^{2}+m^{2})\delta+m^{2}\delta^{2}}$. The dynamics between
fixed points ${\bf r}^{\rm fp}_{-},{\bf r}_{0}^{\rm fp}$, and ${\bf r}^{\rm
fp}_{+}$ can be understood as follows: When $g=0$ we have

$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle mz-\gamma x,$
(45) $\displaystyle\frac{dy}{dt}$ $\displaystyle=$ $\displaystyle-\gamma y,$
(46) $\displaystyle\frac{dz}{dt}$ $\displaystyle=$ $\displaystyle mx-\gamma
z.$ (47)

Note that the $y$ motion is decoupled from $x$ and $z$, and that it is always
stable for $\gamma>0$. Furthermore, the linearized model has an additional
symmetry which becomes explicit after changing variables to $\xi_{\pm}=(z\pm
x)/2$:

$\displaystyle\frac{d\xi_{+}}{dt}$ $\displaystyle=$
$\displaystyle(m-\gamma)\xi_{+},$ (48) $\displaystyle\frac{d\xi_{-}}{dt}$
$\displaystyle=$ $\displaystyle-(m+\gamma)\xi_{-}.$ (49)

The $\xi_{+}$ and $\xi_{-}$ variables are also decoupled. $\xi_{+}$ is the
coordinate along the line $z=x$ mentioned above, and $\xi_{-}$ is the
coordinate along the perpendicular line $z=-x$. Motion in the $\xi_{+}$
direction is stable for $m<\gamma$; in this case each point on the line $z=x$
flows to the fixed point ${\bf r}_{0}^{\rm fp}$ at the origin. However the
$\xi_{+}$ motion becomes unstable when $m>\gamma$. In this regime ${\bf
r}_{0}^{\rm fp}$ is unstable, and each point on the line $z=x$ (other than
$z\\!=\\!x\\!=\\!0$) flows outward to infinity. We can interpret this unstable
case as having two stable fixed points at $(\infty,0,\infty)$ and
$(-\infty,0,-\infty)$, at the ends of the line $z=x$. By contrast, close to
the singularity at $m=\gamma$, the perpendicular $\xi_{-}$ motion is stable
unless $m$ and $\gamma$ are both negative. In this picture, the most important
effect of the nonlinearity is to move the two stable fixed points at infinity
to the finite positions (44). This is illustrated in Fig. 1. In Fig. 2 we plot
the trajectories for a cloud of randomly chosen initial states (red dots)
within the Bloch ball ${\mathbb{B}}_{1}[{\bf 0}]$, close to the bifurcation
but in the $m^{2}<\gamma^{2}$ phase. In Fig. 3 we show the same plot in the
$m^{2}>\gamma^{2}$ phase. These simulations further support the picture
described above.

Figure 1: Illustration of the dynamics in the neighborhood of the origin for
$m,\gamma\geq 0$. When $m<\gamma$, two unstable fixed points at infinity (red)
feed the stable fixed point ${\bf r}^{\rm fp}_{0}$ at the origin (black).
However, if $m>\gamma$, ${\bf r}^{\rm fp}_{0}$ is unstable (red). The
$\xi_{-}$ axis (green) is the separatrix. ${\bf r}^{\rm fp}_{0}$ feeds two new
stable fixed points ${\bf r}^{\rm fp}_{+,-}$ (black). At the critical point
$m=\gamma$, all points on the $\xi_{+}$ axis (blue dots) are fixed points.

