&=&-(F^{c}_{\mu}+F^{d}_{\mu}\rho+ZF^{p}_{\mu})-\left(\nabla^{2}f(\rho)(F_{\mu}^{p})\right)\rho.\end{array}$
(B.5)

####  B.3.3 Preconditioning

The overdetermined linear system in B.5 can be ill-conditioned. We use
diagonal preconditioning, i.e., we let $d_{i}=\|F_{\mu}^{c\prime}(e_{i})\|$,
for unit vectors $e_{i}$ and then column precondition using

$F_{\mu}^{c\prime}\leftarrow F_{\mu}^{c\prime}\operatorname{{Diag}}(d)^{-1}.$

This diagonal preconditioning has been shown to be the optimal diagonal
preconditioning for the so-called $\Omega$-condition number, [30]. It performs
exceptionally well in our tests below.

####  B.3.4 Step Lengths

The GN method is based on a linearization that suggests a step length of one.
However, long step methods are known to be more efficient in practice for
interior point methods for linear SDPs. Typically step lengths are found using
backtracking to ensure primal-dual positive definiteness of $\rho,Z$.

In our case we do not have a linear objective and we typically experience
Maratos type situations, i.e., we get fast convergence for primal feasibility
but slow and no convergence for dual feasibility. However, we do have the
gradient and Hessian of the objective function and therefore can minimize the
quadratic model for the objective function in the search direction
$\Delta\rho$

$\min_{\alpha}f(\rho)+\alpha\langle\nabla
f(\rho),\Delta\rho\rangle+\frac{1}{2}\alpha^{2}\langle\Delta\rho\nabla^{2}f(\rho),\Delta\rho\rangle,\quad\alpha^{*}=-\langle\nabla
f(\rho),\Delta\rho\rangle/\langle\Delta\rho,\nabla^{2}f(\rho)\Delta\rho\rangle.$

Therefore, we begin the backtracking with this step.

Moreover, we take a step length of one as soon as possible, and only after
this do we allow step lengths larger than one. This means that exact primal
feasibility holds for all further steps. This happens relatively early for our
numerical tests.

##  Appendix C Descriptions and Further Numerics of the Protocols

We briefly describe six QKD protocols where we compare our algorithm to other
algorithms. We also describe how the data ($\gamma$ in 2.2) is generated. In
addition, we remark on the level of numerical difficulty for each example. We
consider four variants of the Bennett-Brassard 1984 (BB84) protocol [3]
including single-photon based variants: entanglement-based (Section C.1),
prepare-and-measure (Section C.2), measurement-device-independent [31]
(Section C.3) and a coherent-state based variant with discrete global phase
randomization [32] (Section C.6). We also consider the single-photon version
of the twin-field QKD [33] (Section C.4). Another interesting protocol in our
numerical tests is the quadrature phase-shift keying scheme of discrete-
modulated continuous-variable QKD with heterodyne detection (Section C.5), see
[26, Protocol 2]. These protocols correspond to numerical problems with the
level of difficulty ranging from easy to difficult. In the descriptions below,
we use the Dirac notation for quantum states which are vectors in the
underlying Hilbert space. We skip the description about some common classical
postprocessing steps in a QKD protocol like error correction and privacy
amplification since they are unimportant for our discussions here. We note
that the description of linear maps ${\mathcal{G}}$ and ${\mathcal{Z}}$
directly follow from the protocol description by following the simplification
procedure explained in [26, Appendix A]. We omit those detailed descriptions
here and note that the explicit expressions for some of protocols can also be
found in [6, Appendix D].

###  C.1 Entanglement-Based BB84

We consider this protocol with a single-photon source and restrict our
discussions to the qubit space. In the quantum communication phase, Alice and
Bob each receive one half of a bipartite state. This is supposed to be a two-
qubit maximally entangled state before Eve’s tampering. And the measure in the
$Z$ basis is with probability $p_{z}$, or in $X$ basis with a probability
$1-p_{z}$. In the classical communication phase, they announce their basis
choices for each round and perform sifting to keep those rounds where they
both chose the same basis. In the end, they generate keys from both $Z$ and
$X$ bases.

In the simulation, we assume both bases have the same error rate
$e_{z}=e_{x}=Q$. In particular, $\Gamma$ (in 2.2) contains the $Z$-basis error
rate, $X$-basis error rate constraints as well as one coarse-grained
constraint for each mismatched basis choice scenario. This is to ensure that
Alice and Bob get completely uncorrelated outcomes in that case. In other
words, the data $\gamma$ are determined by $Q$. This test example is supposed
to be numerically easy, since it involves the smallest possible size of $\rho$
for QKD, i.e., four. Moreover, there is no reduced density operator $\rho_{A}$
constraint for this example. In Table 5.1, instances of this test example are
labeled as ebBB84($p_{z},Q$) for different values of $p_{z}$ and $Q$.

###  C.2 Prepare-and-Measure BB84

Another protocol example in our numerical tests is the prepare-and-measure
version of BB84 with a single-photon source. In the quantum communication
phase, Alice chooses the $Z$ basis with a probability $p_{z}$ or the $X$ basis
with a probability $1-p_{z}$. When she chooses the $Z$ basis, she sends either
$\lvert 0\rangle$ or $\lvert 1\rangle$ at random, where $\lvert 0\rangle$ and
$\lvert 1\rangle$ are eigenstates of the Pauli $\sigma_{Z}$ operator. When she
chooses the $X$ basis, she sends either $\lvert+\rangle$ or $\lvert-\rangle$
at random, where $\lvert\pm\rangle=\frac{1}{\sqrt{2}}(\lvert 0\rangle\pm\lvert
1\rangle)$. After Alice sends the state of her choice to Bob, Bob chooses to
measure in the $Z$ basis with a probability $p_{z}$ or the $X$ basis with a
probability $1-p_{z}$. The rest of the protocol is exactly the same as the
entanglement-based BB84 protocol described in Section C.1. We call $\\{\lvert
0\rangle,\lvert 1\rangle,\lvert+\rangle,\lvert-\rangle\\}$ as stated in BB84.

