# Quantum control attack on quantum key distribution systems

Anton Kozubov<EMAIL_ADDRESS>Andrei Gaidash  George Miroshnichenko
Department of Photonics and Optical Information Technology, ITMO
University,199034 Kadetskaya Line 3b, Saint Petersburg, Russia  Quanttelecom
Ltd., Saint Petersburg, 199034 Birzhevaya Line 14, Russia.

(August 27, 2024)

###### Abstract

In this paper we present the quantum control attack on quantum key
distribution systems. The cornerstone of the attack is that Eve can use
unitary (polar) decomposition of her positive-operator valued measure
elements, which allows her to realize the feed-forward operation (quantum
control), change the states in the channel after her measurement and impose
them to Bob. Below we consider the general eavesdropping strategy and the
conditions those should be satisfied to provide the attack successfully.
Moreover we consider several types of the attack, each of them is based on a
different type of discrimination. We also provide the example on two non-
orthogonal states and discuss different strategies in this case.

††preprint: APS/123-QED

Quantum key distribution (QKD) systems Scarani  _et al._ (2009); Diamanti  _et
al._ (2016) in the last decades extends beyond the laboratory research and go
straight ahead to the market. Nevertheless a lot of vulnerabilities for real
QKD systems are not covered in the various researches in both theoretical and
experimental areas. Its is well-known that all implemented equipment is non-
ideal. Due to this fact some attacks can be performed on real QKD systems.
However the common assumption on which most existing theoretical security
proofs are based is that the eavesdropper has no access to receiving and
transmitting equipment. However in real life implementations she can influence
the hardware of legitimate user in some way and one need to take this into
consideration.

Thus possible attacks can generally be separated in two categories. The first
one is the attacks on quantum states in the channel. In this field there are a
huge amount of works Koashi (2004); Renner and Wolf (2005); Dupuis  _et al._
(2016); Arnon-Friedman  _et al._ (2018); Ekert  _et al._ (1994); Kozubov  _et
al._ (2019) that cover different aspects of theoretical security. All of them
are based on consideration of various properties of used states. A lot of
techniques were proposed for estimating the secure key since the very fist
days of QKD era. However some protocols still do not have strict security
proof. Some of them still have open questions, i.e. the security proof for
coherent-one-way (COW) and deferential-phase-shift (DPS) protocols against
coherent attacks still do not exist.

Attacks on the hardware realization or so-called "quantum hacking" belongs to
the second category. There are huge amount of possible attacks on the hardware
realization of QKD systems. One of the most crucial attack, which can be
implemented to the most types of existing systems, is detector blinding or
faked-state attack Lydersen  _et al._ (2010); Makarov (2009); Makarov* and
Hjelme (2005); Vakhitov  _et al._ (2001); Lydersen  _et al._ (2011); Gerhardt
_et al._ (2011). A lot of experimental demonstrations of such attack was
proposed for different QKD systems. Many researchers are trying to find the
appropriate hardware solution. Most of them are based on some modifications of
single-photon detectors (SPD). However there are still no approved
experimental solutions allowing to overcome this attack. The only reliable way
to deal with such Eve’s strategy yet is measurement-device-independent (MDI)
Lo  _et al._ (2012); Braunstein and Pirandola (2012); Ma and Razavi (2012);
Tamaki  _et al._ (2012); Liu  _et al._ (2013); Yin  _et al._ (2016) or twin-
field (TF) Lucamarini  _et al._ (2018) QKD systems.

In this letter we propose the approach of how to take into theoretical
consideration the possibility of eavesdropper’s interaction with Bob’s
detection node by considering quantum control attack assuming Eve may "rule"
Bob’s detector using appropriate states. Our approach is appropriate for
protocols based on any particular kind of used states. Such clarification is
important since we can construct any appropriate positive-operator valued
measure (POVM). Different Eve’s strategies based on various POVM constructions
will be considered in this paper.

General description of the attack. – Proposed in this letter quantum controll
attack relates to the class of intercept-resend attacks. However, similar
attacks was considered earlier for different particular cases, for instance in
the paper Lucamarini  _et al._ (2009) it was considered for the case of single
photons and in Gaidash  _et al._ (2018) for the case of linearly independent
states. In this letter we present generalized approach for any particular kind
of states, however the most crucial case is when linearly independent states
are used in the protocol. The protocol of the attack can be described as
follows:

(i) Eve exchanges the channel with lossless channel and make her initial
states (ancillas) $|\Psi\rangle_{\mathcal{E}}$ interact with the states
prepared by Alice
$\\{|u_{1}\rangle_{\mathcal{A}},...,|u_{n}\rangle_{\mathcal{A}}\\}$ using
unitary operator $U_{\mathcal{AE}}$

(ii) Eve constructs the appropriate POVM and measure her states that were
interacted with Alice’s ones
$\\{|\Psi^{u_{1}}\rangle_{\mathcal{E}},\dots,|\Psi^{u_{n}}\rangle_{\mathcal{E}}\\}$

(iii) According to the measurement result she alter the states in the channel
using appropriate unitary operator from polar decomposition of Kraus operator
and impose them to Bob by sending them directly to him. All distinguished by
Eve states should be altered and resend directly to Bob providing detection
event with probability maximally close to unity. Otherwise the appropriate
unitary operator (the detailed description could be found further) should be
applied to the states in the quantum channel in order to prevent detection
event on Bob’s detector.

It can be easily noted that this attack is quite similar to the detector
blinding attack. Let us clarify the statement about detector blinding. The
main idea of detector blinding attack is that the eavesdropper blinds Bob’s
avalanche detectors by irradiating the receiving equipment with continuous
wave (CW) laser radiation, thus switching them into a linear classical mode
(or appropriate operation to control SNSPD detector). In case when Eve
distinguishes the state she performs detection event at the Bob’s detector.
Otherwise there will be no detection event the detector.

Lets consider several possible strategies of the attack based on implementing
different types of POVM to discriminate the states. In the first case of
linearly independent states Eve can simply use unambiguous state
discrimination measurement and identify states without errors, but with
inconclusive results. The implementation of the attack and countermeasures was
proposed in Kozubov  _et al._ (2019); Gaidash  _et al._ (2019).

Nevertheless, it is not always possible to construct errorless USD POVM. For
example, there can be set of more then two states, which are linearly
dependent and Eve cannot construct USD POVM. Thus she can discriminate those
states with some error probability and without inconclusive results, e.g.
Bennett  _et al._ (1992).

However in general case Eve may construct arbitrary POVM which contains both
errors and inconclusive results (in particular she use the same detection
scheme as Bob does only assuming that her hardware is perfect or at least
better). Moreover this strategy is quite suitable for experimental
verification.

As it was mentioned above the proposed attack is based on the fact that
despite the structure of the states Eve can provide some distinguishing
measurement. Also very important feature here is that we do assume the fact
that efficiency of our detector is not unit and there are always losses in the
channel and at the Bob’s side.

Thus the attack can be described as the extension of (or general) intercept-
resend attack assuming Eve can control the Bob’s detector (equal to detector
blinding). Since the main idea of both attacks (quantum control attack and
detector blinding) is the imposing of discriminated by Eve states to Bob, they
can be considered almost similar. Lets consider it as follows.

