# Linear position measurements with minimum error-disturbance in each minimum
uncertainty state

Kazuya Okamura<EMAIL_ADDRESS>Research Origin for Dressed
Photon, 3-13-19 Moriya-cho Kanagawa-ku, Yokohama, 221-0022, Japan  Graduate
School of Informatics, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan

###### Abstract

In quantum theory, measuring process is an important physical process; it is a
quantum description of the interaction between the system of interest and the
measuring device. Error and disturbance are used to quantitatively check the
performance of the measurement, and are defined by using measuring process.
Uncertainty relations are a general term for relations that provide
constraints on them, and actively studied. However, the true error-disturbance
bound for position measurements is not known yet. Here we concretely construct
linear position measurements with minimum error-disturbance in each minimum
uncertainty state. We focus on an error-disturbance relation (EDR), called the
Branciard-Ozawa EDR, for position measurements. It is based on a quantum root-
mean-square (q-rms) error and a q-rms disturbance. We show the theorem that
gives a necessary and sufficient condition for a linear position measurement
to achieve its lower bound in a minimum uncertainty state, and explicitly give
exactly solvable linear position measurements achieving its lower bound in the
state. We then give probability distributions and states after the measurement
when using them. It is expected to construct measurements with minimum error-
disturbance in a broader class of states in the future, which will lead to a
new understanding of quantum limits, including uncertainty relations.

##  I Introduction

Quantum measurement theory plays a central role both in quantum foundations
and in quantum information science. As well-known, Heisenberg uncertainty
relations give inevitable limitations on measurements for quantum systems and
are treated as one of monumental achievements in early quantum mechanics. In
recent years, quantum measurement technology is rapidly developing and
actively used in various devices. It is now considered that quantum
measurements provide information processing with new computing methods, like
measurement-based quantum computation [1, 2, 3].

Universally valid reformulation of uncertainty relations is a currently
developing topic from theory [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] to
experiment [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. The violation of
the Heisenberg EDR is crucial in this context. A model of position measurement
violating the Heisenberg EDR, called the error-free linear measurement [28,
29], has already founded in the debate on the sensitivity limit of the
gravitational wave detector, called the standard quantum limit, in the 1980s
[30, 31, 32, 33]. This fact motivated Ozawa [4, 5, 6, 7] to formulate
universally valid uncertainty relations. Several experiments [16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27] verify universally valid uncertainty relations
and the violation of the Heisenberg EDR. The Branciard-Ozawa EDR treated in
the paper is a universally valid trade-off relation for the noise-operator
based q-rms error and the disturbance-operator based q-rms disturbance. It is
first proved by Branciard [9] for vector (or pure) states, and is extended to
mixed states by Ozawa [10]. Both the noise-operator based q-rms error and the
disturbance-operator based q-rms disturbance are defined in Sec. II.1. The
goal of the paper is to concretely construct exactly solvable linear position
measurements which achieve the lower bound of the Branciard-Ozawa EDR in each
minimum uncertainty state. Exactly solvable linear measurements are
systematically studied by Ozawa [34, 35] and the key to achieving our goal.

In the paper, we consider a one-dimensional non-relativistic single-particle
system $\mathbf{S}$, whose position $Q_{1}$ and momentum $P_{1}$ at time $0$
are defined as self-adjoint operators on
$\mathcal{H}_{\mathbf{S}}=L^{2}(\mathbb{R})$ and satisfy the canonical
commutation relation (CCR) $[Q_{1},P_{1}]=i\hbar 1$. A unit vector $\psi$ in
$\mathcal{H}_{\mathbf{S}}$ is called a minimum uncertainty state (or a
Gaussian wave packet) if it satisfies $\sigma(Q_{1})\sigma(P_{1})=\hbar/2$,
where $\sigma(Q_{1})$ and $\sigma(P_{1})$ are standard deviations of $Q_{1}$
and $P_{1}$ in $\psi$, respectively. When the state of $\mathbf{S}$ is a
minimum uncertainty state $\psi$, the noise-operator based q-rms error
$\varepsilon(Q_{1})$ of $Q_{1}$ and the disturbance-operator based q-rms
disturbance $\eta(P_{1})$ of $P_{1}$ satisfy

$\varepsilon(Q_{1})^{2}\sigma(P_{1})^{2}+\sigma(Q_{1})^{2}\eta(P_{1})^{2}\geq\hbar^{2}/4$
(1)

for all position measurements. This inequality is the Branciard-Ozawa EDR in
the minimum uncertainty state $\psi$. The main result of the paper is to show
the lower bound of Eq. (1) is achievable by using linear position
measurements. That is to say, we concretely construct linear position
measurements satisfying

$\varepsilon(Q_{1})^{2}\sigma(P_{1})^{2}+\sigma(Q_{1})^{2}\eta(P_{1})^{2}=\hbar^{2}/4$
(2)

in $\psi$. Therefore, when we consider position measurements in each minimum
uncertainty state, no EDR is tighter than the Branciard-Ozawa EDR. Showing
this statement is the main contribution of the paper.

In order to define (exactly solvable) linear position measurements, we use a
one-dimensional non-relativistic single-particle system $\mathbf{P}$, the
probe part of a measuring apparatus $\mathbf{A}$, whose position $Q_{2}$ and
momentum $P_{2}$ are defined on $\mathcal{H}_{\mathbf{P}}=L^{2}(\mathbb{R})$
and satisfy the CCR, $[Q_{2},P_{2}]=i\hbar 1$. The measuring interaction of
linear position measurements for $\mathbf{S}$ is given by

$H_{int}=K[\alpha Q_{1}P_{2}+\beta P_{1}Q_{2}+\gamma(Q_{1}P_{1}-Q_{2}P_{2})],$
(3)

where $K(>0)$ is the coupling constant, and $\alpha$, $\beta$ and $\gamma$ are
real numbers. The famous von Neumann model [36] corresponds to the case where
$(\alpha,\beta,\gamma)=(1,0,0)$. On the other hand, the error-free position
measurement is the case where $(\alpha,\beta,\gamma)=(2,-2,1)/3\sqrt{3}$. We
believe that the rediscovery of exactly solvable linear measurements in the
context of uncertainty relations is another contribution of the paper.

In Sec. II, measuring process, the noise-operator based q-rms error and the
disturbance-operator based q-rms disturbance are defined. Linear position
measurement is then defined. In Sec. III, we first present a theorem that
gives a necessary and sufficient condition for a linear position measurement
to satisfy Eq. (2) in $\psi$. Next, we give three families of linear position
measurements satisfying Eq. (2) in $\psi$. We then investigate probability
distributions and states after the measurement when using linear position
measurements satisfying Eq. (2) in $\psi$. In Sec. IV, the results of the
paper are examined. In particular, the use of Gauss’s error and the noise-
operator base q-rms error is discussed. In Sec. V, we prove the theorem and
show a systematic construction of linear position measurements satisfying Eq.
(2) in $\psi$. Furthermore, we find probability distributions and states after
the measurement when using them via characteristic functions.

### Conventions

Let $\mathcal{H}$, $\mathcal{K}$ be Hilbert spaces and $\phi$ be a unit vector
of $\mathcal{H}$. When $\phi$ is the state of the system, for every self-
adjoint operator $A$ on $\mathcal{H}$, the mean of $A$ and the standard
deviation of $A$ are denoted by $\langle
A\rangle_{\phi}=\langle\phi|A|\phi\rangle=\langle\phi|A\phi\rangle$ and
$\sigma(A\|\phi)=\sqrt{\langle A^{2}\rangle_{\phi}-\langle
A\rangle_{\phi}^{2}}$, respectively. As long as there is no confusion,
$\langle A\rangle_{\phi}$ and $\sigma(A\|\phi)$ are abbreviated as $\langle
A\rangle$ and $\sigma(A)$, respectively. For every linear operator $A$ on
$\mathcal{H}$ and $B$ on $\mathcal{K}$, the tensor product $A\otimes B$ of $A$
and $B$, a linear operator on $\mathcal{H}\otimes\mathcal{K}$, is abbreviated
as $AB$. In particular, we write $A\otimes 1$ and $1\otimes B$ as $A$ and $B$,
respectively, for short. For every self-adjoint operator $A$ on $\mathcal{H}$,
$E^{A}$ denotes the spectral measure of $A$. Let $n$ be a natural number,
$A_{1},\cdots,A_{n}$ mutual commuting self-adjoint operators on $\mathcal{H}$,
and $\phi$ a unit vector of $\mathcal{H}$. The joint probability measure
$\mu^{A_{1},\cdots,A_{n}}_{\phi}$ of $A_{1},\cdots,A_{n}$ in $\phi$ is defined
by

$\mu^{A_{1},\cdots,A_{n}}_{\phi}(J_{1}\times\cdots\times
J_{n})=\langle\phi|E^{A_{1}}(J_{1})\cdots E^{A_{n}}(J_{n})\phi\rangle$ (4)

for all intervals $J_{1},\cdots,J_{n}$ of $\mathbb{R}$. The probability
density function of $\mu^{A_{1},\cdots,A_{n}}_{\phi}$ with respect to the
Lebesgue measure on $\mathbb{R}^{n}$ is denoted by
$p^{A_{1},\cdots,A_{n}}_{\phi}(a_{1},\cdots,a_{n})$ if it exists.

##  II Preliminaries

We consider the one-dimensional non-relativistic single-particle system
$\mathbf{S}$. Throughout the paper, we suppose that the state $\psi$ of the
system $\mathbf{S}$ is a minimum uncertainty state such that $\langle
Q_{1}\rangle_{\psi}=q_{1}$, $\langle P_{1}\rangle_{\psi}=p_{1}$ and
$\sigma(Q_{1}\|\psi)=\sigma_{1}>0$, i.e.,

$\psi(x)=\sqrt[4]{\dfrac{1}{(2\pi)\sigma_{1}^{2}}}e^{-\frac{1}{4\sigma_{1}^{2}}(x-q_{1})^{2}+i\frac{p_{1}}{\hbar}x}$
(5)

in the coordinate representation. Then $\psi$ satisfies
$\sigma(P_{1})=\hbar/(2\sigma_{1})=:\hat{\sigma}_{1}$.

###  II.1 Measuring process, q-rms error and q-rms disturbance

A measuring process for $\mathbf{S}$ is a 4-tuple
$\mathbb{M}_{0}=(\mathcal{K},\zeta,M,U)$ of a Hilbert space $\mathcal{K}$, a
unit vector $\zeta$ of $\mathcal{K}$, a self-adjoint operator $M$ on
$\mathcal{K}$ and a unitary operator $U$ on
$\mathcal{H}_{\mathbf{S}}\otimes\mathcal{K}$. Here self-adjoint operators and
density operators on $\mathcal{K}$ describe observables and states of the
probe part $\mathbf{P}_{0}$ of a measuring appratus $\mathbf{A}_{0}$,
respectively. $\zeta$ is the state of $\mathbf{P}_{0}$, and $M$ is the meter
observable. $U$ then describes the measuring interaction between $\mathbf{S}$
and $\mathbf{P}_{0}$ which turns on at time $0$ and turns off at time
$\tau(>0)$. When using $\mathbb{M}_{0}$, for every observable $X=:X(0)$ of
$\mathbf{S}+\mathbf{P}_{0}$ at time $0$, the observable $X(\tau)$ at time
$\tau$ is given by

$X(\tau)=U^{-1}XU.$ (6)

Then, the noise-operator based q-rms error of $A$ in a vector state $\phi$ is
defined by

$\varepsilon(A,\mathbb{M}_{0},\phi)=\sqrt{\langle
N(A,\mathbb{M}_{0})^{2}\rangle_{\phi\otimes\zeta}},$ (7)

where $N(A,\mathbb{M}_{0})=M(\tau)-A(0)$ is the noise operator for $A$. For
every observable $B$, the disturbance-operator based q-rms disturbance of $B$
in $\phi$ is defined by

$\eta(B,\mathbb{M}_{0},\phi)=\sqrt{\langle
D(B,\mathbb{M}_{0})^{2}\rangle_{\phi\otimes\zeta}},$ (8)

where $D(B,\mathbb{M}_{0})=B(\tau)-B(0)$ is the disturbance operator of $B$.
Unless confusion arises, $\varepsilon(A,\mathbb{M}_{0},\phi)$ and
$\eta(B,\mathbb{M}_{0},\phi)$ are abbreviated as $\varepsilon(A)$ and
$\eta(B)$, respectively. For any density operator $\rho$ on
$\mathcal{H}_{\mathbb{S}}$, the noise-operator based q-rms error
$\varepsilon(A,\mathbb{M}_{0},\rho)$ and the disturbance-operator based q-rms
disturbance $\eta(B,\mathbb{M}_{0},\rho)$ are defined by replacing
$\langle\cdots\rangle_{\phi\otimes\zeta}$ by
$\mathrm{Tr}[(\cdots)(\rho\otimes|\zeta\rangle\langle\zeta|)]$ in Eqs. (7) and
(8), respectively.