For the security analysis, we use the source-replacement scheme [16] to
convert it to its equivalent entanglement-based scheme. Therefore, the main
differences between this example and the one in Section C.1 are: (1) the
dimension of Alice’s system for this example is four due to the source-
replacement scheme, while it is two for the entanglement-based BB84; (2) there
is the reduced density operator constraint $\rho_{A}$ which is of size $4$ and
translated to $16$ linear constraints. In this test example, the size of
$\rho$ is $8$ and the size of ${\mathcal{G}}(\rho)$ is $32$ before FR.

The data simulation is done in a similar way as that in the entanglement-based
BB84 protocol, i.e., $e_{x}=e_{z}=Q$. In Table 5.1, instances of this test
example are labeled as pmBB84($p_{z},Q$).

###  C.3 Measurement-Device-Independent BB84

In the measurement-device-independent variant of BB84 with single-photon
sources, Alice and Bob each prepare one of four BB84 states (with the
probability of choosing the $Z$ basis as $p_{z}$). Then they both send this to
an untrusted third-party Charlie for measurements. He ideally then performs
the Bell-state measurements and announces the outcomes. We consider a setup
where Charlie only uses linear optics, and thus can only measure two out of
four Bell states. In this protocol, Charlie announces either a successful
Bell-state measurement or a failure. If a successful measurement, Charlie then
also announces the Bell state. Therefore, there are three possible
announcement outcomes. After the announcement, Alice and Bob perform the basis
sifting, as well as discard rounds that are linked to unsuccessful events.
They then generate keys from rounds where they both chose the $Z$ basis and
Charlie’s announcement is one of the successful events.

We now consider the measurement-device-independent type of protocols. As
described in [4], the optimization variable $\rho$ involves three parties as
$\rho_{ABC}$. Here, registers $AB$ together serve the role of $A$ in the
reduced density operator constraint set 2.5. The dimension of Alice’s system
is $4$ and so is Bob’s dimension. The register $C$ is a classical register
that stores the announcement outcome. Thus it is three-dimensional with three
possible announcement outcomes. In the data simulation, we assume that each
qubit sent to Charlie goes through a depolarizing channel, with the
depolarizing probability $p$.

In the numerical tests, we label instances of this protocol example as
mdiBB84($p_{z},p$). The size of $\rho$ is $48$ and that of
${\mathcal{G}}(\rho)$ is $96$ before FR.

###  C.4 Twin-Field QKD

As above, this protocol also uses the measurement-device-independent setup.
The exact protocol description can be found in [34, Protocol 1]. In this
protocol, Alice and Bob each prepare a state
$\lvert\phi_{q}\rangle_{Aa}=\sqrt{q}\lvert 0\rangle_{A}\lvert
0\rangle_{a}+\sqrt{1-q}\lvert 1\rangle_{A}\lvert 1\rangle_{a}$
($\lvert\phi_{q}\rangle_{Bb}$) with $0\leq q\leq 1$, where the register $A$ is
a qubit system and the register $a$ is an optical mode with the vacuum state
$\lvert 0\rangle_{a}$ and the single-photon state $\lvert 1\rangle_{a}$. After
they send states to the intermediate station, Charlie at the intermediate
station is supposed to perform the single-photon interference of these two
signal pulses and then announces the measurement outcome for each of two
detectors: click or no-click. Then Alice and Bob each perform the $X$-basis
measurement on their local qubits with a probability $p_{x}$ or the $Z$-basis
measurement with a probability $1-p_{x}$. They generate keys from rounds where
they both choose the $X$ basis and where Charlie announces a successful
measurement outcome, that is, having exactly one of two detectors click.

In the simulation, we consider a lossy channel, with the transmittance
$10^{-0.02L}$, for the distance $L$ in kilometers between Alice and Bob. We
consider the symmetric scenario where Charlie is at an equal distance away
from Alice and Bob. We also consider detector imperfections: each detector at
Charlie’s side has detector efficiency $\eta_{d}=14.5\%$ and dark count
probability $p_{d}=10^{-8}$. In instances of this protocol, data is generated
as a function of: $q$ that appears in the states $\lvert\phi_{q}\rangle_{Aa}$
and $\lvert\phi_{q}\rangle_{Bb}$; the total distance $L$ in kilometers between
Alice and Bob; and the probability of choosing $X$ basis $p_{x}$. The
instances of this test example are labeled as TFQKD($q,L,p_{x}$).

###  C.5 Discrete-Modulated Continuous-Variable QKD

The exact protocol description can be found in [26, Protocol 2]. We use the
same simulation method described in [26, Equation (30)] to generate the data
$\gamma$. In this protocol, Alice sends Bob one of four coherent states
$\lvert\alpha e^{i\theta_{j}}\rangle$, where $\theta_{j}=\frac{j\pi}{2}$ for
$j=0,1,2,3$. And Bob performs the heterodyne measurement, i.e., measuring both
$X$\- and $P$-quadratures after splitting the signal into two halves by a
$50/50$ beamsplitter. The first and second moments of $X$\- and
$P$-quadratures are used to constrain $\rho$. The data simulation uses a
phase-invariant Gaussian channel with transmittance $\eta_{t}$ and excess
noise $\xi$ to generate those values. We use the same photon-number cutoff
assumption used there to truncate the infinite-dimensional Hilbert space. For
the calculation, it is typically sufficient to choose $N_{c}\geq 10$ to
minimize the effects of errors due to the truncation. For simplicity, we
assume the detector at Bob’s side is an ideal detector. The channel
transmittance, $\eta_{t}$, is related to the transmission distance $L$ between
Alice and Bob, by $\eta_{t}=10^{-0.02L}$.

Let $N_{c}$ be an integer that represents the cutoff photon number. Before FR,
the sizes of $\rho$ and ${\mathcal{G}}(\rho)$ are $4(N_{c}+1)$ and
$16(N_{c}+1)$, respectively. In Table 5.1, we label instances of this example
as DMCV($N_{c},L,\xi,\alpha$).