Let $\mathcal{A},\mathcal{B},\mathcal{E}$ be Hilbert spaces of Alice, Bob and
Eve respectively and
$\\{|u_{1}\rangle_{\mathcal{A}},...,|u_{n}\rangle_{\mathcal{A}}\\}$ is the
space of states prepared by Alice. At the beginning Alice’s states and Eve’s
initial states (ancillas) $|\Psi\rangle_{\mathcal{E}}$ do not interact with
each other. To obtain any information about Alice’s states they should somehow
interact with Eve’s ancillas. It can be provided by using some unitary
operation $U_{\mathcal{AE}}$ as follows:

$\displaystyle\begin{cases}|u_{1}\rangle_{\mathcal{A}}\otimes|\Psi\rangle_{\mathcal{E}}\\\
\vdots\\\
|u_{n}\rangle_{\mathcal{A}}\otimes|\Psi\rangle_{\mathcal{E}}\end{cases}\xrightarrow{U_{\mathcal{AE}}}\begin{cases}|\tilde{u}_{1}\rangle_{\mathcal{B}}\otimes|\Psi^{u_{1}}\rangle_{\mathcal{E}}\\\
\vdots\\\
|\tilde{u}_{n}\rangle_{\mathcal{B}}\otimes|\Psi^{u_{n}}\rangle_{\mathcal{E}}\end{cases},$
(1)

where
$\\{|\tilde{u}_{1}\rangle_{\mathcal{B}},...,|\tilde{u}_{n}\rangle_{\mathcal{B}}\\}$
and
$\\{|\Psi^{u_{1}}\rangle_{\mathcal{E}},|\Psi^{u_{n}}\rangle_{\mathcal{E}}\\}$
are the spaces of altered states in quantum channel and Eve’s ancillas
accordingly after Eve implies the unitary operation $U_{\mathcal{AE}}$. To
impose any states Eve has to somehow discriminate them before it (with
inconclusive results, errors, or both of them). According to the fact that
this is intercept-resend attack Eve has to distinguish every single qubit.
Since this Eve needs to construct the POVM with the next family of positive
semi-definite operators $\\{A^{x}_{\mathcal{E}}\\}_{x\in\mathcal{X}}$:

$\displaystyle\mathbb{I}=\sum_{x}A^{x}_{\mathcal{E}}.$ (2)

Since we consider that the attack is equivalent to the detector blinding that
means that Eve wants to control Bob’s detector by altering the states in the
channel. To do so she must resend all the states where she got detection event
and be sure that Bob will also have detection event with probability equal to
unity. All inconclusive results should be blocked. This can be done using
unitary decomposition technique (also called polar decomposition or quantum
control) Shapiro  _et al._ (1987, 1986); Feng  _et al._ (2018); Mendes (2009)
in order to alter the states in the channel according to the measurement
result. Thus Eve can apply unitary decomposition to the elements of POVM as
follows:

$\displaystyle
K_{\mathcal{EB}}^{x}=V^{x}_{\mathcal{BE}}\left(\sqrt{A^{x}_{\mathcal{E}}}\otimes\mathbb{I}_{\mathcal{B}}\right),$
(3)

where $K_{\mathcal{EB}}^{x}$ is the Kraus operator, $V^{x}_{\mathcal{BE}}$ is
the unitary operator, which allows to alter the states after measurement and
index $x$ denotes the set of possible outcomes of implemented POVM. Such
decomposition allows to realize co-called feed-forward operation (quantum
control). It is quite similar to preparing the new states which form
dependence on the measurement results. Let us consider the conditional
probabilities of all possible outcomes according to the generalized
measurement:

$\displaystyle\mathcal{P}(?|u_{i})=\text{Tr}_{\mathcal{E}}\left(A^{?}_{\mathcal{E}}|\Psi^{u_{i}}\rangle_{\mathcal{E}}\langle\Psi^{u_{i}}|\right),$
(4)
$\displaystyle\mathcal{P}(u_{j}|u_{i})=\text{Tr}_{\mathcal{E}}\left(A^{u_{j}}_{\mathcal{E}}|\Psi^{u_{i}}\rangle_{\mathcal{E}}\langle\Psi^{u_{i}}|\right),$
(5)

here $\mathcal{P}(?|u_{i})$ denotes the conditional probabilities of
inconclusive results measuring $u_{i}$, here and further when $i=j$ we assume
the case correct distinguishing, otherwise with errors, thus
$\mathcal{P}(u_{j}|v_{i})$ denotes the conditional probabilities of
unambiguous discrimination if $i=j$, otherwise the conditional probabilities
of errors accordingly.

The most important thing about the discrimination for Eve is the number of her
detection events. Generally it should be lower-bounded with Bob’s detection
rate in order that Eve can maintain the latter:

$\displaystyle\mathcal{G}_{\mathcal{E}}\geq\mathcal{G}_{\mathcal{B}},$ (6)

with

$\mathcal{G}_{\mathcal{E}}=N\cdot\left(\mathcal{P}(u_{i}|u_{i})+\mathcal{P}(u_{j}|u_{i})\right),$
(7)

where $N$ is the number of sent by Alice qubits. The states (in the channel
and at the Eve’s side) after Eve’s measurement can be presented as:

$\displaystyle|\tilde{u}_{i}\rangle_{\mathcal{B}}|\Psi^{u_{i}?}\rangle_{\mathcal{E}}=\frac{\sqrt{A^{?}_{\mathcal{E}}}|\tilde{u_{i}}\rangle_{\mathcal{B}}|\Psi^{u_{i}}\rangle_{\mathcal{E}}}{\sqrt{\mathcal{P}(?|u_{i})}},$
(8)
$\displaystyle|\tilde{u}_{i}\rangle_{\mathcal{B}}|\Psi^{\tilde{u}_{j}}\rangle_{\mathcal{E}}=\frac{\sqrt{A^{u_{j}}_{\mathcal{E}}}|\tilde{u}_{i}\rangle_{\mathcal{B}}|\Psi^{u_{i}}\rangle_{\mathcal{E}}}{\sqrt{\mathcal{P}(u_{j}|u_{i})}},$
(9)

where
$\\{|\Psi^{u_{i}?}\rangle_{\mathcal{E}},|\Psi^{\tilde{u}_{j}}\rangle_{\mathcal{E}}\\}$
are the states of Eve’s ancillas after her measurement. However she do not
obtain any information from $|\Psi^{u_{i}?}\rangle_{\mathcal{E}}$ since all of
them relate to the inconclusive results.