We say that two observables $A$ and $B$ are commuting in a vector state $\phi$
if $[E^{A}(J_{1}),E^{B}(J_{2})]\phi=0$ for all intervals(, more generally,
Borel sets) $J_{1},J_{2}$ of $\mathbb{R}$. If two observables $A$ and $B$ are
commuting in $\phi$, then there exists a probability measure
$\mu^{A,B}_{\phi}$ such that

$\mu^{A,B}_{\phi}(J_{1}\times
J_{2})=\langle\phi|E^{A}(J_{1})E^{B}(J_{2})\phi\rangle.$ (9)

We refer the reader to [37] and references therein for the general treatment
of the state-dependent commutativity.

Let $A$ be an observable of $\mathbf{S}$ and
$\mathbb{M}_{0}=(\mathcal{K},\zeta,M,U)$ a measuring process for $\mathbf{S}$.
If $A(0)$ and $M$ are commuting in $\phi\otimes\zeta$, then the noise-operator
based q-rms error $\varepsilon(A,\mathbb{M}_{0},\phi)$ satisfies

$\varepsilon(A,\mathbb{M}_{0},\phi)=\varepsilon_{G}(\mu^{A(0),M(\tau)}_{\phi\otimes\zeta}),$
(10)

where Gauss’ (rms) error $\varepsilon_{G}(\mu)$ for a probability measure
$\mu$ on $\mathbb{R}^{2}$ is defined by

$\varepsilon_{G}(\mu)=\left(\int_{\mathbb{R}^{2}}(x-y)^{2}\;d\mu(x,y)\right)^{\frac{1}{2}}.$
(11)

A measuring process $\mathbb{M}_{0}$ for $\mathbf{S}$ is called a position
measurement (for $\mathbf{S}$) if it is used to measure the position $Q_{1}$
of $\mathbf{S}$. Every position measurement $\mathbb{M}_{0}$ satisfies Eq. (1)
in $\psi$. In the paper, a position measurement for $\mathbf{S}$ is said to
have the minimum error-disturbace in $\psi$ if it satisfies Eq. (2) in $\psi$.

We omit here the introduction of completely positive instrument, which is a
central concept in quantum measurement theory for describing state changes due
to measurements. We refer the reader to [7, 38, 39] for details. Before
stating the main results of the paper, we shall define linear position
measurements for $\mathbf{S}$.

###  II.2 Linear position measurements

As mentioned in Sec. I, we use the system $\mathbf{P}$ to define linear
position measurements. In considering linear position measurements, we ignore
the intrinsic dynamics of $\mathbf{S}$ and $\mathbf{P}$. The composite system
$\mathbf{S}+\mathbf{P}$ of $\mathbf{S}$ and $\mathbf{P}$ is described by the
tensor product Hilbert space
$\mathcal{H}_{\mathbf{S}}\otimes\mathcal{H}_{\mathbf{P}}\cong
L^{2}(\mathbb{R}^{2})$. The time evolution of the composite system
$\mathbf{S}+\mathbf{P}$, the measuring interaction, is described by the
unitary operator

$U(t)=e^{-\frac{it}{\hbar}H_{int}},\hskip 14.22636ptt\in\mathbb{R},$ (12)

on $\mathcal{H}_{\mathbf{S}}\otimes\mathcal{H}_{\mathbf{P}}$. Here the
interaction Hamiltonian $H_{int}$ is given by Eq. (3). Since we ignore the
intrinsic dynamics of $\mathbf{S}$ and $\mathbf{P}$, $K$ contributes only to
the time scale of the measurement time. For simplicity, we assume $K=1$ in the
paper. Each observable $X(t)$ of $\mathbf{S}+\mathbf{P}$ at time $t$ with the
initial condition $X(0)=X$ is then given by

$X(t)=U(t)^{-1}XU(t).$ (13)

By solving Heisenberg’s equations of motions, we see that $Q_{1}(t)$,
$Q_{2}(t)$, $P_{1}(t)$ and $P_{2}(t)$ satisfy the following relations:

$\displaystyle\left(\begin{array}[]{c}Q_{1}(t)\\\ Q_{2}(t)\end{array}\right)$
$\displaystyle=e^{tS}\left(\begin{array}[]{c}Q_{1}(0)\\\
Q_{2}(0)\end{array}\right),$ (14e)
$\displaystyle\left(\begin{array}[]{c}P_{1}(t)\\\ P_{2}(t)\end{array}\right)$
$\displaystyle=e^{-tS^{T}}\left(\begin{array}[]{c}P_{1}(0)\\\
P_{2}(0)\end{array}\right)$ (14j)

for all $t\in\mathbb{R}$, where

$S=\left(\begin{array}[]{cc}\gamma&\beta\\\ \alpha&-\gamma\end{array}\right)$
(15)

and $S^{T}$ is the transpose of $S$.

We call a 4-tuple $\mathbb{M}=(\mathcal{H}_{\mathbf{P}},\xi,Q_{2},U(\tau))$ a
linear position measurement for $\mathcal{H}_{\mathbf{S}}$ (or for
$\mathbf{S}$) if we use $Q_{2}$ to measure $Q_{1}$, where $\tau(>0)$ is the
time the measurement finishes. Here we adopt the following matrix element of
$e^{\tau S}$:

$\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)=e^{\tau S},$ (16)

which implies

$\left(\begin{array}[]{cc}d&-c\\\ -b&a\end{array}\right)=e^{-\tau S^{T}}$ (17)

since $ad-bc=1$. The q-rms error $\varepsilon(Q_{1})$ of $Q_{1}$ and the q-rms
disturbance $\eta(P_{1})$ of $P_{1}$ then are given by

$\displaystyle\varepsilon(Q_{1})^{2}$
$\displaystyle=\langle\psi\otimes\xi|(Q_{2}(\tau)-Q_{1}(0))^{2}(\psi\otimes\xi)\rangle$
$\displaystyle=$
$\displaystyle(c-1)^{2}\sigma(Q_{1})^{2}+d^{2}\sigma(Q_{2})^{2}+((c-1)\langle
Q_{1}\rangle+d\langle Q_{2}\rangle)^{2},$ (18a) $\displaystyle\eta(P_{1})^{2}$
$\displaystyle=\langle\psi\otimes\xi|(P_{1}(\tau)-P_{1}(0))^{2}(\psi\otimes\xi)\rangle$
$\displaystyle=$
$\displaystyle(d-1)^{2}\sigma(P_{1})^{2}+c^{2}\sigma(P_{2})^{2}+((d-1)\langle
P_{1}\rangle-c\langle P_{2}\rangle)^{2},$ (18b)

respectively. When using a linear position measurement
$\mathbb{M}=(\mathcal{H}_{\mathbf{P}},\xi,Q_{2},U(\tau))$ for $\mathbf{S}$,
$Q_{1}(0)$ and $Q_{2}(\tau)$ are mutually commuting, and so are $P_{1}(0)$ and
$P_{1}(\tau)$, so that

$\displaystyle\varepsilon(Q_{1},\mathbb{M},\psi)$
$\displaystyle=\varepsilon_{G}(\mu^{Q_{1}(0),Q_{2}(\tau)}_{\psi\otimes\xi}),$
(19a) $\displaystyle\eta(P_{1},\mathbb{M},\psi)$
$\displaystyle=\varepsilon_{G}(\mu^{P_{1}(0),P_{1}(\tau)}_{\psi\otimes\xi}).$
(19b)

##  III Results

###  III.1 Characterization theorem

The first main result is summarized as the following theorem:

######  Theorem.

A linear position measurement
$\mathbb{M}=(\mathcal{H}_{\mathbf{P}},\xi,Q_{2},U(\tau))$ for
$L^{2}(\mathbb{R})$ satisfies Eq. (2) in $\psi$ if and only if it satisfies
the following two conditions:
$(i)$ $c>0$, $d>0$ and $c+d=1$.
$(ii)$ $\xi$ is equal to the minimum uncertainty state $\xi_{c}$ with $\langle
Q_{2}\rangle_{\xi_{c}}=q_{1}$, $\langle P_{2}\rangle_{\xi_{c}}=-p_{1}$ and
$\sigma(Q_{2}\|\xi_{c})=\sqrt{\frac{c}{1-c}}\sigma_{1}$, i.e.,

$\xi_{c}(y)=\sqrt[4]{\dfrac{1-c}{(2\pi)c\sigma_{1}^{2}}}e^{-\frac{1-c}{4c\sigma_{1}^{2}}(y-q_{1})^{2}-i\frac{p_{1}}{\hbar}y},\hskip
14.22636pty\in\mathbb{R},$ (20)

in the coordinate representation.
If a linear position measurement $\mathbb{M}$ has the minimum error-
disturbance in $\psi$, then we have

$\varepsilon(Q_{1})^{2}=(1-c)\sigma(Q_{1})^{2}\hskip 8.53581pt\text{and}\hskip
8.53581pt\eta(P_{1})^{2}=c\sigma(P_{1})^{2}.$ (21)

Furthermore, for every $\mu\in(0,1)$, there exists a linear position
measurement for $L^{2}(\mathbb{R})$ satisfying

$\varepsilon(Q_{1})^{2}=(1-\mu)\sigma(Q_{1})^{2}\hskip
8.53581pt\text{and}\hskip 8.53581pt\eta(P_{1})^{2}=\mu\sigma(P_{1})^{2}$ (22)

in $\psi$.

By the above theorem, any linear position measurement with the minimum error-
disturbance in $\psi$ satisfies

$\displaystyle\varepsilon(Q_{1})$
$\displaystyle=\sqrt{1-\mu}\sigma(Q_{1})<\sigma(Q_{1}),$ (23a)
$\displaystyle\eta(P_{1})$
$\displaystyle=\sqrt{\mu}\sigma(P_{1})<\sigma(P_{1}),$ (23b)
$\displaystyle\varepsilon(Q_{1})\eta(P_{1})$
$\displaystyle=\dfrac{\hbar}{2}\sqrt{\dfrac{1}{4}-\left(\mu-\dfrac{1}{2}\right)^{2}}\leq\dfrac{\hbar}{4}<\dfrac{\hbar}{2}.$
(23c)

The region of possible values of the pair $(\varepsilon(Q_{1}),\eta(P_{1}))$
is drawn in FIG. 1, when using position measurements in $\psi$.

Figure 1: When the state of $\mathbf{S}$ is $\psi$, possible values of the
pair $(\varepsilon(Q_{1}),\eta(P_{1}))$ are indicated by the area with a grid
of dotted blue lines and with blue boundary except for two points
$(\sigma(Q_{1}),0)$ and $(0,\sigma(P_{1}))$. By the theorem,
$\varepsilon(Q_{1})^{2}\sigma(P_{1})^{2}+\sigma(Q_{1})^{2}\eta(P_{1})^{2}=\hbar^{2}/4$
($\varepsilon(Q_{1}),\eta(P_{1})>0$), a part of its boundary, is achieved by
linear position measurements, and gives the unbreakable limitation for the
pair $(\varepsilon(Q_{1}),\eta(P_{1}))$. The red line is Heisenberg’s bound,
$\varepsilon(Q_{1})\eta(P_{1})=\hbar/2$. On the other hand, the dashed magenta
line indicates $\varepsilon(Q_{1})\eta(P_{1})=\hbar/4$.

###  III.2 Linear position measurements with the minimum error-disturbance in
$\psi$

Next, we present three families $\\{\mathbb{A}_{\mu}\\}_{\mu\in(0,1)}$,
$\\{\mathbb{B}_{\mu}\\}_{\mu\in(0,1)}$ and
$\\{\mathbb{C}_{\mu}\\}_{\mu\in(0,1)}$ of linear position measurements
satisfying Eq. (22) in $\psi$ for each $\mu\in(0,1)$. $\mathbb{A}_{\mu}$,
$\mathbb{B}_{\mu}$ and $\mathbb{C}_{\mu}$, respectively, are linear position
measurements $(\mathcal{H}_{\mathbf{P}},\xi_{\mu},Q_{2},U(\tau))$ whose
parameters $\tau$, $\alpha$, $\beta$ and $\gamma$ are given in Table 1. Here
$D$ used in Table 1 is defined by $D=\det S=-(\gamma^{2}+\alpha\beta)$.

Table 1:  | $\tau$ | $\alpha$ | $\beta$ | $\gamma$ | $D$
---|---|---|---|---|---
$\mathbb{A}_{\mu}$ | $\arccos(1-\mu)$ | $\sqrt{\frac{\mu}{2-\mu}}$ | $-\sqrt{\frac{2-\mu}{\mu}}$ | $0$ | $1$
$\mathbb{B}_{\mu}$ | $\mu$ | $1$ | $-1$ | $1$ | $0$
$\mathbb{C}_{\mu}$ | $-\log(1-\mu)$ | $\frac{2(1-\mu)}{2-\mu}$ | $0$ | $1$ | $-1$

From Table 1, for every $\mu\in(0,1)$, $e^{\tau S}$ and $e^{-\tau S^{T}}$ for
$\mathbb{A}_{\mu}$, $\mathbb{B}_{\mu}$, and $\mathbb{C}_{\mu}$, respectively,
are then given in Table 2.