When the noise $\xi=0$, this problem can be solved analytically via physical
arguments. The detailed instructions for analytical calculation can be found
in [26, Appendix C]. We use this special case to demonstrate that our
interior-point method can reproduce the analytical results to high precision.

###  C.6 Discrete-Phase-Randomized BB84

We consider the phase-encoding BB84 protocol with $c$ (a parameter) discrete
global phases evenly spaced between [0, $2\pi$] [32]. A detailed protocol
description can also be found in [6, Sec. IV D]. In particular, each of four
BB84 states is realized by a two-mode coherent state $\lvert\alpha
e^{i\theta}\rangle_{r}\lvert\alpha e^{i(\theta+\phi_{A})}\rangle_{s}$, where
the first mode is the phase reference mode and the second mode encodes the
private information. In particular, the $\theta$ is a global phase that
involves discrete phase randomization, i.e.,
$\theta\in\\{\frac{2\pi\ell}{c}:\ell=0,\dots,c-1\\}$. The relative phase for
encoding is $\phi_{A}\in\\{\frac{j\pi}{2}:j=0,1,2,3\\}$, where $\\{0,\pi\\}$
correspond to the $Z$ basis and $\\{\frac{\pi}{2},\frac{3\pi}{2}\\}$
correspond to the $X$ basis.

Data simulation is done in exactly the same way as in [6, Section IV D], where
we consider detector imperfections and a lossy channel with a misalignment
error due to phase drift.

We remark that the instances of this test example become more challenging as
one increases the number of discrete phases $c$, since the size of $\rho$ is
$12c$ and the size of ${\mathcal{G}}(\rho)$ is $48c$ before FR. In all
instances of this protocol, we choose $p_{z}=0.5$ and the data are simulated
with detector efficiency $\eta_{d}=0.045$, dark count probability
$p_{d}=8.5\times 10^{-7}$ and relative phase drift of $11^{\circ}$. The final
key rate values are presented by taking the error correction efficiency as
1.16. In the numerical tests, we label instances of this protocol as
dprBB84($c,\alpha,L$).