When the measurement is provided Eve can implement feed-forward unitary
operation as follows:

$\displaystyle\begin{cases}|\tilde{u}_{i}\rangle_{\mathcal{B}}\otimes|\Psi^{u_{i}?}\rangle_{\mathcal{E}}\xrightarrow{V^{\emptyset}_{\mathcal{BE}}}|\tilde{\tilde{u}}_{i}^{?}\rangle_{\mathcal{B}}\otimes|\tilde{\Psi}^{u_{i}?}\rangle_{\mathcal{E}}\\\
|\tilde{u}_{i}\rangle_{\mathcal{B}}\otimes|\Psi^{u_{j}}\rangle_{\mathcal{E}}\xrightarrow{V_{\mathcal{BE}}}|\tilde{\tilde{u}}_{i}\rangle_{\mathcal{B}}\otimes|\tilde{\Psi}^{u_{j}}\rangle_{\mathcal{E}}\end{cases}$
(10)

here
$\\{|\tilde{\tilde{u}}_{i}\rangle_{\mathcal{B}},|\tilde{\tilde{u}}_{i}^{?}\rangle_{\mathcal{B}}\\}$
and
$\\{|\tilde{\Psi}^{u_{i}?}\rangle_{\mathcal{E}},|\tilde{\Psi}^{u_{j}}\rangle_{\mathcal{E}}\\}$
are altered states after applying unitary operators $V_{\mathcal{BE}}$ and
$V^{\emptyset}_{\mathcal{BE}}$ in quantum channel and at Eve’s side
respectively, where operators $V_{\mathcal{BE}}$ and
$V^{\emptyset}_{\mathcal{BE}}$ denotes unitary operators for conclusive result
of the measurement and inconclusive outcome respectively. In case of
$|\Psi^{u_{i}?}\rangle_{\mathcal{E}}$ Eve apply unitary operator
$V^{\emptyset}_{\mathcal{BE}}$ to $|\tilde{u}_{i}\rangle_{\mathcal{B}}$
changing it with vacuum states, otherwise she increases the absolute value of
their amplitudes. It should be mentioned, $V_{\mathcal{BE}}$ can be chosen in
a way that from the Bob’s point of view there will be no additional errors
introduced by Eve despite she had errors in her measurement results. Thereby
Bob cannot reveal Eve by monitoring the error rate. However, in real-world
implementation there is always errors due to the non-ideality of equipment,
POVM construction etc. Thus Eve should maintain both errors and detection
rate. It can be done by constructing operators $V^{\emptyset}_{\mathcal{BE}}$
and $V_{\mathcal{BE}}$ in the way to maintain error and detection rates at the
Bob’s side:

$\displaystyle\begin{cases}E_{\mathcal{E}}(\xi,\zeta)=E_{\mathcal{B}},\\\
\mathcal{G}_{\mathcal{E}}(\xi,\zeta)=\mathcal{G}_{\mathcal{B}},\end{cases}$
(11)

where $E_{\mathcal{B}}$ and $E_{\mathcal{E}}(\xi,\zeta)$ is the expected error
rate at the Bob’s side and the error rate which should be provided by Eve
respectively. Here $\xi$ and $\zeta$ are the parameters to satisfy the
conditions above (e.g. intensity and coding parameters changes respectively).
According to the overlapping preservation for unitary operation we assume that

${}_{\mathcal{B}}\langle\tilde{u}_{i}|\tilde{u}_{j}\rangle_{\mathcal{BE}}\langle\Psi^{i}|\Psi^{j}\rangle_{\mathcal{E}}=_{\mathcal{B}}\langle\tilde{\tilde{u}}_{k}|\tilde{\tilde{u}}_{l}\rangle_{\mathcal{BE}}\langle\tilde{\Psi}^{k}|\tilde{\Psi}^{l}\rangle_{\mathcal{E}},$
(12)

where indexes $i,j$ denotes possible initial states at the Alice and Eve’s
side, and indexes $k,l$ denotes possible outcomes of the measurement including
inconclusive result, correct distinguishing and error. Using proposed
technique Eve can maintain both detection and error rates regarding to the
presence of errors and inconclusive results at the Bob side (they always
appear due to non-orthogonality of the states, losses in the channel and non-
ideal equipment). Generally also different possible strategies can be applied
to altering of the states. Since in this work we are interested in imposing of
states distinguished by Eve. Here Eve provides the case when she exchanges all
the states when she has inconclusive results with vacuum states (using
appropriate unitary operator) and otherwise alters the states in the way to
provide detection event with probability maximally close to unity. Thus this
attack is closely related to the detector blinding (faked-state) attack, since
Eve impose her discriminated states to Bob directly. However, Eve also can
construct her unitary operation in that way to maintain the states in quantum
channel and pretend that she does not interact with them.

Example. – In this section let us consider the simplest case with two linearly
independent states. The space of states prepared by Alice can be denoted as
$\\{|u\rangle_{\mathcal{A}},|v\rangle_{\mathcal{A}}\\}$. The initial state of
Eve’s states (ancillas) $|\Psi\rangle_{\mathcal{E}}$ do not interact with
states prepared by Alice at the beginning. To obtain any information about
Alice’s states Eve should somehow make them interact with her ancillas. It can
be provided by using some unitary operation $U_{\mathcal{AE}}$ as follows:

$\displaystyle\begin{cases}|u\rangle_{\mathcal{A}}\otimes|\Psi\rangle_{\mathcal{E}}\\\
|v\rangle_{\mathcal{A}}\otimes|\Psi\rangle_{\mathcal{E}}\end{cases}\xrightarrow{U_{\mathcal{AE}}}\begin{cases}|\tilde{u}\rangle_{\mathcal{B}}\otimes|\Psi^{u}\rangle_{\mathcal{E}}\\\
|\tilde{v}\rangle_{\mathcal{B}}\otimes|\Psi^{v}\rangle_{\mathcal{E}}\end{cases},$
(13)

here $\\{|\tilde{u}\rangle_{\mathcal{B}},|\tilde{v}\rangle_{\mathcal{B}}\\}$
and $\\{|\Psi^{u}\rangle_{\mathcal{E}},|\Psi^{v}\rangle_{\mathcal{E}}\\}$ are
the spaces of altered states in quantum channel and Eve’s ancillas accordingly
after Eve implies the unitary operation $U_{\mathcal{AE}}$. Lets also assume
that the overlapping of Eve’s states after applying unitary operation is
denoted as follows:

$w=_{\mathcal{E}}\langle\Psi^{u}|\Psi^{v}\rangle_{\mathcal{E}}.$ (14)

The elements of POVM, since we assume that discrimination probabilities should
be the same for all states, can be constructed using appropriate non-
normalized vectors on the same plane as the signal states, that can be denoted
as follows:

$\displaystyle|\phi_{u}\rangle_{\mathcal{E}}=|\Psi^{u}\rangle_{\mathcal{E}}-\mu|\Psi^{v}\rangle_{\mathcal{E}},$
(15)
$\displaystyle|\phi_{v}\rangle_{\mathcal{E}}=\mu|\Psi^{u}\rangle_{\mathcal{E}}-|\Psi^{v}\rangle_{\mathcal{E}}.$
(16)

Then positive semi-definite operators operators of POVM can be denoted as
follows:

$\displaystyle
A_{\mathcal{E}}^{u}=\frac{1}{\delta}|\phi_{u}\rangle_{\mathcal{E}}\langle\phi_{u}|,$
(17) $\displaystyle
A_{\mathcal{E}}^{v}=\frac{1}{\delta}|\phi_{u}\rangle_{\mathcal{E}}\langle\phi_{u}|,$
(18) $\displaystyle A_{\mathcal{E}}^{?}=1-A_{u}-A_{v},$ (19)

here parameter $\mu$ denotes the measurement regime of POVM and $\delta$
optimizes POVM performance, reducing the probability of inconclusive result,
but maintaining the positive definiteness of operator $A_{\mathcal{E}}^{?}$.
Let us consider the probabilities of all possible outcomes for one of the
states according to the Born rule:

$\displaystyle\mathcal{P}(u|u)=\frac{1}{\delta}|_{\mathcal{E}}\langle\Psi^{u}|\phi_{u}\rangle_{\mathcal{E}}|^{2}=\frac{1}{\delta}(1-\mu\cdot
w)^{2}$ (20)
$\displaystyle\mathcal{P}(v|u)=|\frac{1}{\delta}|_{\mathcal{E}}\langle\Psi^{u}|\phi_{v}\rangle_{\mathcal{E}}|^{2}=\frac{1}{\delta}(w-\mu)^{2}$
(21) $\displaystyle\mathcal{P}(?|u)=1-\mathcal{P}(u|u)-\mathcal{P}(v|u),$ (22)

here $\mathcal{P}(?|u)$ denotes the probability of inconclusive result,
$\mathcal{P}(u|u)$ and denotes the probability of unambiguous discrimination
and $\mathcal{P}(v|u)$ is the probability of errors accordingly. The vectors
are chosen so that the matching conditions are met:

$\displaystyle\mathcal{P}(u|u)=\mathcal{P}(v|v),$ (23)
$\displaystyle\mathcal{P}(v|u)=\mathcal{P}(u|v),$ (24)
$\displaystyle\mathcal{P}(?|u)=\mathcal{P}(?|v).$ (25)

Eve’s detection rate can be obtained applying to the states the next operator
$A_{\mathcal{E}}^{D}$:

$\displaystyle A_{\mathcal{E}}^{D}=A_{\mathcal{E}}^{u}+A_{\mathcal{E}}^{v}.$
(26)

Lets find the maximum eigenvalue $\lambda_{\max}$ of this operator.
Optimization parameter $\delta$ is chosen then according to the next
condition:

$\displaystyle\lambda_{\max}=1.$ (27)

This condition allows us to reduce the average probability of an inconclusive
result and maintain the $A_{\mathcal{E}}^{?}$ operator’s positive
definiteness. Denote the eigenvector of the operator as follows:

$\displaystyle A_{\mathcal{E}}^{D}|\chi\rangle=\lambda|\chi\rangle,$ (28)
$\displaystyle|\chi\rangle=\alpha|\Psi^{u}\rangle_{\mathcal{E}}+\beta|\Psi^{v}\mathcal{E}$
(29)

In the matrix form, the equation can be presented as:

$\displaystyle\frac{1}{\delta}\begin{pmatrix}1+\mu^{2}-2\mu\cdot
w&(1+\mu^{2})w-2\mu\\\ (1+\mu^{2})w-2\mu&1+\mu^{2}-2\mu\cdot
w\end{pmatrix}\begin{pmatrix}\alpha\\\
\beta\end{pmatrix}=\lambda\begin{pmatrix}\alpha\\\ \beta\end{pmatrix}$ (30)

Assuming the following condition

$\frac{2\mu}{1+\mu^{2}}\geq w$ (31)

we obtain the solution:

$\displaystyle\lambda_{\max}=\frac{1}{\delta}(1-w)(1+\mu)^{2}.$ (32)

Thus the probabilities of measurement outcomes can be redefined as follows:

$\displaystyle\mathcal{P}(u|u)=\frac{(1-\mu\cdot w)^{2}}{(1-w)(1+\mu)^{2}},$
(33) $\displaystyle\mathcal{P}(v|u)=\frac{(w-\mu)^{2}}{(1-w)(1+\mu)^{2}},$
(34)
$\displaystyle\mathcal{P}(?|u)=\frac{(1+w)(1+\mu)^{2}}{(1-w)(1+\mu)^{2}}\left(\frac{2\mu}{1+\mu^{2}}-w\right).$
(35)

Lets consider different POVM strategies. The first case is errorless USD POVM
can be obtained if $\mu=w$. Then the probabilities of different outcomes has
the next form:

$\displaystyle\mathcal{P}(u|u)=1-w,$ (36) $\displaystyle\mathcal{P}(v|u)=0,$
(37) $\displaystyle\mathcal{P}(?|u)=w$ (38)

However, if Eve implement such discrimination without errors, she can be
revealed by increasing the number of states, since the discrimination
probability is a function of number of states.

Another degenerate case is when her discrimination POVM does not produce any
inconclusive results (measurement in Breidbard basisBennett  _et al._ (1992))
can be obtained if the following condition is satisfied

$\displaystyle\frac{2\mu}{1+\mu^{2}}=w,$ (39)
$\displaystyle\mu=\frac{1-\sqrt{1-w^{2}}}{w}.$ (40)

In this case the detection rate $\mathcal{G}_{\mathcal{E}}$ is equal to unity,
so she can satisfy the condition in eq. 6. Then the probabilities of different
can be calculated using the following expressions:

$\displaystyle\mathcal{P}(?|u)=0,$ (41)
$\displaystyle\mathcal{P}(u|u)=\frac{w^{2}}{2(1-\sqrt{1-w^{2}})},$ (42)
$\displaystyle\mathcal{P}(v|u)=\frac{w^{2}}{2(1+\sqrt{1-w^{2}})}.$ (43)

Nevertheless in this case Eve produces a huge amount of errors and can be
easily revealed by error rates.

Thus the only way for her to provide the successful attack is to implement the
general case, then her POVM has both errors and inconclusive results. It can
be done by changing a parameter $\mu$. The simplest way to implement such POVM
is just to construct the same as Bob’s one. If she does so the condition in
eq. 6 will be naturally satisfied, since Eve has ideal (or at least better)
equipment. To provide the attack successfully she should satisfy the next
condition:

$\displaystyle N\frac{(1-\mu\cdot
w)^{2}+(w-\mu)^{2}}{(1-w)(1+\mu)^{2}}\geq\mathcal{G}_{\mathcal{B}},$ (44)

It is quite obvious that $\mu$ can be chosen in that way to satisfy the
condition. The states after Eve measurement can be presented as:

$\displaystyle|\tilde{u}\rangle_{\mathcal{B}}|\Psi^{u?}\rangle_{\mathcal{E}}=\frac{\sqrt{A^{?}_{\mathcal{E}}}|\tilde{u}\rangle_{\mathcal{B}}|\Psi^{u}\rangle_{\mathcal{E}}}{\sqrt{\mathcal{P}(?|u)}},$
(45)
$\displaystyle|\tilde{u}\rangle_{\mathcal{B}}|\Psi^{\tilde{u}}\rangle_{\mathcal{E}}=\frac{\sqrt{A^{u}_{\mathcal{E}}}|\tilde{u}\rangle_{\mathcal{B}}|\Psi^{u}\rangle_{\mathcal{E}}}{\sqrt{\mathcal{P}(u|u)}},$
(46)
$\displaystyle|\tilde{u}\rangle_{\mathcal{B}}|\Psi^{\tilde{v}}\rangle_{\mathcal{E}}=\frac{\sqrt{A^{v}_{\mathcal{E}}}|\tilde{u}\rangle_{\mathcal{B}}|\Psi^{u}\rangle_{\mathcal{E}}}{\sqrt{\mathcal{P}(v|u)}},$
(47)