Table 2:  | $e^{\tau S}$ | $e^{-\tau S^{T}}$
---|---|---
$\mathbb{A}_{\mu}$ | $\left(\begin{array}[]{cc}1-\mu&\mu-2\\\ \mu&1-\mu\end{array}\right)$ | $\left(\begin{array}[]{cc}1-\mu&-\mu\\\ 2-\mu&1-\mu\end{array}\right)$
$\mathbb{B}_{\mu}$ | $\left(\begin{array}[]{cc}1+\mu&-\mu\\\ \mu&1-\mu\end{array}\right)$ | $\left(\begin{array}[]{cc}1-\mu&-\mu\\\ \mu&1+\mu\end{array}\right)$
$\mathbb{C}_{\mu}$ | $\left(\begin{array}[]{cc}\frac{1}{1-\mu}&0\\\ \mu&1-\mu\end{array}\right)$ | $\left(\begin{array}[]{cc}1-\mu&-\mu\\\ 0&\frac{1}{1-\mu}\end{array}\right)$

Let $\mu\in(0,1)$. $\mathbb{A}_{\mu}$, $\mathbb{B}_{\mu}$ and
$\mathbb{C}_{\mu}$ satisfy the following relations:

$\displaystyle\hskip
8.53581pt\varepsilon(Q_{1},\mathbb{A}_{\mu},\psi)^{2}=\varepsilon(Q_{1},\mathbb{B}_{\mu},\psi)^{2}$
$\displaystyle=\varepsilon(Q_{1},\mathbb{C}_{\mu},\psi)^{2}=(1-\mu)\sigma(Q_{1})^{2},$
(24a) $\displaystyle\hskip
8.53581pt\eta(Q_{1},\mathbb{A}_{\mu},\psi)^{2}=\eta(Q_{1},\mathbb{B}_{\mu},\psi)^{2}$
$\displaystyle=\eta(Q_{1},\mathbb{C}_{\mu},\psi)^{2}=\mu\sigma(P_{1})^{2}$
(24b)

To quantitatively check the difference of
$\\{\mathbb{A}_{\mu}\\}_{\mu\in(0,1)}$, $\\{\mathbb{B}_{\mu}\\}_{\mu\in(0,1)}$
and $\\{\mathbb{C}_{\mu}\\}_{\mu\in(0,1)}$, the q-rims disturbance
$\eta(Q_{1})$ of $Q_{1}$ is very useful. Since every linear position
measurement $\mathbb{M}$ with the minimum error-disturbance in $\psi$
satisfies

$\eta(Q_{1},\mathbb{M},\psi)^{2}=\left\\{(a-1)^{2}+b^{2}\dfrac{c}{1-c}\right\\}\sigma_{1}^{2}+(a+b-1)^{2}q_{1}^{2},$
(25)

we obtain

$\displaystyle\lim_{\mu\rightarrow
1-0}\varepsilon(Q_{1},\mathbb{A}_{\mu},\psi)$
$\displaystyle\eta(Q_{1},\mathbb{A}_{\mu},\psi)=\sigma_{1}^{2},$ (26a)
$\displaystyle\lim_{\mu\rightarrow+0}\eta(Q_{1},\mathbb{A}_{\mu},\psi)$
$\displaystyle=2|q_{1}|,$ (26b) $\displaystyle\lim_{\mu\rightarrow
1-0}\varepsilon(Q_{1},\mathbb{B}_{\mu},\psi)$
$\displaystyle\eta(Q_{1},\mathbb{B}_{\mu},\psi)=\sigma_{1}^{2},$ (26c)
$\displaystyle\lim_{\mu\rightarrow+0}\eta(Q_{1},\mathbb{B}_{\mu},\psi)$
$\displaystyle=\lim_{\mu\rightarrow+0}\eta(Q_{1},\mathbb{C}_{\mu},\psi)=0,$
(26d) $\displaystyle\lim_{\mu\rightarrow
1-0}\varepsilon(Q_{1},\mathbb{C}_{\mu},\psi)$
$\displaystyle\eta(Q_{1},\mathbb{C}_{\mu},\psi)=+\infty$ (26e)

from Eq. (24a) and Table 2.

###  III.3 Probability distributions and families of posterior states

We have presented concrete linear position measurements with the minimum
error-disturbance in $\psi$. Our next interest is probability distributions
and states after the measurement when using such measurements.

First, we calculate probability distributions related to $\varepsilon(Q_{1})$
and $\eta(P_{1})$. Let
$\mathbb{M}=(\mathcal{H}_{\mathbf{P}},\xi_{c},Q_{2},U(\tau))$ be a linear
position measurement with the minimum error-disturbance in $\psi$, i.e., a
linear position measurement satisfying the conditions in the theorem in
$\psi$. The probability density functions
$p^{Q_{1}(0),Q_{2}(\tau)}_{\psi\otimes\xi_{c}}(x,y)$ and
$p^{P_{1}(0),P_{1}(\tau)}_{\psi\otimes\xi_{c}}(z,w)$ of the joint probability
distributions of $Q_{1}(0)$ and $Q_{2}(\tau)$ and of $P_{1}(0)$ and
$P_{1}(\tau)$ in $\psi\otimes\xi_{c}$, respectively, are given as follows:

$\displaystyle p^{Q_{1}(0),Q_{2}(\tau)}_{\psi\otimes\xi_{c}}(x,y)$
$\displaystyle=p_{(1-c)\sigma_{1}^{2}}(x-y)p_{c\sigma_{1}^{2}}(y-q_{1}),$
(27a) $\displaystyle p^{P_{1}(0),P_{1}(\tau)}_{\psi\otimes\xi_{c}}(z,w)$
$\displaystyle=p_{c\hat{\sigma}_{1}^{2}}(z-w)p_{(1-c)\hat{\sigma}_{1}^{2}}(w-p_{1}),$
(27b)

where $p_{\sigma^{2}}(x)$ denotes the probability density function of the
Gaussian probability measure with mean $0$ and variance $\sigma^{2}$, i.e.,

$p_{\sigma^{2}}(x)=\dfrac{1}{\sqrt{(2\pi)\sigma^{2}}}e^{-\frac{1}{2\sigma^{2}}x^{2}}.$
(28)

We see that both depend on $\psi$ and $0<c<1$. From Eq. (27a), we obtain

$p^{Q_{2}(\tau)}_{\psi\otimes\xi_{c}}(y)=p_{c\sigma_{1}^{2}}(y-q_{1})$ (29)

and, furthermore, the probability density of the conditional probability
distribution of $Q_{1}(0)$ given the value $y$ of $Q_{2}(\tau)$ in
$\psi\otimes\xi_{c}$

$p^{Q_{1}(0)}_{Q_{2}(\tau)=y|\psi\otimes\xi_{c}}(x)=p_{(1-c)\sigma_{1}^{2}}(x-y).$
(30)

By Eq. (21), we see that, given the value $y$ of $Q_{2}(\tau)$, the difference
of the value $x$ and $y$ of $Q_{1}(0)$ and $Q_{2}(\tau)$ obeys the Gaussian
distribution with mean $0$ and standard deviation
$\sqrt{1-c}\sigma_{1}=\varepsilon(Q_{1})$.

Next, we consider family of posterior states, which is set of states after the
measurement for each output value of the meter (see [40, 39] for the general
theory of family of posterior states). Let $\mathbb{M}$ be a linear position
measurement with the minimum error-disturbance in $\psi$. The family
$\\{\psi_{y}\\}_{y\in\mathbb{R}}$ of posterior states for $(\mathbb{M},\psi)$
is the set of the minimum uncertainty state $\psi_{y}$ with $\langle
Q_{1}\rangle_{\psi_{y}}=(a+b)y$, $\langle P_{1}\rangle_{\psi_{y}}=p_{1}$ and
$\sigma(Q_{1}\|\psi_{y})=\sigma_{1}/\sqrt{1-c}$ for all $y\in\mathbb{R}$,
i.e.,

$\psi_{y}(x)=\sqrt[4]{\dfrac{1-c}{(2\pi)\sigma_{1}^{2}}}e^{-\frac{1-c}{4\sigma_{1}^{2}}(x-(a+b)y)^{2}+i\frac{p_{1}}{\hbar}x}$
(31)

for all $y\in\mathbb{R}$ in the coordinate representation. For every interval
$J$ of $\mathbb{R}$, we then obtain the state $\rho_{J}$ after the measurement
under the condition that output values not contained in $J$ is excluded, which
is given by

$\mathrm{Tr}[X\rho_{J}]=\frac{\langle
U(\tau)(\psi\otimes\xi_{c})|XE^{Q_{2}}(J)U(\tau)(\psi\otimes\xi_{c})\rangle}{\langle
U(\tau)(\psi\otimes\xi_{c})|E^{Q_{2}}(J)U(\tau)(\psi\otimes\xi_{c})\rangle}$
(32)

for all bounded operators $X$ on $\mathcal{H}_{\mathbf{S}}$, whenever $\langle
U(\tau)(\psi\otimes\xi_{c})|E^{Q_{2}}(J)U(\tau)(\psi\otimes\xi_{c})\rangle\neq
0$. The family $\\{\psi_{y}\\}_{y\in\mathbb{R}}$ of posterior states for
$(\mathbb{M},\psi)$ then satisfies

$\rho_{J}=\dfrac{1}{\mu_{c\sigma_{1}^{2},q_{1}}(J)}\int_{J}|\psi_{y}\rangle\langle\psi_{y}|\;p_{c\sigma_{1}^{2}}(y-q_{1})\;dy,$
(33)

where $\mu_{c\sigma_{1}^{2},q_{1}}$ is the Gaussian measure with mean $q_{1}$
and variance $c\sigma_{1}^{2}$, i.e.,
$\mu_{c\sigma_{1}^{2},q_{1}}(J)=\int_{J}p_{c\sigma_{1}^{2}}(y-q_{1})dy$ for
all intervals $J$ of $\mathbb{R}$(, more generally, for all Borel sets $J$ of
$\mathbb{R}$).

##  IV Discussion

The Branciard-Ozawa EDR is given by

$\displaystyle\varepsilon(A)^{2}\sigma(B)^{2}+\sigma(A)^{2}\eta(B)^{2}$
$\displaystyle\hskip
8.53581pt+2\varepsilon(A)\eta(B)\sqrt{\sigma(A)^{2}\sigma(B)^{2}-D_{AB}^{2}}\geq
D_{AB}^{2}$ (34)

for observables $A$ and $B$. Here
$\sigma(A)^{2}=\mathrm{Tr}[A^{2}\rho]-(\mathrm{Tr}[A\rho])^{2}$,
$\sigma(B)^{2}=\mathrm{Tr}[B^{2}\rho]-(\mathrm{Tr}[B\rho])^{2}$ and
$D_{AB}=\mathrm{Tr}|\sqrt{\rho}(-i[A,B])\sqrt{\rho}|/2\geq
C_{AB}=|\mathrm{Tr}([A,B]\rho)|/2$, where $\rho$ is the density operator
describing a state of the system. For any vector state
$\rho=|\phi\rangle\langle\phi|$, $D_{AB}$ is equal to $C_{AB}$. Eq. (1) holds
for the case where $A=Q_{1}$, $B=P_{1}$ and $\rho=|\psi\rangle\langle\psi|$.
Spin measurements that achieve the lower bound of the improved version of the
inequality in some class of states have already been constructed [10]. On the
other hand, the first achievement for constructing position measurements with
minimum error-disturbance in some class of states is made in the paper.

In the proof of Heisenberg’s EDR

$\varepsilon(Q_{1})\eta(P_{1})\geq\hbar/2$ (35)

by Heisenberg [41] and Kennard [42], the approximate repeatability hypothesis
(ARH), the approximate version of the repeatable hypothesis (RH), is assumed.
In particular, Heisenberg only dealt with the case where the states after the
measurement are minimum uncertainty states. The (A)RH is a natural assumption
at that time, but is abandoned since the 1980s (see [43] for the details on
the RH and the ARH, and also [44, 36, 45, 46]). Quantum measurement theory has
changed significantly from what it was when quantum mechanics was established.

Following Laplace’s pioneering investigation, Gauss [47] defined his rms error
in 1821. It is now widely used in statistics and experimental science. Gauss’
error is not always applicable to quantum systems due to the noncommutativity
of observables, but we should try it if applicable. In other words, its
universal validity is lost in quantum theory, but it does not mean that it is
always useless. As given in Sec. II.2, its use is reasonable as long as linear
position measurements are considered. In particular, the noise-operator based
q-rms error and Gauss’s error coincide when using linear position
measurements. The author would like to emphasize that the results of the paper
[28] and this paper show that Heisenberg’s EDR is already violated even when
Gauss’s error is available. We must never miss this fact.

A q-rms error is said to be complete if it vanishes only for precise
measurements of observables. As shown in [48, 37] for example, the noise-
operator based q-rms error is not complete. On the other hand, it is recently
verified in [37] that the noise-operator based q-rms error satisfies several
satisfactory conditions except for the completeness from the operational point
of view. Furthermore, several complete q-rms errors, improvements of the
noise-operator based q-rms error, are also defined in [37]. Only the lack of
the completeness does not mean that the noise-operator based q-rms error is
useless. In fact, it is widely applicable and useful enough as a standard. The
results of the paper also contribute to showing its value. The author thinks
that it is better to choose alternatives when there is a problem with the use
of the noise-operator based q-rms error. To this end, it will be increasingly
important to study complete q-rms errors as a candidate for alternatives.