The $\xi_{+}$ dynamics near the unstable fixed point ${\bf r}_{0}^{\rm fp}$
can be used to achieve robust state discrimination with the exponential
speedup supported by expansive nonlinear channels Mielnik (1980); Abrams and
Lloyd (1998); Bechmann-Pasquinucci  _et al._ (1998); Aaronson ; Childs and
Young (2016). Points very close to ${\bf r}^{\rm fp}_{+}$ have $|{\bf r}|\ll
1$, so the nonlinearity can be neglected there. Equations (48) and (49) then
apply to the dynamics near ${\bf r}_{0}^{\rm fp}$ even when $g\neq 0$.
Consider the plane passing through the origin and perpendicular to the
$\xi_{+}$ axis. The velocity field smoothly changes sign across this plane;
i.e., it is a separatrix between basins of attraction for ${\bf r}^{\rm
fp}_{+}$ and ${\bf r}^{\rm fp}_{-}$. Suppose that a qubit is prepared in a
state from the set $\\{X_{\alpha},X_{\beta}\\}$, with $X_{\alpha}$ and
$X_{\beta}$ close in trace distance $\epsilon=\|X_{\alpha}-X_{\beta}\|_{1}$
but on opposite sides of the separatrix. Then we implement a gate by turning
on the nonlinearity for a time $t=O(1/g)$, after which $X_{\alpha}$ and
$X_{\beta}$ flow to different fixed points. After this evolution, the
nonlinearity is turned off and the qubit is measured. This nonlinear gate
would lead to an exponential speedup if the initial separation is
exponentially small: $\epsilon=2^{-k}$ Abrams and Lloyd (1998); Aaronson ;
Childs and Young (2016).

Positivity of the dissipative torsion channel requires that the Bloch vector
${\bf r}$ remain in the Bloch ball ${|\bf r}|\leq 1$. However the master
equation (37) does not itself enforce this condition. Therefore the positivity
condition must be implemented dynamically through control of the qubit
Hamiltonian, and trajectories leaving the Bloch ball are regarded as
unphysical.

Figure 2: Simulation solutions of the torsion channel (37) with $\gamma=1$,
$m=0.9$, and $g=1$, showing attraction to ${\bf r}^{\rm fp}_{0}$. The $x$,
$y$, and $z$ axes are Bloch vector coordinates. Red dots indicate random
initial conditions. The Bloch sphere is outlined in yellow. Figure 3:
Simulation solutions to (37) with $\gamma=1$, $m=1.1$, $g=1$, and
$\delta=0.2$, showing attraction to ${\bf r}^{\rm fp}_{\pm}$.

##  IV Discussion

Although this gate is appealing, it has limitations:

  1. (i)

First, finite experimental resolution and control will limit the smallest
values of $\epsilon$ achievable in practice. If the inputs to the
discriminator are the outputs of a preceding process, they will also come with
errors.

  2. (ii)

Second, the fixed points ${\bf r}^{\rm fp}_{+,-}$ are not perfectly
distinguishible. Although there is considerable flexibility in choosing their
location, in practice they need to be well within the Bloch ball to ensure
positivity. If the ${\bf r}^{\rm fp}_{+,-}$ are too close to the surface
$|{\bf r}|=1$, trajectories approaching them may lead to unphysical solutions
leaving the Bloch ball.

  3. (iii)

And third, there will likely be errors associated with the effective model
itself.

In conclusion, we have discussed three classes of nonlinear PTP channels and
explored their computational power. Engineering both nonlinearity and
dissipation allows one to implement rich dynamics similar to that of classical
nonlinear systems. We identified a bifurcation to a phase where the Bloch ball
separates into two basins of attraction, which can be used to implement fast
quantum state discrimination Abrams and Lloyd (1998); Aaronson ; Childs and
Young (2016) with intrinsic fault-tolerance.

## Acknowledgements

It is a pleasure to thank Andrew Childs for correspondence.