###  C.7 Additional Numerical Results

Problem Data | Gauss-Newton | Frank-Wolfe with FR | Frank-Wolfe w/o FR | cvxquad with FR
---|---|---|---|---
protocol | parameter | size | gap | time | gap | time | gap | time | gap | time
ebBB84 | (0.50,0.01) | (4,16) | 1.14e-12 | 0.42 | 5.96e-05 | 95.32 | 5.88e-05 | 99.09 | 6.37e-01 | 216.75
ebBB84 | (0.50,0.03) | (4,16) | 8.35e-13 | 0.20 | 6.37e-05 | 93.35 | 6.24e-05 | 99.47 | 5.88e-01 | 258.98
ebBB84 | (0.50,0.05) | (4,16) | 5.98e-13 | 0.18 | 1.01e-04 | 95.31 | 1.17e-04 | 101.36 | 5.46e-01 | 213.04
ebBB84 | (0.50,0.07) | (4,16) | 1.05e-12 | 0.19 | 1.66e-04 | 96.92 | 1.65e-04 | 100.46 | 5.07e-01 | 179.38
ebBB84 | (0.50,0.09) | (4,16) | 1.35e-12 | 0.18 | 1.43e-04 | 96.35 | 2.55e-04 | 100.64 | 4.70e-01 | 170.14
ebBB84 | (0.70,0.01) | (4,16) | 1.21e-13 | 0.21 | 7.60e-05 | 96.33 | 7.62e-05 | 99.14 | 7.06e-01 | 172.09
ebBB84 | (0.70,0.03) | (4,16) | 6.36e-13 | 0.18 | 9.16e-05 | 96.77 | 9.15e-05 | 99.78 | 6.59e-01 | 160.88
ebBB84 | (0.70,0.05) | (4,16) | 5.34e-13 | 0.18 | 1.67e-04 | 97.03 | 1.03e-04 | 100.73 | 6.14e-01 | 173.77
ebBB84 | (0.70,0.07) | (4,16) | 1.26e-12 | 0.20 | 1.80e-04 | 96.22 | 1.74e-04 | 100.71 | 5.70e-01 | 261.77
ebBB84 | (0.70,0.09) | (4,16) | 2.06e-13 | 0.18 | 3.85e-04 | 97.08 | 2.61e-04 | 101.55 | 5.26e-01 | 225.94
ebBB84 | (0.90,0.01) | (4,16) | 5.40e-13 | 0.17 | 1.02e-04 | 97.68 | 1.03e-04 | 98.45 | 8.73e-01 | 141.93
ebBB84 | (0.90,0.03) | (4,16) | 7.05e-13 | 0.21 | 1.25e-04 | 97.97 | 1.43e-04 | 99.50 | 8.27e-01 | 164.26
ebBB84 | (0.90,0.05) | (4,16) | 3.48e-13 | 0.18 | 1.48e-04 | 96.91 | 9.81e-05 | 100.99 | 7.83e-01 | 186.87
ebBB84 | (0.90,0.07) | (4,16) | 1.42e-12 | 0.17 | 2.71e-04 | 96.55 | 2.75e-04 | 101.79 | 7.39e-01 | 179.55
ebBB84 | (0.90,0.09) | (4,16) | 1.09e-12 | 0.21 | 3.51e-04 | 96.40 | 3.42e-04 | 101.99 | 6.94e-01 | 228.63
Table C.1: Numerical Report for ebBB84 Instances Problem Data | Gauss-Newton | Frank-Wolfe with FR | Frank-Wolfe w/o FR | cvxquad with FR
---|---|---|---|---
protocol | parameter | size | gap | time | gap | time | gap | time | gap | time
pmBB84 | (0.50,0.01) | (8,32) | 5.96e-13 | 0.38 | 6.19e-06 | 1.95 | 4.64e-04 | 25.18 | 6.30e-01 | 190.28
pmBB84 | (0.50,0.03) | (8,32) | 1.01e-12 | 0.18 | 1.71e-05 | 1.37 | 6.54e-04 | 90.80 | 5.74e-01 | 181.83
pmBB84 | (0.50,0.05) | (8,32) | 5.51e-13 | 0.19 | 1.12e-04 | 1.31 | 6.47e-04 | 1.94 | 5.26e-01 | 159.28
pmBB84 | (0.50,0.07) | (8,32) | 8.88e-14 | 0.16 | 5.89e-05 | 1.30 | 8.77e-04 | 1.98 | 4.81e-01 | 161.49
pmBB84 | (0.50,0.09) | (8,32) | 9.38e-13 | 0.19 | 6.71e-05 | 1.42 | 9.04e-04 | 2.03 | 4.40e-01 | 179.01
pmBB84 | (0.70,0.01) | (8,32) | 7.69e-13 | 0.25 | 7.62e-06 | 1.33 | 2.39e-04 | 130.73 | 7.03e-01 | 213.43
pmBB84 | (0.70,0.03) | (8,32) | 4.75e-13 | 0.18 | 2.38e-05 | 1.32 | 2.68e-04 | 133.52 | 6.51e-01 | 246.50
pmBB84 | (0.70,0.05) | (8,32) | 6.16e-13 | 0.17 | 3.52e-05 | 1.37 | 3.37e-04 | 134.59 | 6.04e-01 | 281.84
pmBB84 | (0.70,0.07) | (8,32) | 6.30e-13 | 0.19 | 7.43e-05 | 1.26 | 3.04e-04 | 141.97 | 5.60e-01 | 259.33
pmBB84 | (0.70,0.09) | (8,32) | 8.47e-13 | 0.16 | 9.25e-05 | 1.29 | 3.55e-04 | 7.07 | 5.18e-01 | 297.32
pmBB84 | (0.90,0.01) | (8,32) | 3.68e-13 | 0.18 | 7.07e-06 | 1.33 | 3.27e-04 | 4.98 | 8.60e-01 | 247.82
pmBB84 | (0.90,0.03) | (8,32) | 1.27e-12 | 0.18 | 2.29e-05 | 1.35 | 5.43e-04 | 137.74 | 7.96e-01 | 230.35
pmBB84 | (0.90,0.05) | (8,32) | 1.36e-12 | 0.16 | 4.62e-05 | 1.35 | 5.96e-04 | 72.04 | 7.38e-01 | 291.83
pmBB84 | (0.90,0.07) | (8,32) | 5.13e-13 | 0.17 | 7.31e-05 | 1.35 | 6.25e-04 | 40.11 | 6.84e-01 | 235.27
pmBB84 | (0.90,0.09) | (8,32) | 7.84e-13 | 0.19 | 1.06e-04 | 1.32 | 7.39e-04 | 142.32 | 6.32e-01 | 244.81