##  V Methods

As in the standard textbook of quantum mechanics, for $j,k=1,2$, $Q_{j}$ and
$P_{k}$ satisfy

$\displaystyle(Q_{j}f)(x_{1},x_{2})$ $\displaystyle=x_{j}f(x_{1},x_{2}),$
$\displaystyle(P_{k}g)(x_{1},x_{2})$
$\displaystyle=\dfrac{\hbar}{i}\dfrac{\partial}{\partial x_{k}}g(x_{1},x_{2})$

in the coordinate representation for suitable functions $f$ and $g$ on
$\mathbb{R}^{2}$. We do not explicitly use these representations here.

###  V.1 Proof of Theorem

Let $\mathbb{M}=(\mathcal{H}_{\mathbf{P}},\xi,Q_{2},U(\tau))$ be a linear
position measurement for $\mathcal{H}_{\mathbf{S}}$. By Eqs. (18a), (18b), the
following evaluation holds for $\mathbb{M}$:

$\displaystyle\hskip
14.22636pt\varepsilon(Q_{1})^{2}\sigma(P_{1})^{2}+\sigma(Q_{1})^{2}\eta(P_{1})^{2}$
$\displaystyle\geq(c-1)^{2}\sigma(Q_{1})^{2}\sigma(P_{1})^{2}+d^{2}\sigma(Q_{2})^{2}\sigma(P_{1})^{2}$
$\displaystyle\hskip
14.22636pt+(d-1)^{2}\sigma(Q_{1})^{2}\sigma(P_{1})^{2}+c^{2}\sigma(Q_{1})^{2}\sigma(P_{2})^{2}$
$\displaystyle=\dfrac{\hbar^{2}}{4}\\{(c-1)^{2}+(d-1)^{2}\\}$
$\displaystyle\hskip
14.22636pt+c^{2}\sigma(Q_{1})^{2}\sigma(P_{2})^{2}+d^{2}\sigma(Q_{2})^{2}\sigma(P_{1})^{2}$
$\displaystyle=\dfrac{\hbar^{2}}{4}\\{(c-1)^{2}+(d-1)^{2}\\}+2|cd|\sigma(Q_{1})\sigma(P_{1})\sigma(Q_{2})\sigma(P_{2})$
$\displaystyle\hskip
14.22636pt+(|c|\sigma(Q_{1})\sigma(P_{2})-|d|\sigma(Q_{2})\sigma(P_{1}))^{2}$
$\displaystyle\geq\dfrac{\hbar^{2}}{4}\\{(c-1)^{2}+(d-1)^{2}\\}+\hbar|cd|\sigma(Q_{2})\sigma(P_{2})$
$\displaystyle\geq\hbar^{2}l(c,d),$ (36)

where $l(c,d)$ is defined by

$l(c,d)=\dfrac{1}{4}\\{(c-1)^{2}+(d-1)^{2}\\}+\dfrac{1}{2}|cd|,$ (37)

and takes the minimal value $1/4$ when $c,d\geq 0$ and $c+d=1$. $\mathbb{M}$
has the minimum error-disturbance in $\psi$ if and only if it satisfies the
following four conditions:
$(1)$ $(c-1)\langle Q_{1}\rangle+d\langle Q_{2}\rangle=0$ and $(d-1)\langle
P_{1}\rangle-c\langle P_{2}\rangle=0$.
$(2)$ $|c|\sigma(Q_{1})\sigma(P_{2})=|d|\sigma(Q_{2})\sigma(P_{1})$.
$(3)$ $\sigma(Q_{2})\sigma(P_{2})=\hbar/2$.
$(4)$ $c,d\geq 0$ and $c+d=1$.
$(3)$ implies that $\xi$ is a minimum uncertainty state. By
$\sigma(Q_{1})\sigma(P_{1})=\sigma(Q_{2})\sigma(P_{2})=\hbar/2$ and $(2)$,

$\sigma(Q_{2})^{2}=\dfrac{\hbar}{2}\dfrac{\sigma(Q_{2})}{\sigma(P_{2})}=\dfrac{\hbar}{2}\dfrac{|c|}{|d|}\dfrac{\sigma(Q_{1})}{\sigma(P_{1})}=\dfrac{|c|}{|d|}\sigma(Q_{1})^{2}$
(38)

This is consistent with the condition $(3)$ when $c,d\neq 0$. Therefore, under
the condition $(i)$ in the theorem, by Eq. (38) and the condition $(1)$, $\xi$
is a minimum uncertainty state such that $\langle Q_{2}\rangle_{\xi}=q_{1}$,
$\langle P_{2}\rangle_{\xi}=-p_{1}$ and
$\sigma(Q_{2}\|\xi)=\sqrt{\dfrac{c}{1-c}}\sigma_{1}$, that is, $\xi=\xi_{c}$.
Under the condition $(i)$ in the theorem, $\xi=\xi_{c}$ implies the conditions
$(1)$, $(2)$ and $(3)$. Then we obtain

$\displaystyle\varepsilon(Q_{1})^{2}$
$\displaystyle=(c-1)^{2}\sigma_{1}^{2}+(1-c)^{2}\dfrac{c\sigma_{1}^{2}}{1-c}=(1-c)\sigma_{1}^{2},$
$\displaystyle\eta(P_{1})^{2}$
$\displaystyle=c^{2}\hat{\sigma}_{1}^{2}+c^{2}\dfrac{1-c}{c}\hat{\sigma}_{1}^{2}=c\hat{\sigma}_{1}^{2}.$

This completes the proof of the first half of the theorem.

For every $\mu\in(0,1)$, we find linear position measurements such that
$c=\mu$ and $d=1-\mu$. To explicitly give linear position measurements
satisfying Eq. (22) in $\psi$ for every $\mu\in(0,1)$, we use a more explicit
formula of $e^{tS}$:

$e^{tS}=\left\\{\begin{array}[]{ll}\displaystyle{(\cos(t\sqrt{D}))I+\dfrac{\sin(t\sqrt{D})}{\sqrt{D}}S},&\quad(D>0),\\\
I+tS,&\quad(D=0),\\\
\displaystyle{(\cosh(t\sqrt{-D}))I+\dfrac{\sinh(t\sqrt{-D})}{\sqrt{-D}}S},&\quad(D<0).\end{array}\right.$
(39)

Since $c=\mu>0$, $\alpha$ must be non-zero, so that $\beta$ is uniquely
determined by

$\beta=-\dfrac{\gamma^{2}+D}{\alpha}.$ (40)

From now on, we take the position that $\alpha$, $\gamma$, and $D$ are the
fundamental variables, and $\beta$ is determined from them. Then, the cases
$D>0$, $D=0$, $D<0$ must be handled separately.

[$D>0$] By Eqs. (16), (39), both $c=\mu$ and $d=1-\mu$ become

$\left\\{\begin{array}[]{l}\displaystyle{\mu=\dfrac{\alpha}{\sqrt{D}}\sin(\tau\sqrt{D})}.\\\
\displaystyle{1-\mu=\cos(\tau\sqrt{D})-\dfrac{\gamma}{\sqrt{D}}\sin(\tau\sqrt{D})}.\end{array}\right.$
(41)

This system of equations is equivalent to

$\left\\{\begin{array}[]{ll}\cos(\tau\sqrt{D})=\dfrac{\gamma}{\alpha}\mu+(1-\mu),&\\\
\sin(\tau\sqrt{D})=\dfrac{\sqrt{D}}{\alpha}\mu.&\end{array}\right.$ (42)

For every $0<\mu<1$, $\gamma\geq 0$ and $D>0$, we have
$\dfrac{\gamma}{\alpha}\mu+(1-\mu)>0$ and $\dfrac{\sqrt{D}}{\alpha}\mu>0$ for
all $\alpha>0$. Moreover, the function

$u_{+}(\alpha)=\left\\{\dfrac{\gamma}{\alpha}\mu+(1-\mu)\right\\}^{2}+\left(\dfrac{\sqrt{D}}{\alpha}\mu\right)^{2}$

on $\mathbb{R}_{+}=\\{\alpha\in\mathbb{R}\;|\;\alpha>0\\}$ is monotone
decreasing and satisfies $\lim_{\alpha\rightarrow+0}u_{+}(\alpha)=+\infty$ and
$\lim_{\alpha\rightarrow+\infty}u_{+}(\alpha)=(1-\mu)^{2}<1$, so that there
uniquely exists $\alpha>0$ such that $u_{+}(\alpha)=1$. There then uniquely
exists $0<\tau<\dfrac{\pi}{2\sqrt{D}}$ satisfying Eq. (42). For every
$0<\mu<1$, $D>0$ and $\gamma\geq 0$, there uniquely exist $\alpha>0$ and
$0<\tau<\dfrac{\pi}{2\sqrt{D}}$ such that Eq. (42). For every $\mu\in(0,1)$,
the model $\mathbb{A}_{\mu}$ is included in this case. This completes the
proof of the theorem.

[$D=0$] By Eqs. (16), (39), both $c=\mu$ and $d=1-\mu$ become

$\left\\{\begin{array}[]{l}\displaystyle{\mu=\alpha\tau}\\\
\displaystyle{1-\mu=1-\gamma\tau},\end{array}\right.$ (43)

This system of equations and $D=0$ then imply

$\alpha=-\beta=\gamma=\dfrac{\mu}{\tau}.$ (44)

For every $0<\mu<1$ and $\gamma>0$, there uniquely exist $\alpha>0$, $\beta<0$
and $\tau>0$ such that Eq. (44). For every $\mu\in(0,1)$, the model
$\mathbb{B}_{\mu}$ is included in this case.

[$D<0$] By Eqs. (16), (39), both $c=\mu$ and $d=1-\mu$ become

$\left\\{\begin{array}[]{l}\displaystyle{\mu=\dfrac{\alpha}{\sqrt{-D}}\sinh(\tau\sqrt{-D})},\\\
\displaystyle{1-\mu=\cosh(\tau\sqrt{-D})-\dfrac{\gamma}{\sqrt{-D}}\sinh(\tau\sqrt{-D})},\end{array}\right.$
(45)

so that $\alpha\neq 0$. This system of equations is equivalent to

$\left\\{\begin{array}[]{ll}\cosh(\tau\sqrt{-D})=\dfrac{\gamma}{\alpha}\mu+(1-\mu),&\\\
\sinh(\tau\sqrt{-D})=\dfrac{\sqrt{-D}}{\alpha}\mu.&\end{array}\right.$ (46)

Here we assume that $\alpha>0$ and $\gamma>0$. The function $u_{-}$ on
$\mathbb{R}_{+}$ is then defined by

$u_{-}(\alpha)=\left\\{\dfrac{\gamma}{\alpha}\mu+(1-\mu)\right\\}^{2}-\left(\dfrac{\sqrt{-D}}{\alpha}\mu\right)^{2}.$
(47)

There exists a solution $\alpha>0$ of $u_{-}(\alpha)=1$ if and only if

$(2-\mu)\alpha^{2}-2(1-\mu)\gamma\alpha-\mu(\gamma^{2}+D)=0.$ (48)

Since $1-\mu>0$, $\gamma>0$ and $1-(1-\mu)^{2}>0$ are always satisfied, the
above quadratic equation has a unique solution $\alpha>0$ only when
$-\mu(\gamma^{2}+D)\leq 0$. Thus, $\gamma$ must satisfy $\gamma\geq\sqrt{-D}$.
For every $0<\mu<1$, $D<0$ and $\gamma\geq\sqrt{-D}$, there uniquely exist
$\alpha>0$ and $\tau>0$ such that Eq. (46). For every $\mu\in(0,1)$, the model
$\mathbb{C}_{\mu}$ is included in this case.

###  V.2 Probability distributions and families of posterior states

The characteristic function $\lambda$ of the probability measure $\mu$ on
$\mathbb{R}^{d}$ is defined as the inverse Fourier transform of $\mu$: For
every $k\in\mathbb{R}^{d}$,

$\lambda(k)=\int_{\mathbb{R}^{d}}e^{i\langle x,k\rangle}\;d\mu(x),$ (49)

where $\langle\cdot,\cdot\rangle$ is the inner product on $\mathbb{R}^{d}$. A
probability measure on $\mathbb{R}^{d}$ is a Gaussian measure (also called a
multivariate normal distribution)

$\mu_{V,m}(dx)=\dfrac{1}{\sqrt{(2\pi)^{d}\det(V)}}e^{-\frac{1}{2}\langle
x-m,V^{-1}(x-m)\rangle}\;dx$ (50)

if and only if its characteristic function has the form

$\lambda_{V,m}(k)=e^{i\langle m,k\rangle-\frac{1}{2}\langle
k,Vk\rangle},\hskip 14.22636ptk\in\mathbb{R}^{d},$ (51)

where $V>0$ is a covariance matrix, $m\in\mathbb{R}^{d}$ is a mean vector. For
more on this basic fact, see the section on characteristic functions in the
standard textbook of probability theory and statistics.