## References

  * Mielnik (1980) B. Mielnik, “Mobility of nonlinear systems,” J. Math. Phys. 21, 44 (1980).
  * Abrams and Lloyd (1998) D. S. Abrams and S. Lloyd, “Nonlinear quantum mechanics implies polynomial-time solution for NP-Complete and $\\#$P problems,” Phys. Rev. Lett. 81, 3992–3995 (1998).
  * Bechmann-Pasquinucci  _et al._ (1998) H. Bechmann-Pasquinucci, B. Huttner, and N. Gisin, “Nonlinear quantum state transformation of spin-1/2,” Phys. Lett. A 242, 198 (1998).
  * Czachor (1998) M. Czachor, “Notes on nonlinear quantum algorithms,” Acta Phys. Slov. 48, 157 (1998).
  * (5) M. Czachor, “Local modification of the Abrams-Lloyd nonlinear algorithm,” quant-ph/9803019.
  * Bacon (2004) D. Bacon, “Quantum computational complexity in the presence of closed timelike curves,” Phys. Rev. A 70, 032309 (2004).
  * (7) S. Aaronson, “NP-complete problems and physical reality,” arXiv: quant-ph/0502072.
  * Brun  _et al._ (2009) T. A. Brun, J. Harrington, and M. M. Wilde, “Localized closed timelike curves can perfectly distinguish quantum states,” Phys. Rev. Lett. 102, 210402 (2009).
  * Bennett  _et al._ (2009) C. H. Bennett, D. Leung, G. Smith, and J. A. Smolin, “Can closed timelike curves or nonlinear quantum mechanics improve quantum state discrimination or help solve hard problems?” Phys. Rev. Lett. 103, 170502 (2009).
  * Kahou and Feder (2013) M. E. Kahou and D. L. Feder, “Quantum search with interacting Bose-Einstein condensates,” Phys. Rev. A 88, 032310 (2013).
  * Meyer and Wong (2013) D. A. Meyer and T. G. Wong, “Nonlinear quantum search using the Gross-Pitaevskii equation,” New J. Phys. 15, 063014 (2013).
  * Meyer and Wong (2014) D. A. Meyer and T. G. Wong, “Quantum search with general nonlinearities,” Phys. Rev. A 89, 012312 (2014).
  * Childs and Young (2016) A. M. Childs and J. Young, “Optimal state discrimination and unstructured search in nonlinear quantum mechanics,” Phys. Rev. A 93, 022314 (2016).
  * (14) G. Di Molfetta and B. Herzog, “Searching via nonlinear quantum walk on the 2D-grid,” arXiv: 2009.07800.
  * Bollinger  _et al._ (1989) J. J. Bollinger, D. J. Heinzen, W. M. Itano, S. L. Gilbert, and D. J. Wineland, “Test of the linearity of quantum mechanics by rf spectroscopy of the 9Be+ ground state,” Phys. Rev. Lett. 63, 1031 (1989).
  * Chupp and Hoare (1990) T. E. Chupp and R. J. Hoare, “Coherence in freely precessing ${}^{21}\mathrm{Ne}$ and a test of linearity of quantum mechanics,” Phys. Rev. Lett. 64, 2261 (1990).
  * Walsworth  _et al._ (1990) R. L. Walsworth, I. F. Silvera, E. M. Mattison, and R. F. C. Vessot, “Test of the linearity of quantum mechanics in an atomic system with a hydrogen maser,” Phys. Rev. Lett. 64, 2599 (1990).
  * Majumder  _et al._ (1990) P. K. Majumder, B. J. Venema, S. K. Lamoreaux, B. R. Heckel, and E. N. Fortson, “Test of the linearity of quantum mechanics in optically pumped 201Hg,” Phys. Rev. Lett. 65, 2931 (1990).
  * Kibble (1978) T. W. B. Kibble, “Relativistic models of nonlinear quantum mechanics,” Comm. Math. Phys. 64, 73 (1978).
  * Weinberg (1989) S. Weinberg, “Precision tests of quantum mechanics,” Phys. Rev. Lett. 62, 485 (1989).
  * Weinberg (1998) S. Weinberg, “Testing quantum mechanics,” Ann. Phys. 194, 336 (1998).
  * Gisin (1989) N. Gisin, “Stochastic quantum dynamics and relativity,” Helv. Phys. Acta 62, 363–71 (1989).
  * Gisin (1990) N. Gisin, “Weinberg’s nonlinear quantum mechanics and supraluminal communication,” Phys. Lett. A 143, 1 (1990).