Table C.2: Numerical Report for pmBB84 Instances Problem Data | Gauss-Newton | Frank-Wolfe with FR | Frank-Wolfe w/o FR | cvxquad with FR
---|---|---|---|---
protocol | parameter | size | gap | time | gap | time | gap | time | gap | time
mdiBB84 | (0.50,0.01) | (48,96) | 1.25e-12 | 1.27 | 1.64e-05 | 109.60 | 3.75e-04 | 498.82 | 2.11e-01 | 719.99
mdiBB84 | (0.50,0.03) | (48,96) | 8.37e-13 | 0.81 | 3.53e-05 | 107.90 | 5.31e-04 | 2811.15 | 1.95e-01 | 630.89
mdiBB84 | (0.50,0.05) | (48,96) | 1.14e-12 | 0.83 | 4.99e-05 | 111.08 | 5.22e-04 | 464.06 | 1.82e-01 | 586.22
mdiBB84 | (0.50,0.07) | (48,96) | 1.35e-12 | 1.08 | 4.87e-05 | 335.57 | 4.60e-04 | 2169.93 | 1.71e-01 | 569.14
mdiBB84 | (0.50,0.09) | (48,96) | 1.25e-12 | 0.91 | 8.27e-05 | 342.04 | 4.37e-04 | 829.61 | 1.60e-01 | 568.97
mdiBB84 | (0.70,0.01) | (48,96) | 1.20e-12 | 0.68 | 2.20e-05 | 5.68 | 5.53e-04 | 901.29 | 3.79e-01 | 719.27
mdiBB84 | (0.70,0.03) | (48,96) | 4.24e-13 | 1.05 | 1.11e-04 | 5.83 | 1.21e-03 | 256.79 | 3.55e-01 | 670.49
mdiBB84 | (0.70,0.05) | (48,96) | 1.06e-12 | 1.10 | 6.92e-05 | 340.21 | 1.75e-03 | 865.68 | 3.31e-01 | 651.44
mdiBB84 | (0.70,0.07) | (48,96) | 5.71e-13 | 0.95 | 1.37e-04 | 337.17 | 1.55e-03 | 864.25 | 3.09e-01 | 604.94
mdiBB84 | (0.70,0.09) | (48,96) | 1.57e-13 | 0.92 | 1.61e-04 | 347.21 | 2.24e-03 | 872.37 | 2.88e-01 | 604.71
mdiBB84 | (0.90,0.01) | (48,96) | 8.44e-13 | 0.66 | 4.42e-05 | 343.38 | 3.21e-03 | 175.20 | 5.53e-01 | 710.65
mdiBB84 | (0.90,0.03) | (48,96) | 1.39e-12 | 0.90 | 9.15e-05 | 343.08 | 3.66e-03 | 301.29 | 5.16e-01 | 671.95
mdiBB84 | (0.90,0.05) | (48,96) | 9.88e-13 | 0.84 | 1.73e-04 | 345.79 | 4.64e-03 | 519.80 | 4.85e-01 | 644.23
mdiBB84 | (0.90,0.07) | (48,96) | 2.96e-13 | 1.05 | 2.04e-04 | 338.15 | 2.85e-03 | 148.24 | 4.57e-01 | 580.20
mdiBB84 | (0.90,0.09) | (48,96) | 5.21e-13 | 1.02 | 2.52e-04 | 343.95 | 3.26e-03 | 193.04 | 4.31e-01 | 595.07
Table C.3: Numerical Report for mdiBB84 Instances Problem Data | Gauss-Newton | Frank-Wolfe with FR | Frank-Wolfe without FR
---|---|---|---
protocol | parameter | size | gap | time | gap | time | gap | time
TFQKD | (0.75,50.00,0.70) | (12,24) | 8.45e-13 | 1.33 | 2.72e-09 | 2.04 | 1.83e-03 | 155.08
TFQKD | (0.75,100.00,0.70) | (12,24) | 1.42e-12 | 0.86 | 3.75e-09 | 1.35 | 1.53e-03 | 153.57
TFQKD | (0.75,150.00,0.70) | (12,24) | 9.94e-13 | 0.78 | 2.82e-09 | 1.39 | 7.82e-04 | 162.13
TFQKD | (0.75,200.00,0.70) | (12,24) | 1.15e-12 | 1.02 | 3.98e-09 | 1.37 | 7.19e-04 | 155.11
TFQKD | (0.75,250.00,0.70) | (12,24) | 6.79e-13 | 0.79 | 8.21e-09 | 1.32 | 3.14e-04 | 172.53
TFQKD | (0.80,50.00,0.70) | (12,24) | 1.15e-12 | 0.77 | 2.82e-09 | 1.39 | 1.52e-03 | 156.17
TFQKD | (0.80,100.00,0.70) | (12,24) | 1.15e-12 | 1.08 | 2.60e-09 | 1.33 | 1.57e-03 | 156.11
TFQKD | (0.80,150.00,0.70) | (12,24) | 1.25e-12 | 0.94 | 2.97e-09 | 1.29 | 8.30e-04 | 158.12
TFQKD | (0.80,200.00,0.70) | (12,24) | 9.23e-13 | 0.73 | 4.23e-09 | 1.32 | 5.60e-04 | 155.45
TFQKD | (0.80,250.00,0.70) | (12,24) | 3.91e-13 | 0.55 | 2.22e-09 | 1.35 | 1.97e-04 | 164.83
TFQKD | (0.90,50.00,0.70) | (12,24) | 8.08e-13 | 0.79 | 4.30e-09 | 1.30 | 1.55e-03 | 156.38
TFQKD | (0.90,100.00,0.70) | (12,24) | 1.38e-12 | 0.37 | 3.62e-09 | 1.31 | 1.23e-03 | 154.08
TFQKD | (0.90,150.00,0.70) | (12,24) | 1.14e-12 | 0.61 | 2.87e-09 | 1.28 | 6.02e-04 | 161.73
TFQKD | (0.90,200.00,0.70) | (12,24) | 1.04e-12 | 0.34 | 3.98e-09 | 1.36 | 1.68e-04 | 2.63
TFQKD | (0.90,250.00,0.70) | (12,24) | 5.77e-13 | 0.43 | 2.77e-09 | 1.37 | 1.08e-05 | 2.07
TFQKD | (0.95,50.00,0.70) | (12,24) | 9.84e-13 | 0.77 | 4.00e-09 | 1.43 | 1.38e-03 | 156.09
TFQKD | (0.95,100.00,0.70) | (12,24) | 1.06e-12 | 0.70 | 4.36e-09 | 1.41 | 7.47e-04 | 153.19
TFQKD | (0.95,150.00,0.70) | (12,24) | 1.36e-12 | 0.71 | 3.92e-09 | 1.35 | 8.38e-04 | 149.27