After measuring her states Eve should directly resend all her discriminated
states (both with error or without) with probability equal or close to unity.
As it was shown in previous section she should apply unitary trace-preserving
operators $V_{\mathcal{BE}}$ and $V^{\emptyset}_{\mathcal{BE}}$ which should
be constructed according to eq. 11 as follows:

$\displaystyle\begin{cases}|\tilde{u}\rangle_{\mathcal{B}}\otimes|\Psi^{u}\rangle_{\mathcal{E}}\xrightarrow{V_{\mathcal{BE}}}|\tilde{\tilde{u}}\rangle_{\mathcal{B}}\otimes|\tilde{\Psi}^{u}\rangle_{\mathcal{E}}\\\
|\tilde{u}\rangle_{\mathcal{B}}\otimes|\Psi^{u?}\rangle_{\mathcal{E}}\xrightarrow{V^{\emptyset}_{\mathcal{BE}}}|\tilde{\tilde{u}}^{?}\rangle_{\mathcal{B}}\otimes|\tilde{\Psi}^{u?}\rangle_{\mathcal{E}}\\\
|\tilde{u}\rangle_{\mathcal{B}}\otimes|\Psi^{v}\rangle_{\mathcal{E}}\xrightarrow{V_{\mathcal{BE}}}|\tilde{\tilde{u}}\rangle_{\mathcal{B}}\otimes|\tilde{\Psi}^{v}\rangle_{\mathcal{E}}\end{cases},$
(48)

here
$\\{|\tilde{\tilde{u}}\rangle_{\mathcal{B}},|\tilde{\tilde{u}}^{?}\rangle_{\mathcal{B}}\\}$
and
$\\{|\tilde{\Psi}^{u}\rangle_{\mathcal{E}},|\tilde{\Psi}^{u?}\rangle_{\mathcal{E}},|\tilde{\Psi}^{v}\rangle_{\mathcal{E}}\\}$
are the states after applying operators $V_{\mathcal{BE}}$ and
$V^{\emptyset}_{\mathcal{BE}}$ in quantum channel and Eve’s ancillas
accordingly to the measurement outcome as it was shown in previous section.
The operator can alter the states in any appropriate way.

The possible countermeasures against such attack are closely related with
countermeasures against faked-state attack. Since Eve should maintain the
overlapping and detection rate at the Bob’s side she should increase the of
the states. Thereby the possible way to reveal Eve is to use photon-number-
resolving (PNR) detectors which allows to monitor the statistics of the
radiation (e.g. the common approach can be seen in Gaidash  _et al._ (2016)).
Also the countermeasure from Chistiakov  _et al._ (2019) can be implemented.
Another possible solution is presented in Ko  _et al._ (2018), where
coincidence detection events are monitored.

Conclusion. – In this paper we present the technique which allows Eve to
provide so-called quantum control attack. The attack is based on the fact,
that Eve wants to impose discriminated states to Bob and thus obtain up to
100% of the shared key. We consider the general case then Eve can implement
any appropriate POVM which allows her (or not) to obtain the necessary number
detection events to maintain both detection and error rates at the Bob’s side.
The cornerstone of this paper is that considered attack can be implemented to
any states. However not every suggested strategy can be realized successfully
if the proposed conditions are not satisfied by Eve. Nevertheless, such
technique allows us to take into theoretical consideration the detector
blinding quantum hacking attack. Moreover we propose the possible
countermeasures for different types of the attack. This approach allows us to
make theoretical security proofs more accurate not only for ideal-world
considerations but the real-world implementations.

This work was financially supported by Government of Russian Federation (Grant
08-08).

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# Nova Sagittarii 1943 (V1148 Sgr): A Luminous Red Nova?

Howard E. Bond Department of Astronomy & Astrophysics, Pennsylvania State
University, University Park, PA 16802, USA  Space Telescope Science Institute,
3700 San Martin Dr., Baltimore, MD 21218, USA  Jessica Mink Center for
Astrophysics — Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138,
USA  Alison Doane  Center for Astrophysics — Harvard & Smithsonian, 60 Garden
Street, Cambridge, MA 02138, USA  Deceased 2017 October 29  Sarah Lavallee
Center for Astrophysics — Harvard & Smithsonian, 60 Garden Street, Cambridge,
MA 02138, USA Howard E. Bond<EMAIL_ADDRESS>
###### Abstract

Nova Sagittarii 1943 (V1148 Sgr) was an 8th-mag optical transient that was
unusual in having a late-type spectrum during its outburst, in striking
contrast to the normal high-excitation spectra seen in classical novae.
Unfortunately, only an approximate position was given in the discovery
announcement, hampering follow-up attempts to observe its remnant. We have
identified the nova on two photographic plates in the Harvard archive,
allowing us to determine a precise astrometric position. Apart from these two
plates, obtained in 1943 and 1944, none of the photographs in the Harvard
collection, from 1897 to 1950, show V1148 Sgr to limits as faint as $g\simeq
18.3$. Modern deep images show a candidate remnant at $i\simeq 19.2$, lying
only $0\farcs 26$ from the site of the nova. V1148 Sgr may have been a
luminous red nova (LRN), only the sixth one known in the Milky Way. However,
it lacks the near- and mid-infrared excesses, and millimeter-wave emission,
seen in other LRNe, leaving its nature uncertain. We urge spectroscopy of the
candidate remnant.

##  1 The Unusual Optical Transient
Nova Sagittarii 1943

The 8th-magnitude optical transient Nova Sagittarii 1943 (V1148 Sgr) was
discovered on material in the Harvard College Observatory’s Astronomical
Photographic Glass Plate Collection111For more information:
https://library.cfa.harvard.edu/plate-stacks by Margaret Mayall.
Unfortunately, her discovery was presented only briefly in a meeting abstract
several years later (Mayall, 1949), and to our knowledge no further
information was ever published. Nevertheless, V1148 Sgr has remained of
considerable interest because of the unusual spectroscopic characteristics
that she described. Mayall’s discovery was made on objective-prism plates,
obtained with the 10-inch Metcalf telescope at Bloemfontein, South Africa. She
described the spectrum of V1148 Sgr on the discovery photograph, taken on 1943
August 19, as resembling that of a late K-type star, with strong absorption
lines at Ca I 4226 Å and Ca II H and K, and possibly absorption bands of TiO.
Five days later, the Ca I and Ca II absorption continued to be strong, but the
spectrum now had a “banded” appearance, with wide Balmer emission features.
The final spectrum, on 1943 August 29, showed “wide bright bands.” These
findings are extremely unusual for a classical nova (CN), which develops a
high-excitation emission-line spectrum, without prominent absorption features,
soon after the outburst (e.g., Williams et al., 1991; Walter et al., 2012). In
retrospect, it seems possible that the bright emission bands described by
Mayall at the final epoch were actually a misinterpretation of a spectrum of a
very cool object with strong molecular absorption bands, as further discussed
below.222Mayall was very familiar with the normal spectral evolution of CNe,
having announced her objective-prism discoveries of five other novae in her
1949 abstract, along with many others during her long career. Thus it is
significant that she did not describe any of her three plates of V1148 Sgr as
exhibiting a normal nova spectrum.