Let $\mathbb{M}=(\mathcal{H}_{\mathbf{P}},\xi_{c},Q_{2},U(\tau))$ be a linear
position measurement with the minimum error-disturbance in $\psi$. For every
$k=\left(\begin{array}[]{c}k_{1}\\\ k_{2}\end{array}\right)\in\mathbb{R}^{2}$,
the characteristic function of
$\mu^{Q_{1}(0),Q_{2}(\tau)}_{\psi\otimes\xi_{c}}$ is given by

$\displaystyle\lambda^{Q_{1}(0),Q_{2}(\tau)}_{\psi\otimes\xi_{c}}(k)$
$\displaystyle=\langle\psi\otimes\xi_{c}|e^{ik_{1}Q_{1}(0)+ik_{2}Q_{2}(\tau)}(\psi\otimes\xi_{c})\rangle$
$\displaystyle=\langle\psi|e^{i[k_{1}+ck_{2}]Q_{1}}\psi\rangle\langle\xi_{c}|e^{i[dk_{2}]Q_{2}}\xi_{c}\rangle$
$\displaystyle=e^{iq_{1}(k_{1}+ck_{2})-\frac{1}{2}\sigma_{1}^{2}(k_{1}+ck_{2})^{2}}$
$\displaystyle\hskip 14.22636pt\times
e^{iq_{1}(dk_{2})-\frac{1}{2}\frac{c}{1-c}\sigma_{1}^{2}(dk_{2})^{2}}$
$\displaystyle=e^{iq_{1}k_{1}+iq_{1}k_{2}-\frac{1}{2}\langle k,Wk\rangle},$
(52)

where $W=\sigma_{1}^{2}\left(\begin{array}[]{cc}1&c\\\ c&c\end{array}\right)$.
We obtain Eq. (27a) from $\det(W)=\sigma_{1}^{4}c(1-c)$ and

$W^{-1}=\dfrac{1}{(1-c)\sigma_{1}^{2}}\left(\begin{array}[]{cc}1&-1\\\
-1&1\end{array}\right)+\dfrac{1}{c\sigma_{1}^{2}}\left(\begin{array}[]{cc}0&0\\\
0&1\end{array}\right).$ (53)

Eq. (27b) is derived in the same way.

Let $\mathbb{M}=(\mathcal{H}_{\mathbf{P}},\xi_{c},Q_{2},U(\tau))$ be a linear
position measurement with the minimum error-disturbance in $\psi$. In order to
find family of posterior states for $(\mathbb{M},\psi)$, we also check the
following probability density functions via their characteristic functions:

$\displaystyle p^{Q_{1}(\tau),Q_{2}(\tau)}_{\psi\otimes\xi_{c}}(x,y)$
$\displaystyle=p_{\frac{\sigma_{1}^{2}}{1-c}}(x-(a+b)y)p_{c\sigma_{1}^{2}}(y-q_{1}),$
(54a) $\displaystyle p^{P_{1}(\tau),Q_{2}(\tau)}_{\psi\otimes\xi_{c}}(z,y)$
$\displaystyle=p_{(1-c)\hat{\sigma}_{1}^{2}}(z-p_{1})p_{c\sigma_{1}^{2}}(y-q_{1}).$
(54b)

The formula $e^{iuQ_{j}+ivP_{j}}=e^{i\frac{\hbar}{2}uv}e^{iuQ_{j}}e^{ivP_{j}}$
for all $u,v\in\mathbb{R}$ and $j=1,2$ is applied to calculate the
characteristic functions correponding to the probability distributions that
have the above densities. By Eqs. (54a), (54b) and (29), the density functions
$p^{Q_{1}(\tau)}_{Q_{2}(\tau)=y|\psi\otimes\xi_{c}}(x)$ and
$p^{P_{1}(\tau)}_{Q_{2}(\tau)=y|\psi\otimes\xi_{c}}(z)$ of the conditional
probability distributions of $Q_{1}(\tau)$ and $P_{1}(\tau)$ given the value
$y$ of $Q_{2}(\tau)$ in $\psi\otimes\xi_{c}$ are given by

$\displaystyle p^{Q_{1}(\tau)}_{Q_{2}(\tau)=y|\psi\otimes\xi_{c}}(x)$
$\displaystyle=p_{\frac{\sigma_{1}^{2}}{1-c}}(x-(a+b)y),$ (55a) $\displaystyle
p^{P_{1}(\tau)}_{Q_{2}(\tau)=y|\psi\otimes\xi_{c}}(z)$
$\displaystyle=p_{(1-c)\hat{\sigma}_{1}^{2}}(z-p_{1}),$ (55b)

respectively. The relation
$\dfrac{\sigma_{1}^{2}}{1-c}\cdot(1-c)\hat{\sigma}_{1}^{2}=\dfrac{\hbar^{2}}{4}$
implies that the family of posterior states for $(\mathbb{M},\psi)$ is given
by Eq. (31) and is unique up to phase.

##  VI Summary and perspective

In the paper, linear position measurements with the minimum error-disturbance
in each minimum uncertainty state have been constructed for each possible
error value. This is nothing but the result showing that the lower bound of
the Branciard-Ozawa inequality for the q-rms error of position and the q-rms
disturbance of momentum in each minimum uncertainty state is achievable. It is
also the first achievement for position measurements since the reformulation
of uncertainty relations has started. Moreover, we have obtained joint
probability distributions and families of posterior states when using them. In
order to quantitatively examine the effect of the measurement, the q-rms
disturbance of position has also been analyzed. It is expected to construct
measurements with minimum error-disturbance in a broader class of states in
the future, which will lead to a new understanding of quantum limits,
including uncertainty relations. Linear simultaneous measurements of position
and momentum with minimum error-trade-off in each minimum uncertainty state
will be discussed in the subsequent paper [49].

###### Acknowledgements.

The author thanks Prof. Motoichi Ohtsu and Prof. Fumio Hiroshima for their
warmful encouragements. He also thanks Prof. Masanao Ozawa for useful
comments.

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# A review on two types of sonic interfaces

Myoungjean Bae  Department of Mathematical Sciences, KAIST, 291 Daehak-ro,
Yuseong-gu, Daejeon, 43141, Korea<EMAIL_ADDRESS>
###### Abstract.

In this paper, two examples of sonic interfaces ([2, 3, 4, 5, 6]) are
presented. The first example shows the case of sonic interfaces as weak
discontinuities in self-similar shock configurations of unsteady Euler system.
The second example shows the case of sonic interfaces as regular interfaces in
accelerating transonic flows governed by the steady Euler-Poisson system with
self-generated electric forces. And, we discuss analytic differences of the
two examples, and introduce an open problem on decelerating transonic solution
to the steady Euler-Poisson system.

###### Key words and phrases:

Keldysh type, transonic, sonic interface, weak discontinuity, regular
interface

###### 2010 Mathematics Subject Classification:

35C06, 35M10, 35M30, 35Q31, 76H05, 76N10

##  1\. A sonic interface as a weak discontinuity

Fix a constant $\varepsilon_{0}>0$. Given a function
$f:[0,\varepsilon_{0}]\rightarrow\mathbb{R}_{+}$ with

(1.1) $\|f\|_{C^{1,1}([0,\varepsilon_{0}])}<\infty,\quad
f(0)>0,\quad\text{and}\quad\frac{df}{dx}\geq\omega>0\,\forall 0\leq
x\leq\varepsilon_{0},$

set

$P_{0}:=(0,f(0)),$

and define a domain

(1.2)
$\mathcal{Q}_{\varepsilon_{0}}^{f}:=\\{(x,y):0<x<\varepsilon_{0},\,\,0<y<f(x)\\}.$

For each $t\in(0,f(0))$, set

$\mathcal{R}_{t}:=(0,\frac{\varepsilon_{0}}{2})\times(0,f(0)-t).$

Given two constants $a>0$ and $b>0$, and functions $\beta_{k}\in
C(\partial\mathcal{Q}_{\varepsilon_{0}}^{f}\cap\\{y=f(x)\\})$ for $k=1,2,3$,
consider the equation

(1.3)
$(2x-a\psi_{x}+O_{1})\psi_{xx}+O_{2}\psi_{xy}+(b+O_{3})\psi_{yy}-(1+O_{4})\psi_{x}+O_{5}\psi_{y}=0\quad\mbox{in
$\mathcal{Q}_{\varepsilon_{0}}^{f}$},$

and the boundary conditions

(1.4) $\displaystyle\psi=0\quad$ on
$\partial\mathcal{Q}_{\varepsilon_{0}}^{f}\cap\\{x=0\\}$ (1.5)
$\displaystyle\partial_{y}\psi=0\quad$ on
$\partial\mathcal{Q}_{\varepsilon_{0}}^{f}\cap\\{y=0\\}$ (1.6)
$\displaystyle\beta_{1}(x,y)\psi_{x}+\beta_{2}(x,y)\psi_{y}+\psi=0\quad$
$\displaystyle\mbox{on
$\partial\mathcal{Q}_{\varepsilon_{0}}^{f}\cap\\{y=f(x)\\}$}.$

In addition, assume that

(1.7)
$\beta_{1}(x,y)\geq\lambda,|\beta_{2}(x,y),\,\beta_{3}(x,y)|\leq\frac{1}{\lambda}\quad\text{on
$\partial\mathcal{Q}_{\varepsilon_{0}}^{f}\cap\\{y=f(x)\\}$}$

for some constant $\lambda>0$.

######  Theorem 1 (Theorem 3.1 in [1]).

Suppose that a function
$\psi:\overline{\mathcal{Q}_{\varepsilon_{0}}^{f}}\rightarrow\mathbb{R}$
satisfies the following conditions:

  * (i)

$\displaystyle{\psi\in C^{2}(\mathcal{Q}_{\varepsilon_{0}}^{f})\cap
C^{1,1}(\overline{\mathcal{Q}_{\varepsilon_{0}}^{f}})};$

  * (ii)

$\psi>0$ in $\mathcal{Q}_{\varepsilon_{0}}^{f}$;

  * (iii)

there exist constants $\mu>0$ and $\delta\in(0,1)$ such that

$-\mu\leq\frac{\psi_{x}(x,y)}{x}\leq\frac{2-\delta}{a}\quad\text{in
$\mathcal{Q}_{\varepsilon_{0}}^{f}$};$

  * (iv)

$\psi$ satisfies (1.3)–(1.6).

In addition, assume that the terms $O_{i}(x,y),i=1,\cdots,5$ are continuously
differentiable, and that there exists a constant $N>0$ satisfying

(1.8) $\displaystyle\frac{|O_{1}(x,y)|}{x^{2}},\frac{|O_{i}(x,y)|}{x}\leq N$
$\displaystyle\mbox{for $i=2,\cdots,5$},$
$\displaystyle\frac{|DO_{1}(x,y)|}{x},|DO_{k}(x,y)|\leq N$ for $i=2,\cdots,5$

in $\\{x>0\\}$.

Then, the following properties hold:

  * (a)

$\displaystyle{\forall t\in(0,f(0)),\,\,\psi\in
C^{2,\alpha}(\overline{\mathcal{R}_{t}})\,\,\forall\alpha\in(0,1)};$

  * (b)

$\displaystyle{\psi_{xx}(0,y)=\frac{1}{a},\quad\psi_{xy}(0,y)=\psi_{yy}(0,y)=0\quad\text{for
all}\quad 0\leq y<f(0)};$

  * (c)

there are two sequences $\\{(x_{m}^{(1)},y_{m}^{(1)})\\}$ and
$\\{(x_{m}^{(2)},y_{m}^{(2)})\\}$ in $\mathcal{Q}_{\varepsilon_{0}}^{f}$ such
that

$\displaystyle\lim_{m\to\infty}(x_{m}^{(1)},y_{m}^{(1)})=\lim_{m\to\infty}(x_{m}^{(2)},y_{m}^{(2)})=P_{0}\quad\text{for
$j=1,2$},$
$\displaystyle\lim_{m\to\infty}\psi_{xx}(x_{m}^{(1)},y_{m}^{(1)})=\frac{1}{a},\quad\lim_{m\to\infty}\psi_{xx}(x_{m}^{(2)},y_{m}^{(2)})=0.$

Now we demonstrate an application of Theorem 1.