TFQKD | (0.95,200.00,0.70) | (12,24) | 7.63e-13 | 0.65 | 2.60e-09 | 1.36 | 3.13e-04 | 150.75
TFQKD | (0.95,250.00,0.70) | (12,24) | 1.02e-12 | 0.64 | 3.61e-09 | 1.42 | 5.56e-06 | 1.81
Table C.4: Numerical Report for TFQKD Instances Problem Data | Gauss-Newton | Frank-Wolfe with FR | Frank-Wolfe without FR
---|---|---|---
protocol | parameter | size | gap | time | gap | time | gap | time
DMCV | (10.00,60.00,0.05,0.35) | (44,176) | 2.71e-09 | 1016.19 | 4.35e-06 | 612.62 | 3.57e-06 | 919.84
DMCV | (10.00,120.00,0.05,0.35) | (44,176) | 2.70e-09 | 1090.41 | 2.27e-06 | 216.61 | 2.16e-06 | 277.47
DMCV | (10.00,180.00,0.05,0.35) | (44,176) | 2.87e-09 | 1173.14 | 1.90e-07 | 23.64 | 1.36e-07 | 32.50
DMCV | (11.00,60.00,0.05,0.35) | (48,192) | 3.05e-09 | 1419.71 | 2.50e-06 | 768.52 | 1.43e-06 | 1095.78
DMCV | (11.00,120.00,0.05,0.35) | (48,192) | 3.24e-09 | 1484.12 | 2.35e-06 | 261.89 | 2.15e-06 | 380.63
DMCV | (11.00,180.00,0.05,0.35) | (48,192) | 3.40e-09 | 1796.16 | 2.53e-07 | 26.98 | 1.59e-07 | 38.62
DMCV | (10.00,150.00,0.02,0.70) | (44,176) | 2.07e-09 | 1143.23 | 1.67e-06 | 83.32 | 1.82e-06 | 99.96
DMCV | (10.00,200.00,0.02,0.70) | (44,176) | 1.96e-09 | 1214.62 | 7.29e-07 | 21.26 | 7.06e-07 | 28.04
DMCV | (10.00,150.00,0.02,0.80) | (44,176) | 3.27e-09 | 1113.17 | 1.10e-06 | 105.03 | 1.85e-06 | 106.68
DMCV | (10.00,200.00,0.02,0.80) | (44,176) | 1.63e-09 | 1106.38 | 3.18e-07 | 19.61 | 2.90e-07 | 26.11
DMCV | (11.00,150.00,0.02,0.70) | (48,192) | 3.31e-09 | 1607.78 | 1.72e-06 | 103.49 | 1.36e-06 | 150.76
DMCV | (11.00,200.00,0.02,0.70) | (48,192) | 3.05e-09 | 1573.12 | 7.12e-07 | 27.40 | 6.68e-07 | 35.62
DMCV | (11.00,150.00,0.02,0.80) | (48,192) | 3.37e-09 | 1597.15 | 1.36e-06 | 93.86 | 1.57e-06 | 115.21
DMCV | (11.00,200.00,0.02,0.80) | (48,192) | 3.38e-09 | 1541.27 | 3.38e-07 | 25.34 | 2.99e-07 | 34.36
Table C.5: Numerical Report for DMCV Instances Problem Data | Gauss-Newton | Frank-Wolfe with FR | Frank-Wolfe without FR
---|---|---|---
protocol | parameter | size | gap | time | gap | time | gap | time
dprBB84 | (1.00,0.08,15.00) | (12,48) | 9.42e-13 | 1.66 | 3.85e-06 | 117.56 | 1.03e-04 | 189.23
dprBB84 | (1.00,0.08,30.00) | (12,48) | 4.92e-13 | 1.70 | 3.85e-06 | 114.97 | 9.43e-05 | 178.82
dprBB84 | (1.00,0.14,15.00) | (12,48) | 2.96e-13 | 0.91 | 3.63e-04 | 115.55 | 1.16e-02 | 178.91
dprBB84 | (1.00,0.14,30.00) | (12,48) | 5.21e-13 | 1.43 | 2.60e-04 | 1.89 | 8.31e-03 | 2.51
dprBB84 | (2.00,0.08,15.00) | (24,96) | 1.10e-12 | 22.06 | 9.58e-05 | 41.11 | 2.11e-03 | 82.96
dprBB84 | (2.00,0.08,30.00) | (24,96) | 9.58e-13 | 22.26 | 1.17e-04 | 26.10 | 7.32e-06 | 114.66
dprBB84 | (2.00,0.14,15.00) | (24,96) | 1.35e-12 | 24.93 | 1.89e-05 | 7.56 | 5.11e-04 | 16.59
dprBB84 | (2.00,0.14,30.00) | (24,96) | 1.04e-12 | 24.63 | 5.71e-06 | 20.79 | 5.38e-06 | 42.06
dprBB84 | (2.00,0.14,30.00) | (24,96) | 1.04e-12 | 24.82 | 5.71e-06 | 20.19 | 5.38e-06 | 44.72
dprBB84 | (3.00,0.08,15.00) | (36,144) | 1.38e-12 | 129.39 | 2.36e-04 | 12.94 | 7.41e-03 | 33.59
dprBB84 | (3.00,0.08,30.00) | (36,144) | 6.33e-13 | 139.34 | 2.26e-04 | 12.54 | 7.04e-03 | 36.53
dprBB84 | (3.00,0.14,15.00) | (36,144) | 1.30e-12 | 118.84 | 4.32e-05 | 41.49 | 1.43e-04 | 50.07
dprBB84 | (3.00,0.14,30.00) | (36,144) | 3.32e-13 | 127.94 | 5.80e-06 | 11.32 | 5.74e-06 | 37.63
dprBB84 | (4.00,0.08,15.00) | (48,192) | 6.98e-09 | 766.17 | 2.88e-04 | 61.19 | 8.10e-03 | 235.02
dprBB84 | (4.00,0.08,30.00) | (48,192) | 2.13e-09 | 786.01 | 2.97e-04 | 21.23 | 8.45e-03 | 232.35
dprBB84 | (4.00,0.14,15.00) | (48,192) | 2.85e-12 | 539.08 | 1.29e-04 | 18.50 | 3.85e-03 | 201.96
dprBB84 | (4.00,0.14,30.00) | (48,192) | 1.17e-12 | 545.04 | 1.26e-04 | 25.67 | 3.73e-03 | 209.05
Table C.6: Numerical Report for dprBB84 Instances