Unfortunately, Mayall’s three objective-prism plates are no longer available
in the Harvard Plate Collection, so her spectroscopic findings cannot be
investigated. Moreover, Mayall provided only approximate coordinates for V1148
Sgr, which lies in an extremely crowded field in the Galactic bulge [Galactic
coordinates $(l,b)=(5\fdg 2,-3\fdg 0)$]. Thus, until now, its precise location
has been unknown. However, Mayall reported that the nova had also been
detected on three direct plates in the Harvard collection, raising the
possibility that it could be recovered from that material.

##  2 Astrometry

Of the three direct plates mentioned by Mayall, two are available in the
Harvard archive. One of them is plate SB4269, obtained on 1944 August 20
(about one year after the light-curve maximum) with the 60-inch Rockefeller
Reflector at Bloemfontein, which is of excellent quality. It clearly shows the
nova, albeit at a faint level. The transient is marked in ink on the glass
side of the plate, presumably by Mayall herself. We digitized this plate with
a UMAX Technologies scanner, as described by Mink et al. (2006). Using the
resulting .fits image, we identified about a dozen nearby reference stars that
are contained in the recent Gaia Early Data Release 3 (EDR3; Gaia
Collaboration et al. 2016, 2021), and measured their $(x,y)$ positions using
the imexamine task in IRAF.333IRAF was distributed by the National Optical
Astronomy Observatories, operated by AURA, Inc., under cooperative agreement
with the National Science Foundation. We corrected their EDR3 positions from
the Gaia 2000 epoch444We obtained the Gaia astrometry from the Vizier service
at https://vizier.u-strasbg.fr/viz-bin/VizieR-3?-source=I/350/gaiaedr3, which
provides positions adjusted to the 2000 epoch using the measured proper
motions. to the epoch of the plate, using the proper motions given in the Gaia
catalog, and employed them to place a precise astrometric frame on the image.
The rms scatter of the registration was $0\farcs 027$ and $0\farcs 019$ in
right ascension and declination, respectively. The derived position of V1148
Sgr (J2000 equinox, 1944.6 epoch) is:

18:09:07.801, $-$25:59:13.41 .

This position differs by some $24^{\prime\prime}$ from the approximate one
given by Mayall. We note that Duerbeck (1987) had attempted to identify the
remnant of V1148 Sgr, and gave coordinates for a 14th-mag star; this was the
brightest object close to the Mayall position. He stated that it is “very
probably not the exnova.” This object is detected on the SB4249 plate (and was
one of the Gaia reference stars555Duerbeck’s star is at J2000 equinox and
epoch position 18:09:05.847, $-$25:59:08.08, which is $27\farcs 52$ away from
the location of V1148 Sgr. used in our astrometric solution); it is clearly
distinct from the nova.

##  3 Photometry

The Digital Access to a Sky Century at Harvard (DASCH) project (Grindlay et
al., 2009) is a long-term effort aimed at digitizing many of the plates in the
Harvard collection, and performing photometry on the images. The DASCH
Lightcurve Access website666http://dasch.rc.fas.harvard.edu/lightcurve.php
provides the photometry (calibrated to the Sloan $g$ band) and image cutouts.
We searched for plates covering the location of V1148 Sgr, finding some 191
images. Of these only one plate, RB12347, obtained on 1943 August 24, shows
the nova. Stellar images on this plate are elongated, but nevertheless the
nova is conspicuous, at magnitude $g=9.55\pm 0.34$, according to the
photometry provided by the DASCH website. The star’s position, magnitude, and
observation date are consistent with the data presented by Mayall. This bright
object coincides with the transient seen a year later, but considerably
fainter, on SB4269, as described in the previous section. Thus it is clearly
V1148 Sgr.

The 60-inch Rockefeller plates are not being scanned at this time by the DASCH
project, so the DASCH website does not provide a magnitude determination for
the 1944 observation recorded on SB4269. Instead, we performed aperture
photometry on the nova and about a dozen neighboring stars, using the image
scan described in the previous section. We then calibrated the data using $g$
magnitudes from the Pan-STARRS1 catalog777Obtained using the image-access tool
at https://ps1images.stsci.edu/cgi-bin/ps1cutouts. Note that the Harvard
plates were obtained using unfiltered blue-sensitive emulsions, so that the
$g$ bandpass is a fairly close approximation to their sensitivity function.
(Chambers et al., 2016) for the neighbor stars. This analysis yielded a
magnitude of $g=16.72\pm 0.35$ for V1148 Sgr, at the date 1944 August 20.

There are 138 pre-eruption plates available from the DASCH website, covering
the range 1897 August 4 to 1943 August 2. We examined the image cutouts for
these plates, and V1148 Sgr was not detected on any of them. After the
outburst there are 51 plates, from 1944 August 23 to 1950 July 17, which
likewise do not show the transient. The limiting $g$ magnitudes vary widely
for these plates, but generally range from $\sim$15.5 to $\sim$16.5 for most
of them. However, from 1930 to 1939, some 74 plates of the site were obtained
with the 24-inch Bruce Doublet, and these plates reach as faint as $\sim$18.0
to 18.2 mag for the best ones. The last two plates taken in 1943 before the
detection near maximum on August 19 were obtained on 1943 May 15 and August 2.
The limiting $g$ magnitudes on these two plates are $\sim$15.9 and 15.5,
respectively. The DASCH database contains no plates between the one on 1943
August 24 showing the nova near maximum and the SB4269 plate obtained on 1944
August 20. Two more plates were obtained in 1944, both of them on August 23;
neither one shows the nova; however, their limiting magnitudes are only
$\sim$15.5 and 16.3. The best post-outburst plate, A26840, was obtained on
1949 June 8, and does not show the nova to a limiting magnitude of $\sim$18.3.

Figure 1:  Image cutouts from digital scans of four plates from the Harvard
College Observatory’s Astronomical Photographic Glass Plate Collection,
centered on the site of the transient V1148 Sgr. Each frame is
$350^{\prime\prime}$ high; north at the top, east on the left. Outskirts of
the globular cluster NGC 6553 are visible at the northeast corners of the
frames. Left panel: 1935 June 4. V1148 Sgr is not detected to a $g$ magnitude
of $\sim$17.3. Second panel: 1943 August 24. The nova is near maximum
brightness, at magnitude 9.55. Third panel: 1944 August 20. V1148 Sgr (between
the two ink marks, and encircled in red) is detected at 16.7 mag. Fourth
panel: post-outburst plate, taken on 1949 June 8. V1148 Sgr is fainter than
$g\simeq 18.3$ mag.