An irrotational flow of inviscid compressible polytropic gas is governed by
the Euler equation for potential flow:

(1.9) $\displaystyle\partial_{t}\rho+\nabla_{\rm{\bf
x}}\cdot(\rho\nabla_{\rm{\bf x}}\Phi)=0,$
$\displaystyle\partial_{t}\Phi+\frac{1}{2}|\nabla_{\rm{\bf
x}}\Phi|^{2}+\frac{\rho^{\gamma-1}-1}{\gamma-1}=B_{0}$

for an adiabatic exponent $\gamma>1$. The density and the velocity potential
of the flow are represented as $\rho$ and $\Phi$, respectively. And, the term
$B_{0}>0$ represents the Bernoulli constant which is determined by the initial
data. For $\theta_{w}\in(0,\frac{\pi}{2})$, define a symmetric wedge $W$ in
$\mathbb{R}^{2}$ by

(1.10) $W:=\\{\rm{\bf
x}=(x_{1},x_{2})\in\mathbb{R}^{2}:|x_{2}|<x_{1}\tan\theta_{w},\,\,x_{1}>0\\}.$

Suppose that $(\rho,\Phi)$ solves (1.9) in $\mathbb{R}^{2}\setminus W$, and
satisfies the slip boundary condition

(1.11) $\nabla_{\rm{\bf x}}\Phi\cdot{\bf n}_{w}=0\quad\text{on $\partial W$}$

for the exterior unit normal ${\bf n}$ to $\partial W$. Then, for any constant
$\alpha>0$, it can be directly checked that $(\tilde{\rho},\tilde{\Phi})$
given by

$(\tilde{\rho},\tilde{\Phi})(\rm{\bf
x},t):=(\rho,\frac{1}{\alpha}\Phi)(\alpha\rm{\bf x},\alpha t)$

satisfies (1.9) in $\mathbb{R}^{2}\setminus W$, and (1.11) on $\partial W$.
Owing to the scaling invariance, one may seek for a self-similar solution in
the form of

$(\rho,\Phi)(\rm{\bf
x},t)=(\varrho({\bm{\xi}}),t\Psi(\bm{\xi}))\quad\text{for}\,\,\bm{\xi}=(\xi_{1},\xi_{2})\,\,\text{with}\,\,(\xi_{1},\xi_{2}):=\frac{1}{t}(x_{1},x_{2}).$

With a pseudo-potential function $\varphi$ given by

$\varphi(\bm{\xi}):=-\frac{1}{2}|\bm{\xi}|^{2}+\Psi(\bm{\xi}),$

one can rewrite (1.9) as

(1.12) $\nabla\cdot(\varrho\nabla\varphi)+2\varrho=0$

with

(1.13)
$\varrho^{\gamma-1}=(\gamma-1)B_{0}+1-(\gamma-1)\left(\varphi+\frac{1}{2}|\nabla\varphi|^{2}\right).$

With (1.12), the global-in-time existence of weak solutions to (1.9) of
various shock structures are investigated in [2, 3, 5, 6, 7, 8]. When a plane
shock hits the symmetric wedge $W$ head-on, it is proved in [5, 6] that if
$\theta_{w}\in(\theta_{d},\frac{\pi}{2})$ for the critical angle(=the
detachment angle uniquely determined by a shock polar), then (1.9) has a weak
solution of a self-similar regular shock reflection configuration (Fig.1).

Figure 1. Self-similar regular shock reflection ($S_{\infty}$: incoming plane
shock, $\overline{S_{0}\cup\Gamma_{\rm{shock}}}$: reflected shock,
$\theta_{w}\in(\theta_{\rm sonic},\frac{\pi}{2})$)

When a supersonic flow with a constant density moves horizontally toward the
symmetric wedge $W$ at a constant speed, it is proved in [3, 8] that there
exists a global-in-time weak solution of a self-similar weak shock
configuration for $\theta_{w}\in(0,\theta_{d})$ (Fig.2).

Figure 2. Self-similar weak shock configuration past a wedge (Left:
$\theta_{w}<\theta_{{\rm sonic}}$, Right: $\theta_{{\rm
sonic}}\leq\theta_{w}<\theta_{d}$)

In Fig.1 and Fig.2, the straight shocks $S_{0}$ and the corresponding
downstream state in $\Omega_{0}$ are given as the weak shock state on the
shock polar (Fig.3).

Figure 3. The shock polar for steady potential flow equation(${\bf u}_{0}$:
the weak shock state in $\Omega_{0}$ for $0<\theta_{w}<\theta_{\rm sonic}$)

Considering that the incoming supersonic flow has a constant density and a
constant velocity, the straight normal shock $S_{1}$ and the corresponding
downstream state in $\Omega_{1}$ (see Fig.2) can be easily computed from the
Rankine-Hugoniot jump conditions.

Since the density $\rho$ in $\Omega_{0}$(Fig.2) is a constant, it follows from
(1.12) and (1.13) that the pseudo-potential function $\varphi$ satisfies

$\begin{cases}\Delta\varphi+2=0\\\
\varphi+\frac{1}{2}|\nabla\varphi|^{2}=\text{a
constant}\end{cases}\quad\text{in $\Omega_{0}$},$

and this yields

(1.14) $\varphi(\bm{\xi})=-\frac{1}{2}|\bm{\xi}|^{2}+{\bf
u}_{0}\cdot{\bm{\xi}}+k\quad\text{in $\Omega_{0}$}$

for the constant vector ${\bf u}_{0}$ given on the shock polar (Fig.3), and
for a constant $k$.

Take the representation of $\varrho$ in terms of $(\varphi,\nabla\varphi)$,
directly given from (1.13). Substituting the representation
$\varrho(\varphi,\nabla\varphi)$ into (1.12), we are given with a second order
quasi-linear equation

(1.15)
$\nabla\cdot\left(\varrho(\varphi,\nabla\varphi)\nabla\varphi\right)+2\varrho(\varphi,\nabla\varphi)=0.$

If $\varrho^{\gamma-1}-|\nabla\varphi|^{2}<0$, the equation (1.15) is
hyperbolic, and if $\varrho^{\gamma-1}-|\nabla\varphi|^{2}>0$, it is elliptic.

Back to the uniform state in $\Omega_{0}$(Fig.1, Fig.2), let $\rho_{0}>0$
represent the constant density in $\Omega_{0}$. Then, for the pseudo-potential
function $\varphi$ given by (1.14), we have

$\mathcal{C}_{0}:=\\{\bm{\xi}:|\nabla\varphi|^{2}=\varrho_{0}^{\gamma-1}\\}=\\{|\bm{\xi}-{\bf
u}_{0}|^{2}=\varrho_{0}^{\gamma-1}\\}.$

The boundary portion $\Gamma_{0}:=\partial{\Omega_{0}}\cap\mathcal{C}_{0}$ is
called a _pseudo-sonic arc_ associated with the state of the density
$\rho_{0}$, and the velocity ${\bf u}_{0}$. In $\Omega_{0}$, the equation
(1.15) is hyperbolic, and the hyperbolicity is degenerate on the pseudo-sonic
arc $\partial{\Omega_{0}}\cap\mathcal{C}_{0}$. Note that the pseudo-supersonic
region $\Omega_{0}$ bounded by a straight oblique shock $S_{0}$ is shown for
$\theta_{w}\in(\theta_{\rm sonic},\frac{\pi}{2}]$ in Fig.1, and for
$\theta_{w}\in(0,\theta_{\rm sonic})$ in Fig.2. And, this region shrinks to a
point as $\theta_{w}$ approaches to the sonic angle $\theta_{\rm sonic}$,
which is uniquely determined by the shock polar associated with the incoming
flow state.

The region in which the equation (1.15) is
elliptic($\varrho^{\gamma-1}-|\nabla_{\bm{\xi}}\varphi|^{2}>0$) is indicated
as $\Omega$. For the admissible solution $\varphi$ constructed in [3, 5, 6],
one of the essential properties is that there exists a constant $\mu>0$
satisfying

$\frac{|\nabla\varphi(\bm{\xi})|}{\varrho^{\frac{\gamma-1}{2}}(\nabla\varphi(\bm{\xi}),\varphi(\bm{\xi}))}\leq
1-\mu\,{\rm dist}(\bm{\xi},\mathcal{C}_{0})\quad\text{for $\bm{\xi}\in\Omega$
near $\Gamma_{0}$}$

as long as the pseudo-supersonic region $\Omega_{0}$ appears (Fig.1,
Fig.2(left)).

Let $\varphi_{0}$ be the pseudo-potential function given by (1.14). Define a
polar coordinate system by

$\bm{\xi}-{\bf u}_{0}:=r(\cos\theta,\sin\theta).$

And, define a new coordinate system by

$(x,y):=\begin{cases}(\varrho_{0}^{\frac{\gamma-1}{2}}-r,\theta-\theta_{w})\quad&\mbox{for
$\bm{\xi}$ near $\Gamma_{0}$ in Fig. \ref{fig1}}\\\
(\varrho_{0}^{\frac{\gamma-1}{2}}-r,\pi+\theta_{w}-\theta)\quad&\mbox{for
$\bm{\xi}$ near $\Gamma_{0}$ in Fig. \ref{fig2}}\end{cases}$

Finally, define

(1.16) $\psi(x,y):=\varphi(\bm{\xi})-\varphi_{0}(\bm{\xi})\quad\text{in
$\Omega$ near $\Gamma_{0}$}.$

The admissible solutions constructed in [3, 5, 6] have the following
properties: if $\theta_{w}\in(\theta_{\rm sonic},\frac{\pi}{2}]$ in Fig. 1, or
if $\theta_{w}\in(0,\theta_{\rm sonic})$ in Fig. 2, then

  * (i)

$\exists$ a (small) constant $\varepsilon_{0}>0$ and a function
$f:[0,\varepsilon_{0}]\rightarrow\mathbb{R}_{+}$ such that

$\Omega\cap\\{{\rm
dist}(\bm{\xi},\Gamma_{0})<\varepsilon_{0}\\}=\mathcal{Q}_{\varepsilon_{0}}^{f}$

for the domain $\mathcal{Q}_{\varepsilon_{0}}^{f}$ defined by (1.2);

  * (ii)

such a function $f$, representing the curved pseudo-transonic shock
$\Gamma_{\rm{shock}}$ (Fig. 1, Fig. 2), satisfies all the properties stated in
(1.1);

  * (iii)

the equation (1.15) is rewritten as

$(2x-(\gamma+1)+O_{1})\psi_{xx}+O_{2}\psi_{xy}+\left(\frac{1}{\varrho_{0}^{\frac{\gamma-1}{2}}}+O_{3}\right)\psi_{yy}-(1+O_{4})\psi_{x}+O_{5}\psi_{y}=0$

for the terms $O_{j}(x,\psi,\psi_{x},\psi_{y})$ satisfying all the properties
stated in (1.8);

  * (iv)

$\psi$ satisfies the boundary conditions (1.4)–(1.6) with (1.7) holding;

  * (v)

$\displaystyle{\psi\in C^{2}(\mathcal{Q}_{\varepsilon_{0}}^{f}\cap
C^{1,1}(\overline{\mathcal{Q}_{\varepsilon_{0}}^{f}}))}$;

  * (vi)

$\displaystyle{0\leq\psi(x,y)\leq Lx^{2}}$ in
$\mathcal{Q}^{f}_{\varepsilon_{0}}$ for some constant $L>0$;

  * (vii)

$\displaystyle{0\leq\frac{\psi_{x}}{x}\leq\frac{2-\delta}{\gamma+1}}$ in
$\mathcal{Q}^{f}_{\varepsilon_{0}}$ for some constant
$\delta\in(0,\frac{1}{2})$.

Then it directly follows from Theorem 1 that

  * (a)

$\forall y\in[0,f(0))$,

$\lim_{x\to 0+}\psi_{xx}(x,y)=\frac{1}{\gamma+1};$

  * (b)

$\displaystyle{\psi|_{\overline{\mathcal{Q}^{f}_{\varepsilon_{0}}}}}$ is not
$C^{2}$ at the point $(0,f(0))$.

According to the statement (a), the radial derivative of the flow
velocity$(=\nabla_{\rm{\bf x}}\Phi)$ on the pseudo-sonic arc
$\Gamma_{0}$(Fig.1, Fig.2(left)) is nonzero. While the velocity field
$\nabla_{\rm{\bf x}}\Phi$ is discontinuous on the shock
$\overline{S_{0}\cup\Gamma_{\rm{shock}}}$, it is continuous on $\Gamma_{0}$.
What the statement (a) indicates, however, is that the pseudo-sonic arc
$\Gamma_{0}$ is a _weak discontinuity_ in the sense that a derivative of the
velocity field is discontinuous on $\Gamma_{0}$. We also point out that
another pseudo-sonic arc $\Gamma_{1}$ in Fig.2 due to the presence of a normal
shock state in $\Omega_{1}$ is also a _a weak discontinuity_. Now, a question
arises naturally:

_Given a second order equation with a degeneracy of Keldysh type, does the
degeneracy always result in a discontinuity in a second order derivative of
its solutions on the degenerate interface?_

Another example introduced in the next section indicates that the answer to
the question is ‘no’.

##  2\. A sonic interface as a regular interface

Given a constant $L>0$, define

$\Omega_{L}:=\\{{\rm{\bf
x}}=(x_{1},x_{2})\in\mathbb{R}^{2}:0<x_{1}<L,\,\,|x_{2}|<1\\}.$

The boundary $\partial\Omega_{L}$ consists of the entrance
$\Gamma_{0}=\\{0\\}\times[-1,1]$, $\Gamma_{w}:=(0,L)\times\\{\pm 1\\}$, and
the exit $\Gamma_{L}:=\\{L\\}\times[-1,1]$.

For two fixed constants $\gamma>1$ and $\bar{\rho}_{I}>0$, consider the steady
Euler-Poisson system

(2.1) $\displaystyle\nabla\cdot(\rho{\bf u})=0$
$\displaystyle\nabla\cdot(\rho{\bf u}\otimes{\bf u})+\nabla p=\rho\nabla\Phi$
$\displaystyle\nabla\cdot(\rho\mathcal{B}{\bf u})=\rho{\bf u}\cdot\nabla\Phi$
$\displaystyle\Delta\Phi=\rho-\bar{\rho}_{I}$

with

$\mathcal{B}=\frac{1}{2}|{\bf u}|^{2}+\frac{\gamma p}{(\gamma-1)\rho}$

for a fixed adiabatic exponent $\gamma>1$.