We remark on the behavior of the Frank-Wolfe method when used without FR.
Table 5.1 shows that two instances of the TFQKD protocol have vastly different
running times. Similar situations without FR also occur in the tables
presented here. The Frank-Wolfe method that is used and described in [5]
adopts a two-step procedure. The running time of this method is mainly
dependent on the number of iterations, since each iteration solves a linear
SDP problem using CVX. There are two stopping conditions: the first depends on
the value of the gradient information; the second is the maximum number of
iterations, currently set at $300$. For the instances with short running time,
termination typically occurred due to the gradient condition; the other
instances stopped after reaching the maximum number of iterations. A likely
reason for early termination is the perturbation approach used to preserve
positive definiteness of the matrices involved in the objective function. We
emphasize that early termination still leads to reliable (but pessimistically
lower) key rates. More explanations can be found in [5]. As discussed in this
paper, a general motivation for FR is to avoid numerical instability
associated with such perturbation approaches, i.e., without FR, we typically
have the erratic behavior due to ill-posedness of the problems.

## Index

  * $[x,y]$, line segment §2.5
  * $\widehat{\mathcal{G}}$ §3.3
  * $\widehat{\mathcal{Z}}$ §3.3
  * $\|X\|_{F}$, Frobenius norm §2.3
  * $\otimes$, Kronecker product §2.1
  * $\unlhd$ §2.5, §2.5
  * $\cdot^{\dagger}$, conjugate transpose §2.3
  * $\mathbbm{1}_{B}\in\mathbb{H}^{n_{B}}$ §2.1, §3.2.1
  * adjoint §2.4
  * adjoint of ${\mathcal{L}}$, ${\mathcal{L}}^{\dagger}$ §2.4
  * algorithm, GN interior point for QKD §4.4
  * $\operatorname{{BlkDiag}}$ §2.2
  * $\operatorname{{BlkDiag}}(A_{1},A_{2})$, block diagonal matrix with diagonal blocks $A_{1},A_{2}$ §2.2
  * compact spectral decomposition §2.5, §3.2.2
  * conjugate transpose, $\cdot^{\dagger}$ §2.3
  * constraint sets §2.1
    * observational, $S_{O}$ §2.1
    * reduced density, $S_{R}$ §2.1
  * convex cone §2.5
  * density matrices §2.1
  * ${d_{\scriptscriptstyle GN}}$, GN -direction §4.2, §4.3.1
  * dual cone §2.5
  * dual cone, $S^{\dagger}$ §2.5
  * exposing vector §A.4, §2.5, §3.2.2
  * $f(\delta,\sigma)=\operatorname{{Tr}}\left(\delta(\log\delta-\log\sigma)\right)$ §2.1
  * face Definition 2.1
  * $\operatorname{face}(X)$, minimal face §2.5
  * facially reduced reduced density operator constraint §3.2.1
  * Fréchet derivative of $\log$, $\log^{\prime}$ §A.3
  * Frobenius norm, $\|X\|_{F}$ §2.3
  * $F_{\mu}^{\prime}$, Jacobian of $F_{\mu}$ §4.2
  * ${\mathcal{G}}:\,\mathbb{H}^{n}\to\,\mathbb{H}^{k}$ Definition 3.1
  * $g(\rho,y,Z)$, nonlinear least square function §4.2
  * Gauss-Newton direction, ${d_{\scriptscriptstyle GN}}$ §4.2
  * generalized inverse, ${\Gamma_{\scriptscriptstyle V}}^{-1}$ §4.5.1
  * GN -direction, ${d_{\scriptscriptstyle GN}}$ §4.2, §4.3.1
  * gradient of $f$ Corollary A.4
  * $\,\mathbb{H}_{++}^{n}$, positive definite cone of $n$-by-$n$ Hermitian matrices §2.2
  * $\,\mathbb{H}_{+}^{n}$, positive semidefinite cone of $n$-by-$n$ Hermitian matrices §2.2
  * Hessian at $\rho\in\,\mathbb{H}^{n}_{+}$ acting on the direction $\Delta\rho\in\,\mathbb{H}^{n}$ Corollary A.4
  * $\,\mathbb{H}^{n}$, set of $n$-by-$n$ Hermitian matrices §3
  * $\operatorname{{\Im}}(X)$, imaginary part of $X$ §2.2
  * imaginary part of $X$, $\operatorname{{\Im}}(X)$ §2.2
  * Jacobian of $F_{\mu}$, $F_{\mu}^{\prime}$ §4.2
  * Kraus representation Definition 3.1
  * Kronecker product, $\otimes$ §2.1
  * $k_{\delta}$, $k_{\sigma}$ §3.2.3
  * ${\mathcal{L}}^{\dagger}$, adjoint of ${\mathcal{L}}$ §2.4
  * Lagrangian dual item 2
  * line segment, $[x,y]$ §2.5
  * $\log^{\prime}$, Fréchet derivative of $\log$ §A.3
  * $m$, number of linear constraints §2.1
  * minimal face, $\operatorname{face}(X)$ §2.5
  * ${m_{\scriptscriptstyle V}}$ §5.3
  * ${m_{\scriptscriptstyle V}}$, number of linear constraints after FR §3.3
  * ${{\mathcal{M}}_{\scriptscriptstyle Z}}(\Delta X)=Z\Delta X$ §4.2
  * ${{\mathcal{M}}_{\scriptscriptstyle\rho}}(\Delta X)=\Delta X\rho$ §4.2
  * ${\mathcal{N}}^{\dagger}:\mathbb{R}^{n_{\rho}^{2}-{m_{\scriptscriptstyle V}}}\to\mathbb{H}^{n_{\rho}}$ Theorem 4.5
  * $n=n_{A}n_{B}$ which is the size of $\rho$ §2.1
  * $n_{A},n_{B}$ §2.1
  * nonlinear least square function, $g(\rho,y,Z)$ §4.2
  * $\operatorname{null}(X)$, nullspace of $X$ §2.2
  * nullspace §2.2
  * $n_{\rho}$ §5.3
  * observational constraints §2.1
  * $\mathcal{P}_{C}(X)$, projection of $X$ onto $C$ §2.2
  * perturbed complementarity equations §4.1
  * positive definite cone of $n$-by-$n$ Hermitian matrices, $\,\mathbb{H}_{++}^{n}$ §2.2
  * positive definite cone of $n$-by-$n$ real symmetric matrices, $\mathbb{S}_{++}^{n}$ §2.2
  * positive semidefinite cone of $n$-by-$n$ Hermitian matrices, $\,\mathbb{H}_{+}^{n}$ §2.2
  * positive semidefinite cone of $n$-by-$n$ real symmetric matrices, $\mathbb{S}_{+}^{n}$ §2.2
  * $p_{\text{pass}}$ §2.1
  * QKD, quantum key distribution §1
  * Quantum key distribution, QKD §1
  * quantum relative entropy function §3, §3.1
  * $r(cp)=P_{cp}r$ §B.3.2
  * $\operatorname{{\Re}}(X)$, real part of $X$ §2.2
  * $r=P_{icp}r(cp)$ §B.3.2
  * range §2.2
  * $\operatorname{range}(X)$, range of $X$ §2.2
  * real inner product in $\mathbb{C}^{n\times n}$ §2.3
  * real part of $X$, $\operatorname{{\Re}}(X)$ §2.2
  * reduced density operator constraint, $S_{R}$ §2.1, §3.2.1
  * $\mathbb{R}^{n}$, vector space of real $n$-coordinates §2.2
  * ${\mathcal{R}}_{\delta}$ item 3
  * ${\mathcal{R}}_{\rho}$ item 2
  * $\operatorname{{{\mathcal{S}}}}$, symmetrization linear map §A.1
  * $S^{\dagger}$, dual cone §2.5
  * $\mathbb{S}_{++}^{n}$, positive definite cone of $n$-by-$n$ real symmetric matrices §2.2
  * $\mathbb{S}_{+}^{n}$, positive semidefinite cone of $n$-by-$n$ real symmetric matrices §2.2
  * set of $n$-by-$n$ Hermitian matrices, $\,\mathbb{H}^{n}$ §3
  * set of real symmetric $n$-by-$n$ matrices, $\mathbb{S}^{n}$ §2.2
  * size of $\rho$, $n=n_{A}n_{B}$ §2.1
  * $\operatorname{{\mathcal{SK}}}$, skew-symmetrization linear map §A.1
  * skew-symmetrization linear map, $\operatorname{{\mathcal{SK}}}$ §A.1
  * $\operatorname{{sMat}}$ §2.4
  * $\mathbb{S}^{n}$, set of real symmetric $n$-by-$n$ matrices §2.2
  * $S_{O}$, observational constraints §2.1
  * spectrahedron §3
  * spectral resolution of $I$ §3.1
  * $S_{R}$, reduced density operator constraint §2.1, §3.2.1
  * subdifferential, $\partial f$ §3.3
  * subgradient §3.3
  * $\operatorname{{svec}}$ §2.4
  * symmetrization linear map, $\operatorname{{{\mathcal{S}}}}$ §A.1
  * $t(n)=n(n+1)/2$, triangular number §2.4
  * $\operatorname{{Tr}}_{B}(\rho)=\rho_{A}$ §2.1
  * triangular number, $t(n)=n(n+1)/2$ §2.4
  * vector space of real $n$-coordinates, $\mathbb{R}^{n}$ §2.2
  * $V_{\delta}$ §3.2.2
  * $V_{\rho}$ §3.2.2
  * $V_{\sigma}$ §3.2.2
  * $X\succ 0$ §2.2
  * $X\succeq 0$ §2.2
  * $\mathcal{Z}:\,\mathbb{H}^{k}\to\,\mathbb{H}^{k}$ Definition 3.2
  * ${\Gamma_{\scriptscriptstyle V}}^{-1}$, generalized inverse Proposition 4.7
  * $\delta\in\,\mathbb{H}_{+}^{k}$ 3.8
  * $\delta_{\text{EC}}$ §2.1
  * $\rho$, state §2.1
  * $\rho\in\,\mathbb{H}_{+}^{n}$ 3.8
  * $\rho\in\mathbb{H}_{+}^{n_{\rho}}$ 3.19
  * $\sigma\in\,\mathbb{H}_{+}^{k}$ 3.8