  * Polchinski (1991) J. Polchinski, “Weinberg’s nonlinear quantum mechanics and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. Lett. 66, 397 (1991).
  * Czachor (1996) M. Czachor, “Nonlinear Schrödinger equation and two-level atoms,” Phys. Rev. A 53, 1310 (1996).
  * Czachor (1995) M. Czachor, “Elements of nonlinear quantum mechanics (ii): Triple bracket generalization of quantum mechanics,” (1995), quant-ph/9501008.
  * Gisin and Rigo (1995) N. Gisin and M Rigo, “Relevant and irrelevant nonlinear Schrodinger equations,” J. Phys. A 28, 7375 (1995).
  * Czachor and Kuna (1998) M. Czachor and M. Kuna, “Complete positivity of nonlinear evolution: A case study,” Phys. Rev. A 58, 128 (1998).
  * (29) M. Kuna, M. Czachor, and S. B. Lebl, “Nonlinear von Neumann-type equations: Darboux invariance and spectra,” arXiv: 9810023\.
  * Mielnik (2001) B. Mielnik, “Nonlinear quantum mechanics: A conflict with the Ptolomean structure?” Phys. Lett. A 289, 1 (2001).
  * Czachor and Doebner (2002) M. Czachor and H.-D. Doebner, “Correlation experiments in nonlinear quantum mechanics,” Phys. Lett. A 301, 139 (2002).
  * Kent (2005) A. Kent, “Nonlinearity without superluminality,” Phys. Rev. A 72, 012108 (2005).
  * Jordan (2006) T. F. Jordan, “Assumptions that imply quantum dynamics is linear,” Phys. Rev. A 73, 022101 (2006).
  * (34) T. F. Jordan, “Why quantum dynamics is linear,” arXiv: 0702171.
  * Jordan (2010) T. F. Jordan, “Fundamental significance of tests that quantum dynamics is linear,” Phys. Rev. A 82, 032103 (2010).
  * (36) C. M. Ho and S. D. H. Hsu, “Locality and nonlinear quantum mechanics,” arXiv: 1401.7018.
  * Bassi and Hejazi (2015) A. Bassi and K. Hejazi, “No faster than light signaling implies linear evolution: A rederivation,” Eur. J. Phys. 36, 055027 (2015).
  * (38) L. Diósi, “Nonlinear Schrödinger equation in foundations: Summary of 4 catches,” arXiv: 1602.03772.
  * Rembieliński and Caban (2020a) J. Rembieliński and P. Caban, “Nonlinear evolution and signaling,” Phys. Rev. Res. 2, 012027 (2020a).
  * (40) A. Budiyono and H. K. Dipojono, “Nonlinear Schrödinger equations and generalized Heisenberg uncertainty principle violating the principle of estimation independence,” arXiv: 2009.06422.
  * (41) D. E. Kaplan and S. Rajendran, “A causal framework for nonlinear quantum mechanics,” arXiv: 2106.10576.
  * Alicki and Messer (1983) R. Alicki and J. Messer, “Nonlinear quantum dynamical semigroups for many-body open systems,” J. Stat. Phys. 32, 299 (1983).
  * Man’ko and Marmo (1992) V. I. Man’ko and G. Marmo, “Generalized reduction procedure and nonlinear nonstationary dynamical systems,” Mod. Phys. Lett. A 7, 3411 (1992).
  * Ciaglia  _et al._ (2019) F. M. Ciaglia, F. Di Cosmo, A. Figueroa, V. I. Man’ko, G. Marmo, L. Schiavone, F. Ventriglia, and P. Vitale, “Nonlinear dynamics from linear quantum evolutions,” Ann. Phys. 411, 167957 (2019).
  * (45) B. Drossel, “What condensed matter physics and statistical physics teach us about the limits of unitary time evolution,” arXiv: 1908.10145.
  * Fernengel and Drossel (2020) B. Fernengel and B. Drossel, “Bifurcations and chaos in nonlinear Lindblad equations,” J. Phys. A: Math. Theor. 53, 385701 (2020).
  * Breuer and Petruccione (2002) H.-P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_ (Oxford University Press, 2002).
  * Gisin (1981) N. Gisin, “A simple nonlinear dissipative quantum evolution equation,” J. Phys. A: Math. Gen. 14, 2259 (1981).
  * Gisin and Cibils (1992) N. Gisin and M. B. Cibils, “Quantum diffusions, quantum dissipation and spin relaxation,” J. Phys. A 25, 5165 (1992).