Figure 1 shows four image cutouts from Harvard plates. The left panel shows
the nova site on plate A17599 (24-inch Bruce Doublet, 1935 June 4). The
limiting magnitude on this plate is about 17.3, based on reference to the Pan-
STARRS1 sequence described above. There is no object detected at the nova
position. The second frame in the figure is from the plate RB12347 described
above (3-inch Ross Fecker telescope, 1943 August 24). Here the nova is
conspicuous, near maximum light, in spite of the shallow limiting magnitude of
$\sim$14.6 because of the trailed stellar images on this plate. The cutout in
the third panel of Figure 1 shows the nova (between the two ink marks) on
plate SB4269 (1944 August 20); this is the plate used for the astrometry and
photometry described above. The fourth panel in the figure shows a cutout from
the excellent plate A26840, obtained on 1949 June 8. As noted above, the
limiting detection magnitude on this image is about 18.3, and the nova is not
seen.

In summary, there are two Harvard direct plates showing V1148 Sgr, obtained on
1943 August 24 (mag 9.55) and 1944 August 20 (mag 16.72). These are in
addition to the three objective-prism detections reported by Mayall (1949) on
1943 August 19, 24, and 29, at a reported brightest magnitude of $\sim$8\. The
rise to maximum occurred sometime between 1943 August 2 and 19. The deepest
pre-outburst plates do not show a progenitor brighter than $\sim$18.0–18.2
mag. By mid-1949 the nova was fainter than $g$ magnitude 18.3.

##  4 A Candidate Remnant

Figure 2 shows a modern deep color optical image of the site of V1148 Sgr,
obtained from the Pan-STARRS1 website cited above. The red cross at the center
of the frame marks the astrometric position of the nova, given in §2.
Immediately southwest of the position is a faint star. This object is
contained in the Gaia EDR3 catalog, with magnitudes of $G=19.81$ and $G_{\rm
RP}=18.41$, lying at a J2000 position of

18 09 07.782, $-$25 59 13.47 ,

corresponding to a separation of only $0\farcs 26$ from the astrometric
position of the nova. This star is thus a candidate remnant of V1148 Sgr.
Unfortunately, Gaia EDR3 does not list a parallax or proper motion for the
object. The star is also contained in the Pan-STARRS1 photometric catalog,
with magnitudes of $(r,i,z,y)=(19.98,19.18,18.58,18.29)$.

Figure 2:  Color rendition of the field surrounding the site of V1148 Sgr,
created using the PanSTARRS-1 Image Access tool. Frame is $55^{\prime\prime}$
high, and has north at the top, east on the left. Red, green, and blue
correspond to the $y$, $i$, and $g$ filters, respectively. A small red cross
marks the astrometric position of V1148 Sgr. The immediately adjacent star to
the southwest, with $i$ magnitude 19.18, is a candidate remnant, as discussed
in the text.

There is also a fainter star $0\farcs 97$ north-northwest of, and partially
blended with, the candidate in Figure 2. Neither the candidate nor the
neighbor appear to have unusually red (or blue) colors. As the figure shows,
the Galactic-bulge field in which V1148 Sgr lies is extremely crowded, and it
is entirely possible that these two stars are merely chance interlopers.
Spectroscopic observations would provide further constraints.

Note that, because of the unknown proper motion of the nova, its current
position could be slightly offset from the epoch 1944.6 location. For example,
assuming a tangential velocity of $100\,{\>\rm km\>s^{-1}}$, the displacement
from 1944.6 to 2015 would be $\sim$$1\farcs 5$ and $\sim$$0\farcs 2$ for
distances of 1000 and 8000 pc, respectively.

##  5 Membership in Globular Cluster NGC 6553?

As noted by Mayall (1949), V1148 Sgr lies near the Galactic globular cluster
NGC 6553,888Due to a typographical error, Mayall designated the cluster NGC
6533. whose outskirts can be seen at the northeast corner of the frames in
Figure 1. This raises the question whether the nova could be a member of the
cluster. Only one, or possibly two, CNe are known to have occurred in the
Galactic globular-cluster system (e.g., Doyle et al., 2019, and references
therein).

V1148 Sgr is located $300^{\prime\prime}$ from the center of NGC 6553. The
tidal radius of the cluster, calculated from information in Harris
(2010),999Online version of 2010 December, at
http://physwww.mcmaster.ca/~harris/mwgc.dat is $\sim$$460^{\prime\prime}$.
Thus cluster membership is not ruled out. However, as Figure 2 shows, the nova
site is immersed in a rich Galactic-bulge field population. Color-magnitude
and proper-motion diagrams of stars in Gaia EDR3 lying within
$90^{\prime\prime}$ of V1148 Sgr show a nearly pure bulge population, with
very few cluster members.101010The color-magnitude diagram of the cluster,
from the Gaia catalog, exhibits a clump of very red horizontal-branch stars,
and the Gaia proper motions of the cluster stars are tightly concentrated.
These make it easy to distinguish the members from the field.

A tighter constraint on cluster membership would be provided by the proper
motion of the nova. If the faint star noted above is the remnant, then by
combining its Gaia position with our astrometry from the 1944 plate, we
calculate a proper motion of $(\mu_{\alpha},\mu_{\delta})=(-4.5,-1.1)\,\rm
mas\,yr^{-1}$. This is in poor agreement with the mean proper motion of the
cluster, for which we find $(\mu_{\alpha},\mu_{\delta})=(+0.4,-0.5)\,\rm
mas\,yr^{-1}$ from data in the Gaia catalog.

In summary, the sparse available information casts doubt on, but does not
absolutely rule out, a physical association of V1148 Sgr with the globular
cluster NGC 6553. It is possible that a future data release from Gaia may
provide a parallax, and a proper motion independent of the astrometry of the
nova, for the candidate remnant. This could produce a more critical membership
test—but only if evidence emerges that the candidate described in §4 is in
fact the remnant.

##  6 Was V1148 Sgr a Luminous Red Nova?

Luminous red novae (LRNe) are a class of dust-forming astrophysical transients
that has been recognized in recent years. CNe become increasingly blue as
their eruptions proceed and their ejecta become optically thin, exposing a hot
central source. LRNe erupt suddenly, like CNe, but they become extremely cool
and red after their outbursts, due to an expanding photosphere and dust
formation in the outflow. LRNe are generally considered to be the result of
binaries undergoing common-envelope interactions and stellar mergers (see, for
example, recent papers by Ivanova et al. 2013, Pastorello et al. 2019, and
Howitt et al. 2020, and references therein); this was definitely the case for
the Galactic LRN V1309 Sco, which was shown to have been a close binary with a
rapidly decreasing orbital period before its eruption (Tylenda et al., 2011).

The known LRNe that have occurred in the Milky Way are V4332 Sgr (Martini et
al., 1999; Bond, 2018), V838 Mon (Sparks et al., 2008; Woodward et al., 2021),
OGLE-2002-BLG-360 (Tylenda et al., 2013), CK Vul (Nova Vul 1670; Kamiński et
al., 2021a), and V1309 Sco. In the Local Group, three LRNe have been recorded
in M31: M31 RV (Bond, 2011), M31LRN 2015 (MacLeod et al., 2017), and AT
2019zhd (Pastorello et al., 2021). LRNe are often associated with old
populations, including elliptical galaxies and the bulges of the Milky Way and
other spiral galaxies; however, V838 Mon arose from a young open star cluster
(Afşar & Bond, 2007), and several extragalactic LRNe appear to have originated
from massive stars (e.g., Smith et al., 2016; Blagorodnova et al., 2017,
2021). (There is an extensive literature on LRNe, and we have cited only a few
relevant and recent papers.)