Any one-dimensional solution $(\rho,{\bf
u},p,\Phi)=(\bar{\rho},\bar{u}_{1}{\bf e}_{1},\bar{p},\bar{\Phi})(x_{1})$ with
$\bar{\rho}>0$ and $\bar{u}_{1}>0$ can be given as

$\displaystyle\bar{\rho}(x_{1})$ $\displaystyle=\frac{J}{\bar{u}_{1}(x_{1})}$
$\displaystyle\bar{p}(x_{1})$ $\displaystyle=S_{0}\bar{\rho}^{\gamma}(x_{1})$
$\displaystyle\bar{\Phi}(x_{1})$
$\displaystyle=\frac{u_{0}^{2}}{2}+\frac{\gamma
S_{0}}{\gamma-1}\left(\frac{J}{u_{0}}\right)^{\gamma-1}+\int_{0}^{x_{1}}\bar{E}(t)\,dt$

with $(\bar{u}_{1},\bar{E})$ solving

(2.2)
$\begin{cases}\bar{u}_{1}^{\prime}=\frac{\bar{E}\bar{u}_{1}^{\gamma}}{\bar{u}_{1}^{\gamma+1}-u_{s}^{\gamma+1}}\\\
\bar{E}^{\prime}=\frac{J}{\bar{u}_{1}}-\bar{\rho}_{I}\end{cases}\quad\text{for
$x_{1}>0$},\quad(\bar{u}_{1},\bar{E})(0)=(u_{0},E_{0})$

for constants $S_{0}>0$, $J>0$, $u_{0}>0$ and $E_{0}\in\mathbb{R}$. Here, the
constant $u_{s}$ represents the speed at the sonic state, and is explicitly
given by

$u_{s}=(\gamma S_{0}J^{\gamma-1})^{\frac{1}{\gamma+1}}.$

Define
$(\bar{u}_{I},\zeta_{0}):=(\frac{J}{\bar{\rho}_{I}},\frac{\bar{u}_{I}}{u_{s}})$,
and assume that

$\zeta_{0}>1.$

For $H:(0,\infty)\rightarrow\mathbb{R}$ given by

$H(u):=\frac{J}{\bar{u}_{I}}\int_{u_{s}}^{u}\frac{1}{t^{\gamma+1}}(t^{\gamma+1}-u_{s}^{\gamma+1})(\bar{u}_{I}-t)\,dt,$

if $(\bar{u}_{1},\bar{E})(x_{1})$ is a $C^{1}$-solution to (2.2), then it can
be directly checked from (2.2) that

$\frac{1}{2}\bar{E}^{2}-H(\bar{u}_{1})=\frac{1}{2}E_{0}^{2}-H(u_{0})\quad\text{for
$x_{1}>0$}$

as long as the solution exists.

We call a set

(2.3) $\mathcal{T}:=\\{(u,E):\frac{1}{2}E^{2}-H(u)=0\\}$

_the critical trajectory_ on the $uE$-plane. Further, we call a set

$\mathcal{T}_{\rm acc}:=\\{(u,E)\in\mathcal{T}:(u-u_{s})E\geq 0\\}$

_the critical trajectory with an acceleration_(Fig.4).

Figure 4. The critical trajectory

For the initial data $(u_{0},E_{0})$ in (2.2), suppose

(2.4) $(u_{0},E_{0})\in\mathcal{T}_{\rm acc},\quad u_{0}<u_{s}.$

######  Lemma 2.1 (Lemma 1.1 in [4]).

The initial value problem (2.2) with $(u_{0},E_{0})$ satisfying (2.4) has a
unique smooth solution $(\bar{u}_{1},\bar{E})$ with the following properties:

  * (i)

there exists a finite constant $l_{\rm max}>0$ such that

$\bar{u}_{1}^{\prime}(x_{1})>0\quad\text{for $x_{1}\in[0,l_{\rm max})$};$

  * (ii)

$\displaystyle{\lim_{x\to l_{\rm max}-}}\bar{u}_{1}^{\prime}(x_{1})=0$;

  * (iii)

$\mathcal{T}_{\rm acc}\cap\\{(\bar{u}_{1},\bar{E})(x_{1}):0\leq x_{1}\leq
l_{\rm max}\\}=\mathcal{T}_{\rm acc}\cap\\{(u,E):u\geq u_{0}\\}$ (Fig.4);

  * (iv)

there exists a unique constant $l_{\rm s}\in(0,l_{\rm max})$(Fig.5) such that

$\bar{u}_{1}(x_{1})\begin{cases}<u_{s}\quad&\mbox{for $x_{1}<l_{\rm s}$}\\\
=u_{s}\quad&\mbox{at $x_{1}=l_{\rm s}$}\\\ >u_{s}\quad&\mbox{for $x_{1}>l_{\rm
s}$}\end{cases}.$

Figure 5.

For the solution $(\bar{u}_{1},\bar{E})$ given in Lemma 2.1, we can write as

(2.5) $\bar{E}={\rm
sgn}(\bar{u}_{1}-u_{s})\sqrt{2H(\bar{u}_{1})}\quad\text{for $x_{1}\in[0,l_{\rm
max}]$}.$

Set

$\bar{\varphi}(x_{1}):=\int_{0}^{x_{1}}\bar{u}_{1}(t)\,dt.$

It easily follows from (2.2) and (2.5) that

$\left((\partial_{1}\bar{\varphi})^{\gamma+1}-u_{s}^{\gamma+1}\right)\partial_{11}\bar{\varphi}-{\rm
sgn}(\partial_{1}\bar{\varphi}-u_{s})\sqrt{2H(\partial_{1}\bar{\varphi})}(\partial_{1}\bar{\varphi})^{\gamma}=0\quad\text{for
$x_{1}\in[0,l_{\rm max}]$}.$

Therefore, $\bar{\varphi}$ can be regarded as a solution to a second order
equation with a degeneracy of Keldysh type, and the smoothness of
$\bar{\varphi}$ indicates that the degenerate interface $x_{1}=l_{s}$ is not a
weak discontinuity. So we are given with an example for a degenerate interface
of a new type for a second order equation with a degeneracy of Keldysh type.
Naturally, the following questions arise:

  * -

_Does there exist a multi-dimensional solution to ( 2.1) with a degenerate
interface?_

  * -

_If so, what is the regularity of the solution across the degenerate
interface?_

Let us assume that

$\mathcal{B}-\Phi=0,$

which we call _a pseudo Bernoulli’s law_ , and let us write the pressure
function $p$ as

$p=S\rho^{\gamma}.$

For $\rho>0$ and ${\bf u}\cdot{\bf e}_{1}(=:u_{1})>0$, the system (2.1) can be
rewritten as

(2.6) $\left\\{\begin{aligned} &\nabla\cdot({\rho{\bf u}})=0\\\
&\nabla\times{\bf u}=\frac{\rho^{\gamma-1}\partial_{2}S}{(\gamma-1)u_{1}}\\\
&{\rho{\bf u}}\cdot\nabla S=0\\\ &\frac{1}{2}|{\bf u}|^{2}+\frac{\gamma
S\rho^{\gamma-1}}{\gamma-1}=\Phi\\\
&\Delta\Phi=\rho-\bar{\rho}_{I}.\end{aligned}\right.$

Take the one-dimensional solution $(\bar{\rho},\bar{u}_{1}{\bf
e}_{1},\bar{p}(=S_{0}\bar{\rho}^{\gamma}),\bar{\Phi})$ with
$(\bar{u}_{1},\bar{E})$ given from Lemma 2.1. For a two-dimensional velocity
field ${\bf u}$, we represent as

${\bf u}=\nabla\varphi+\nabla^{\perp}\phi,$

and define

$(\psi,\Psi):=(\varphi,\Phi)-(\bar{\varphi},\bar{\Phi}).$

Then we can further rewrite (2.6) as a nonlinear system for
$(\psi,\Psi,\phi,S)$ as follows:

(2.7)
$\displaystyle\sum_{i,j=1}^{2}a_{ij}\partial_{ij}\psi+b_{1}\partial_{1}\psi+\bar{u}_{1}\partial_{1}\Psi+(\gamma-1)\bar{u}_{1}^{\prime}\Psi={f}_{1}(S,\nabla\psi,\nabla^{\perp}\phi,D(\nabla^{\perp}\phi)\Psi,\nabla\Psi)$
(2.8) $\displaystyle\Delta\Psi-\frac{1}{\gamma
S_{0}\bar{\rho}^{\gamma-2}}\Psi+\frac{\bar{u}_{1}}{\gamma
S_{0}\bar{\rho}^{\gamma-2}}\partial_{1}\psi={f}_{2}(S,\nabla\psi,\nabla^{\perp}\phi,\Psi,\nabla\Psi)$
(2.9)
$\displaystyle-\Delta\phi={f}_{3}(S,\nabla\psi,\nabla^{\perp}\phi,\Psi,\nabla\Psi)$
(2.10)
$\displaystyle(\nabla\bar{\varphi}+\nabla\psi+\nabla^{\perp}\phi)\cdot\nabla
S=0$

for

$\displaystyle a_{ij}$
$\displaystyle=(\gamma+1)\left(\bar{\Phi}+\Psi-\frac{1}{2}|{\bf
v}|^{2}\right)\delta_{ij}-({\bf v}\cdot{\bf e}_{i})({\bf v}\cdot{\bf
e}_{j})\,\,\text{with ${\bf
v}=\nabla\bar{\varphi}+\nabla\psi+\nabla^{\perp}\phi$}$ $\displaystyle b_{1}$
$\displaystyle=\bar{E}-(\gamma+1)\bar{u}_{1}^{\prime}\bar{u}_{1}.$

To find a two-dimensional solution to (2.1) as a small perturbation of
$(\bar{\rho},\bar{u}_{1}{\bf e}_{1},\bar{p},\bar{\Phi})$. we prescribe the
following boundary conditions:

(2.11) $\displaystyle{\bf u}\cdot{\bf e}_{2}=w_{\rm en},\quad S=S_{\rm
en},\quad\partial_{1}\Phi=E_{\rm en}$ on $\Gamma_{0}$ $\displaystyle{\bf
u}\cdot{\bf e}_{2}=0,\quad\partial_{2}\Phi=0$ on $\Gamma_{w}$
$\displaystyle\Phi=\bar{\Phi}$ on $\Gamma_{L}$

for three functions $w_{\rm en},\,S_{\rm en},\,E_{\rm
en}:[-1,1]\rightarrow\mathbb{R}$ satisfying

(2.12) $\mathcal{P}(w_{\rm en},S_{\rm en},E_{\rm en}):=\|w_{\rm
en}\|_{C}^{5}([-1,1])+\|(S_{\rm en},E_{\rm
en})-(S_{0},E_{0})\|_{C^{4}([-1,1])}\leq\sigma$

for some small constant $\sigma>0$, and satisfying the compatibility
conditions

$\displaystyle\left(\frac{d}{dx_{2}}\right)^{k}E_{\rm
en}=\left(\frac{d}{dx_{2}}\right)^{k}S_{\rm en}=0$ $\displaystyle\text{at
$|x_{2}|=1$ for $k=1,3$},$
$\displaystyle\left(\frac{d}{dx_{2}}\right)^{l}w_{\rm en}=0$ at $|x_{2}|=1$
for $l=0,2,4$.

In the framework of the Helmhotz decomposition, (2.11) becomes

(2.13) $\displaystyle\partial_{2}\psi=\omega_{\rm
en},\,\,\partial_{1}\phi=0,\,\,S=S_{\rm en},\,\,\partial_{1}\Psi=E_{\rm
en}-E_{0}$ on $\Gamma_{0}$
$\displaystyle\partial_{2}\psi=0,\,\,\phi=0,\,\,\partial_{2}\Psi=0$ on
$\Gamma_{w}$ $\displaystyle\phi=0,\,\,\Psi=0$ $\displaystyle\mbox{on
$\Gamma_{L}$}.$

The boundary condition $\phi=0$ on $\Gamma_{L}$ is added in (2.13) for the
well-posedness of the boundary value problem (2.7)–(2.10) with (2.13) for
$(\psi,\Psi,\phi,S)$. As we seek for a solution $(\psi,\Psi,\phi,S)$ with
$\|(\psi,\Psi,\phi)\|_{W^{1,\infty}}$ being small, we first investigate a
modified equation of (2.7):

(2.14)
$\bar{a}_{11}\partial_{11}w+\bar{a}_{22}\partial_{22}w+b_{1}\partial_{1}w=f$

for $(\bar{a}_{11},\bar{a}_{22})=(a_{11},a_{22})$ with $\psi=\phi=\Psi=0$. For
the normalized coefficients

(2.15)
$\alpha_{11}:=\frac{\bar{a}_{11}}{\bar{a}_{22}}=1-\left(\frac{\bar{u}_{1}}{u_{s}}\right)^{\gamma+1},\quad\beta_{1}:=\frac{b_{1}}{\bar{a}_{22}}=\frac{(\bar{E}-(\gamma+1)\bar{u}_{1}^{\prime}\bar{u}_{1})\bar{u}_{1}^{\gamma-1}}{u_{s}^{\gamma+1}},$

define a linear differential operator

(2.16)
$\mathcal{L}w:=\alpha_{11}\partial_{11}w+\partial_{22}w+\beta_{1}\partial_{1}w.$

According to Lemma 2.1(iv), the operator $\mathcal{L}$ is elliptic in
$\Omega_{L}\cap\\{x_{1}<l_{s}\\}$, hyperbolic in
$\Omega_{L}\cap\\{x_{1}>l_{s}\\}$, and degenerate on
$\Omega_{L}\cap\\{x_{1}=l_{s}\\}$. Therefore, it is a mixed type operator with
a degeneracy of Keldysh type if $L>l_{s}$. From this, it is easy to see that
the equation (2.7), as a second order equation for $\psi$, is a mixed type
with a degeneracy of Keldysh type.