  * Kowalski and Rembieliński (2019) K. Kowalski and J. Rembieliński, “Integrable nonlinear evolution of the qubit,” Ann. Phys. 411, 167955 (2019).
  * Kowalski (2020) K. Kowalski, “Linear and integrable nonlinear evolution of the qutrit,” Quant. Inf. Proc. 19, 145 (2020).
  * Fonseca Romero  _et al._ (2004) K. M. Fonseca Romero, P. Talkner, and P. Hänggi, “Is the dynamics of open quantum systems always linear?” Phys. Rev. A 69, 052109 (2004).
  * Rembieliński and Caban (2020b) J. Rembieliński and P. Caban, “Nonlinear extension of the quantum dynamical semigroup,” Quantum 5, 420 (2020b), arXiv: 2003.09170.
  * Gao  _et al._ (2015) T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Höfling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554 (2015).
  * (55) P. A. Kalozoumis and D. Petrosyan, “Self-organized PT-symmetry of exciton-polariton condensate in a double-well potential,” arXiv: 2107.01675.
  * Lloyd and Slotine (2000) S. Lloyd and J.-J. E. Slotine, “Quantum feedback with weak measurements,” Phys. Rev. A 62, 012307 (2000).
  * Kiss  _et al._ (2006) T. Kiss, I. Jex, G. Alber, and Vymětal, “Complex chaos in the conditional dynamics of qubits,” Phys. Rev. A 74, 040301 (2006).
  * Kiss  _et al._ (2011) T. Kiss, S. Vymětal, L. D. Tóth, A. Gábris, I. Jex, and G. Alber, “Measurement-induced chaos with entangled states,” Phys. Rev. Lett. 107, 100501 (2011).
  * Muñoz-Arias  _et al._ (2020) M. H. Muñoz-Arias, P. M. Poggi, P. S. Jessen, and I. H. Deutsch, “Simulating nonlinear dynamics of collective spins via quantum measurement and feedback,” Phys. Rev. Lett. 124, 110503 (2020).
  * Sudarshan  _et al._ (1961) E. C. G. Sudarshan, P. M. Mathews, and Rau J., “Stochastic dynamics of quantum-mechanical systems,” Phys. Rev. 121, 920 (1961).
  * Kraus (1971) K. Kraus, “General state changes in quantum theory,” Ann. Phys. 64, 311 (1971).
  * Choi (1975) M.-D. Choi, “Completely positive linear maps on complex matrices,” Lin. Alg. App. 10, 285 (1975).
  * Ando and Choi (1986) T. Ando and M.-D. Choi, “Nonlinear completely positive maps,” in  _Aspects of Positivity in Functional Analysis_ , edited by R. Nagel, U. Schlotterbeck, and M. P. H. Wolff (Elsevier, 1986).
  * Pechukas (1994) P. Pechukas, “Reduced dynamics need not be completely positive,” Phys. Rev. Lett. 73, 1060 (1994).
  * Shaji and Sudarshan (2005) A. Shaji and E. C. G. Sudarshan, “Who’s afraid of not completely positive maps?” Phys. Lett. A 341, 48 (2005).
  * Carteret  _et al._ (2008) H. A. Carteret, D. R. Terno, and K. Życzkowski, “Dynamics beyond completely positive maps: Some properties and applications,” Phys. Rev. A 77, 042113 (2008).
  * Dominy  _et al._ (2016) J. M. Dominy, A. Shabani, and D. A. Lidar, “A general framework for complete positivity,” Quantum Inf. Process. 15, 465 (2016).
  * Peres (1996) A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. 77, 1413 (1996).
  * Gregoratti and Werner (2004) M. Gregoratti and R. F. Werner, “On quantum error correction by classical feedback in discrete time,” J. Math. Phys. 45, 2600 (2004).
  * Hayden and King (2005) P. Hayden and C. King, “Correcting quantum channels by measuring the environment,” Quant. Inf. Comp. 5, 156 (2005).
  * (71) A. Winter, “On environment-assisted capacities of quantum channels,” arXiv:quant-ph/0507045.
  * Gorini  _et al._ (1976) V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups of N-level systems,” J. Math. Phys. 17, 821 (1976).
  * Lindblad (1976) G. Lindblad, “On the generators of quantum dynamical semigroups,” Comm. Math. Phys. 48, 119–30 (1976).