In the case of V1148 Sgr, the late-type spectrum at the discovery epoch
described by Mayall (1949) may make it a candidate for another Galactic
LRN,111111A Galactic-bulge microlensing event is ruled out by the changing
spectrum reported by Mayall. similar to prototypes like V4332 Sgr and V838
Mon—if so, only the sixth one known in the Milky Way. Indeed, several authors
have made the suggestion that V1148 Sgr was a LRN, including Bond & Siegel
(2006), Retter et al. (2006), and Smith et al. (2016). The apparent similarity
of V1148 Sgr to V838 Mon and other red transients was noted by Kimeswenger
(2007), who made an unsuccessful attempt to identify its remnant by searching
the vicinity for H$\alpha$-emitting objects. However, our candidate, described
in §4, is considerably fainter than the objects he considered.

The fragmentary observations discussed in §3 indicate that V1148 Sgr declined
some $\sim$10 magnitudes over the first year since its outburst, at blue
wavelengths. This is not dramatically different from V838 Mon’s drop of about
9 mag in $B$ in the first year after its eruption (e.g., Figure 5 in Sparks et
al., 2008).

Mayall’s description of the nova’s spectrum, at the final objective-prism
observation, as having bright emission bands (see §1) remains a puzzle. A
normal CN does rapidly develop a spectrum with broad, high-excitation emission
lines (see the references cited in §1). However, Mayall did not characterize
the spectrum as being that of a normal CN, indicating that it was unusual. It
is conceivable that in fact the spectrum had, by late August 1943, become that
of an extremely cool object, as is the case for LRNe. For example, Barsukova
et al. (2007, their Figure 1) and Munari et al. (2007, their Figure 2) show
late-time blue-region spectra of V838 Mon that are dominated by extremely
strong molecular absorption. These can give the spectrum a superficial
appearance of having strong emission bands, in the form of the continuum
levels between the molecular bands. Unfortunately, since Mayall’s spectrum
plates are now lost, it is not possible to verify this interpretation.

An argument against V1148 Sgr being a LRN comes from the fact that all of the
known recent Galactic LRNe became conspicuous near- and mid-infrared sources
following their outbursts: V4332 Sgr (Banerjee et al., 2015), V838 Mon
(Woodward et al., 2021), OGLE-2002-BLG-360 (Tylenda et al., 2013), and V1309
Sco (Tylenda & Kamiński, 2016). The remnant of the much older event CK Vul is
not bright in the mid-infrared, but shows a far-infrared flux excess (Evans et
al., 2002).

Figure 3:  Images of the site of V1148 Sgr at the optical $r$ band (from
PanSTARRS-1) and at five near- and mid-infrared wavelengths (from the Spitzer
Space Telesceope). Frames are centered on the position of the transient, have
the same $55^{\prime\prime}$ height as in Figure 2, and have north at the top,
east on the left. No near- or mid-infrared counterpart of the nova is
detected.

To investigate whether there is a near- or mid-infrared source at the site of
V1148 Sgr, we downloaded “Super-Mosaic” images obtained with the Spitzer Space
Telescope from the NASA/IPAC Infrared Science
Archive.121212https://irsa.ipac.caltech.edu/data/SPITZER/Enhanced/SEIP/overview.html
In Figure 3 we show frames with the same $55^{\prime\prime}$ angular height as
in Figure 2. The left-hand image is in the optical ground-based $r$ band, from
the PanSTARRS-1 website, centered on the coordinates of the nova. The
remaining panels show Spitzer frames from the Infrared Array Camera at 3.6,
4.5, 5.8, and $8.0\,\mu$m, and from the Multi-Band Imaging Photometer at
$24\,\mu$m. These images demonstrate that there is no bright near- or mid-
infrared source at the location of V1148 Sgr. The candidate remnant discussed
in §4 is only marginally detected in the two shortest-wavelength Spitzer
frames.

In addition, CK Vul, along with V4332 Sgr, V838 Mon, and V1309 Sco, all lie at
the centers of cool, compact molecular nebulae, which are prominent at
millimeter wavelengths (Kamiński et al., 2018, 2021a, 2021b). Kamiński et al.
(2022) attempted to detect CO emission from V1148 Sgr with the Atacama
Pathfinder Experiment (APEX) 12-m submillimeter telescope, with unsuccessful
results. Unfortunately, they pointed at the Duerbeck position discussed in §2.
This would have placed the nova outside the APEX beam size, which ranged from
$13\farcs 2$ to $18\farcs 7$. However, Kamiński has informed us privately that
his team has also observed V1148 Sgr with the Submillimeter
Array131313https://lweb.cfa.harvard.edu/sma/ (SMA), with a beam large enough
to include the revised position reported here. No detection was made in the
SMA data.

##  7 Summary and Future Studies

We have used Harvard plate material to determine a precise astrometric
position for V1148 Sgr, an optical transient that appears to have had some
spectroscopic features consistent with the handful of other members of the
class of LRNe. However, this interpretation remains questionable because of
the absence of a near- and mid-infrared or millimeter source at its location.
Thus the nature of this transient remains unclear.

We have identified a faint candidate remnant of the event for which a
spectroscopic study would be useful. However, the observation would be
challenging because of the source’s faintness, as well as a neighboring star
$\sim$$1^{\prime\prime}$ away (which itself may be another candidate
counterpart).

The LRN CK Vul is surrounded by a faint “hourglass” nebula seen in narrow-band
H$\alpha$ and [N II] images (Kamiński et al., 2021a, and references therein).
Thus deep optical imaging of V1148 Sgr in these bandpasses would be of
interest.

H.E.B. remembers with gratitude his introduction to the Harvard plate
collection, nearly five decades ago, by Dr. Martha L. Hazen, who became a
cherished friend. We thank Tomek Kamiński for providing results from his SMA
observations in advance of publication. The DASCH project at Harvard is
grateful for partial support from National Science Foundation grants
AST-0407380, AST-0909073, and AST-1313370. This research has made use of the
VizieR catalogue access tool, CDS, Strasbourg, France (DOI :
10.26093/cds/vizier). The original description of the VizieR service was
published in 2000, A&AS 143, 23. This work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. The Pan-STARRS1 Surveys
(PS1) and the PS1 public science archive have been made possible through
contributions by the Institute for Astronomy, the University of Hawaii, the
Pan-STARRS Project Office, the Max-Planck Society and its participating
institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max
Planck Institute for Extraterrestrial Physics, Garching, Johns Hopkins
University, Durham University, the University of Edinburgh, the Queen’s
University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las
Cumbres Observatory Global Telescope Network Incorporated, the National
Central University of Taiwan, the Space Telescope Science Institute, the
National Aeronautics and Space Administration under Grant No. NNX08AR22G
issued through the Planetary Science Division of the NASA Science Mission
Directorate, National Science Foundation Grant No. AST-1238877, the University
of Maryland, Eotvos Lorand University (ELTE), Los Alamos National Laboratory,
and the Gordon and Betty Moore Foundation. This research has made use of the
NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration.

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