One of differences between the two equations (1.3) and (2.7) (or (2.14)) is
that the equation (1.3) is hyperbolic before($x<0$) the degenerate
boundary(=sonic arc), and turns to be elliptic after($x>0$) the degenerate
interface, while the equation (2.14) changes its type from being elliptic to
being hyperbolic across the degenerate interface. Does this difference lead to
a different result on the regularity of solutions to the boundary value
problem (2.7)–(2.10) with (2.13)?

######  Theorem 2 (Theorem 2 in [4]).

Given constants $(\gamma,\zeta_{0},J,S_{0},E_{0})$ satisfying

$\gamma>1,\quad\zeta_{0}>1,\quad S_{0}>0,\quad E_{0}<0,$

suppose that $(u_{0},E_{0})\in\mathcal{T}_{\rm acc}$, which is equivalent to
$0<u_{0}<u_{s}$. Then one can fix two constants $\bar{J}$ and $\underline{J}$
depending only on $(\gamma,\zeta_{0},S_{0})$ with
$0<\bar{J}<1<\underline{J}<\infty$ so that whenever the background momentum
density $J(=\bar{\rho}\bar{u}_{1})$ satisfies

either $0<J\leq\bar{J}$ or $\underline{J}\leq J<\infty$,

there exists a constant $d\in(0,1)$ depending on $(\gamma,\zeta_{0},S_{0},J)$
so that if the two constants $u_{0}$ and $L$ are fixed to satisfy

(2.17) $1-d\leq\frac{u_{0}}{u_{s}}<1<\frac{\bar{u}_{1}(L)}{u_{s}}\leq 1+d,$

and if the constant $\sigma>0$ in (2.12) is fixed sufficiently small depending
only on $(\gamma,\zeta_{0},S_{0},E_{0},J,L)$, then the boundary value problem
(2.7)–(2.10) with (2.13) has a unique solution $(\psi,\phi,\Psi,S)$ that
satisfies the estimate

(2.18) $\begin{split}\|(\psi,\phi,\Psi,S-S_{0})\|_{H^{4}(\Omega_{L})}\leq
C\mathcal{P}(S_{\rm en},E_{\rm en},w_{\rm en})\end{split}$

for some constant $C>0$ depending on $(\gamma,\zeta_{0},S_{0},E_{0},J,L)$.

Furthermore, there exists a function $\mathfrak{f}_{\rm
sn}:[-1,1]\rightarrow(0,L)$ such that

(2.19) $\frac{|{\bf u}|}{\sqrt{\gamma
S\rho^{\gamma-1}}}\begin{cases}<1\quad&\mbox{for $x_{1}<\mathfrak{f}_{\rm
sn}(x_{2})$}\\\ =1\quad&\mbox{for $x_{1}=\mathfrak{f}_{\rm sn}(x_{2})$}\\\
>1\quad&\mbox{for $x_{1}>\mathfrak{f}_{\rm sn}(x_{2})$}\end{cases},$

and the function $\mathfrak{f}_{\rm sn}$ satisfies

(2.20) $\|\mathfrak{f}_{\rm sn}-l_{\rm
s}\|_{H^{2}((-1,1))}+\|\mathfrak{f}_{\rm sn}-l_{\rm s}\|_{C^{1}([-1,1])}\leq
C\mathcal{P}(S_{\rm en},E_{\rm en},w_{\rm en})$

for some constant $C>0$ depending on $(\gamma,\zeta_{0},S_{0},E_{0},J,L)$.

We are given from Theorem 2 with a classical solution $(\rho,{\bf u},p)$ of
(2.1) with the sonic interface $x_{1}=\mathfrak{f}_{\rm sn}(x_{2})$ across
which the velocity field ${\bf u}$ is not only continuous, but also its
derivative is continuous. In other words, $x_{1}=\mathfrak{f}_{\rm sn}(x_{2})$
is a degenerate interface but not a weak discontinuity. We call this sonic
interface as a _regular interface_ in the sense that the solution $(\rho,{\bf
u},p)$ is a classical solution across the sonic interface.

##  3\. Discussion

In §1, we show an example of a (pseudo) sonic interface as a weak
discontinuity in the sense that a velocity field is continuous but its
derivative is discontinuous on the interface. Such an example is given from a
solution to a mixed-type equation with a degeneracy of Keldysh type. More
precisely, the type of the equation changes from being hyperbolic to being
elliptic through a degenerate interface of codimension one. In §2, we show an
example of a sonic interface as a regular interface on which a velocity field
is $C^{1}$. This example is also given from a solution to a degenerate
equation of Keldysh type, where the equation changes its type from being
elliptic to being hyperbolic.

The interesting point is that the equations given in §1–2 are both Keldysh
type, and that the lower order derivative terms ‘$(-1+O_{4})\psi_{x}$’ from
(1.3), and ‘$b_{1}\partial_{1}\psi$’ from (2.7) have significant contributions
to Theorem 1 and Theorem 2, respectively. In spite of those similarities, the
equations (1.3) and (2.7) give different regularity results. Why?

###  3.1. The existence of a special smooth solution

In §2, the classical solution of (2.1) with a sonic interface as a regular
interface(Theorem 2) is constructed as a small perturbation of a smooth one-
dimensional solution(Lemma 2.1). Therefore, one may suggest that the existence
of a special smooth solution can be a clue to determine a destiny of a sonic
interface in general.

It is true that a smooth one-dimensional solution comes in handy to
reformulate (2.1) into (2.7)–(2.10) so that an iteration method can be
applied. But still, it is another matter to establish the well-posedness of
the boundary value problem (2.7)–(2.10) with (2.13). According to the work in
[4], the essential ingredient used to prove Theorem 2 is the strictly
increasing property of $\bar{u}_{1}$ stated in Lemma 2.1(i). This monotonicity
property combined with the method developed in [9] yields the well-posedness
of a linearized boundary value problem derived from (2.7)–(2.10) with (2.11),
and this yields Theorem 2.

Figure 6. Self-similar normal shock reflection ($S_{0}$: reflected normal
shock, $\varphi=\varphi_{0}$ in $\Omega_{0}\cup\Gamma_{0}\cup\Omega$, see [5])

But, there is another example that shows that the existence of a special
smooth solution does not necessarily tells the destiny of a sonic interface in
general. As repeatedly pointed out, the sonic arc $\Gamma_{0}$ in a self-
similar regular shock reflection is a weak discontinuity for
$\theta_{w}\in(\theta_{\rm sonic},\frac{\pi}{2})$(see §1). For the wedge-angle
$\theta_{w}=\frac{\pi}{2}$, however, the sonic arc is not a weak
discontinuity(Fig. 6) because the state behind the reflected normal shock
$S_{0}$ is simply given by $\varphi=\varphi_{0}$ for a quadratic polynomial
function $\varphi_{0}$ of self-similar variables $\bm{\xi}=(\xi_{1},\xi_{2})$
in the form of (1.14). In [5], a self-similar regular shock reflection
solution for $\theta_{w}\in(\frac{\pi}{2}-\sigma,\frac{\pi}{2})$ is
constructed as a small perturbation of the normal shock reflection solution,
but the nature of the sonic arc $\Gamma_{0}$ for $\theta_{w}<\frac{\pi}{2}$ is
not determined by the arc $\Gamma_{0}$ for $\theta_{w}=\frac{\pi}{2}$.

###  3.2. Acceleration VS. deceleration in a transonic transition

Notice that the self-similar flow(Fig.1 and Fig.2(left)) introduced in §1
turns from being (pseudo) supersonic in $\Omega_{0}$ to being (pseudo)
subsonic in $\Omega$ through the (pseudo) sonic arc $\Gamma_{0}$. This is a
decelerating transition on $\Gamma_{0}$, and Theorem 1 implies that
$\Gamma_{0}$ is a weak discontinuity. On the other hand, the $C^{1}$-transonic
solution of Euler-Poisson system (2.1) given by Theorem 2 in §2 has an
accelerating transition on the sonic interface $x_{1}=\mathfrak{f}_{\rm
sn}(x_{2})$.

######  Question.

Does a decelerating transonic transition always yield a sonic interface as a
weak discontinuity, while accelerating transonic transition does not?

Here is another interesting example to investigate. For the set $\mathcal{T}$
given by (2.3), define

$\mathcal{T}_{\rm dec}:=\\{(u,E)\in\mathcal{T}:(u-u_{s})E\leq 0\\}.$

We call the set $\mathcal{T}_{\rm dec}$ _the critical trajectory with a
deceleration_. By modifying the proof of Lemma 2.1 given in [4], one obtains
the following lemma:

######  Lemma 3.1.

For the initial data $(u_{0},E_{0})$ in (2.2), suppose

$(u_{0},E_{0})\in\mathcal{T}_{\rm dec},\quad u_{0}>u_{s}.$

Then, the initial value problem (2.2) has a unique smooth solution
$(\bar{u}_{1},\bar{E})$ with the following properties:
(Case 1) For $1<\gamma<2$,

  * (i)

there exists a finite constant $\tilde{l}_{\rm max}>0$ such that

$\bar{u}_{1}^{\prime}(x_{1})<0\quad\text{for $x_{1}\in[0,\tilde{l}_{\rm
max})$};$

  * (ii)

$\displaystyle{\lim_{x_{1}\to\tilde{l}_{\rm
max}-}\bar{u}_{1}^{\prime}(x_{1})=0\,\,\text{and}\,\,\lim_{x_{1}\to\tilde{l}_{\rm
max}-}\bar{E}(x_{1})=\infty}$;

  * (iii)

$\mathcal{T}_{\rm dec}\cap\\{(\bar{u}_{1},\bar{E})(x_{1}):0\leq
x_{1}\leq\tilde{l}_{\rm max}\\}=\mathcal{T}_{\rm dec}\cap\\{(u,E):u\leq
u_{0}\\}$ (Fig.4);

  * (iv)

there exists a unique constant $\tilde{l}_{s}\in(0,\tilde{l}_{\rm max})$ such
that

$\bar{u}_{1}(x_{1})\begin{cases}>u_{s}\quad&\mbox{for
$x_{1}<\tilde{l}_{s}$}\\\ =u_{s}\quad&\mbox{at $x_{1}=\tilde{l}_{s}$}\\\
<u_{s}\quad&\mbox{for $x_{1}>\tilde{l}_{s}$}\end{cases}.$

(Case 2) For $\gamma\geq 2$, the statements given in (i)–(iv) hold with
$\tilde{l}_{\rm max}=\infty$.

The smooth one-dimensional solution of the Euler-Poisson system (2.1) has a
decelerating speed and a sonic interface at $x_{1}=\tilde{l}_{s}$. Similarly
to Theorem 2, would it be possible to establish the existence of a classical
solution to (2.1) as a small perturbation of this one-dimensional decelerating
smooth transonic solution? In proving Theorem 2, the key property used in [4]
is that there exists a constant $\lambda_{L}>0$ satisfying

(3.1) $-2\beta_{1}-(2m-1)\partial_{1}\alpha_{11}\geq\lambda_{L}\quad\text{in
$\Omega_{L}$ for $m=0,1,2,3$}$

for the coefficients $\alpha_{11}$ and $\beta_{1}$ given by (2.15). This
inequality enables to apply [9, Theorem 1.7] to achieve a priori $H^{m+1}$
estimates of solutions obtained from Theorem 2 for $m=0,1,2,3$. More
precisely, a direct computation yields the representation

$-2\beta_{1}-(2m-1)\partial_{1}\alpha_{11}=\frac{\bar{u}_{1}^{\prime}}{u_{s}^{\gamma+1}}\left(2m(\gamma+1)\bar{u}_{1}^{\gamma}+(\gamma-1)\bar{u}_{1}^{\gamma}+2\frac{u_{s}^{\gamma+1}}{\bar{u}_{1}}\right).$

And, the inequality (3.1) is obtained by Lemma 2.1.

For one-dimensional smooth transonic solution $(\bar{u}_{1},\bar{E})$ lying on
the critical trajectory $\mathcal{T}_{\rm dec}$ with a deceleration, Lemma 3.1
implies that the inequality (3.1) does not hold because
$\bar{u}_{1}^{\prime}<0$. So one cannot apply [9, Theorem 1.7] to establish
the existence of a classical solution to (2.1) as a small perturbation of
$(\bar{u}_{1},\bar{E})$. This opens to a possibility that a multi-dimensional
solution to (2.1) given as a small perturbation of $(\bar{u}_{1},\bar{E})$
with $\bar{u}_{1}^{\prime}<0$ may contain a sonic interface as a weak
discontinuity. This is an open problem to be investigated in the future.

Acknowledgements: The author shows her sincere respect to Prof. Gui-Qiang
Chen’s for his ambition and contribution to mathematics. The research of
Myoungjean Bae was supported in part by Samsung Science and Technology
Foundation under Project Number SSTF-BA1502-51.

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