## References

  * Scarani et al. [2009] V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev.  The security of practical quantum key distribution.  _Rev. Mod. Phys._ , 81:1301, 2009.  doi: 10.1103/RevModPhys.81.1301.
  * Xu et al. [2020] F. Xu, X. Ma, Q. Zhang, H.-K. Lo, and J.-W. Pan.  Secure quantum key distribution with realistic devices.  _Rev. Mod. Phys._ , 92:025002, 2020.  doi: 10.1103/RevModPhys.92.025002.
  * Bennett and Brassard [1984] C. H. Bennett and G. Brassard.  Quantum cryptography: Public key distribution and coin tossing.  In _International Conference on Computers, Systems & Signal Processing, Bangalore, India, Dec 9-12, 1984_, pages 175–179, 1984.  doi: 10.1016/j.tcs.2014.05.025.  Reprint of the 1984 original.
  * Coles et al. [2016] P. J. Coles, E. M. Metodiev, and N. Lütkenhaus.  Numerical approach for unstructured quantum key distribution.  _Nat. Commun._ , 7:11712, 2016.  doi: 10.1038/ncomms11712.
  * Winick et al. [2018] A. Winick, N. Lütkenhaus, and P. J. Coles.  Reliable numerical key rates for quantum key distribution.  _Quantum_ , 2:77, 2018.  doi: 10.22331/q-2018-07-26-77.
  * George et al. [2021] I. George, J. Lin, and N. Lütkenhaus.  Numerical calculations of the finite key rate for general quantum key distribution protocols.  _Physical Review Research_ , 3:013274, 2021.  doi: 10.1103/PhysRevResearch.3.013274.
  * Zhang et al. [2021] Y. Zhang, P. J. Coles, A. Winick, J. Lin, and N. Lütkenhaus.  Security proof of practical quantum key distribution with detection-efficiency mismatch.  _Phys. Rev. Research_ , 3:013076, 2021.  doi: 10.1103/PhysRevResearch.3.013076.
  * Upadhyaya et al. [2021] T. Upadhyaya, T. van Himbeeck, J. Lin, and N. Lütkenhaus.  Dimension reduction in quantum key distribution for continuous- and discrete-variable protocols.  _PRX Quantum_ , 2:020325, 2021.  doi: 10.1103/PRXQuantum.2.020325.
  * Li and Lütkenhaus [2020] N. K. H. Li and N. Lütkenhaus.  Improving key rates of the unbalanced phase-encoded bb84 protocol using the flag-state squashing model.  _Phys. Rev. Research_ , 2:043172, 2020.  doi: 10.1103/PhysRevResearch.2.043172.
  * Faybusovich and Zhou [2019] L. Faybusovich and C. Zhou.  Long-step path-following algorithm for solving symmetric programming problems with nonlinear objective functions.  _Computational Optimization and Applications_ , 72(3):769–795, 2019.  ISSN 15732894.  doi: 10.1007/s10589-018-0054-7.
  * Devetak and Winter [2005] I. Devetak and A. Winter.  Distillation of secret key and entanglement from quantum states.  _Proc. R. Soc. A_ , 461:207–235, 2005.  doi: 10.1098/rspa.2004.1372.
  * Nielsen and Chuang [2000] M. A. Nielsen and I. L. Chuang, editors.  _Quantum Computation and Quantum Information_.  Cambridge University Press, Cambridge, UK, 2000.  doi: 10.1017/CBO9780511976667.
  * Drusvyatskiy and Wolkowicz [2017] D. Drusvyatskiy and H. Wolkowicz.  The many faces of degeneracy in conic optimization.  _Foundations and Trends ® in Optimization_, 3(2):77–170, 2017.  ISSN 2167-3888.  doi: /10.1561/2400000011.
  * Watrous [2018] J. Watrous.  _The Theory of Quantum Information_.  Cambridge University Press, Cambridge, UK, 2018.  ISBN 1107180562.  doi: 10.1017/9781316848142.
  * Coles [2012] P. J. Coles.  Unification of different views of decoherence and discord.  _Phys. Rev. A_ , 85:042103, 2012.  doi: 10.1103/PhysRevA.85.042103.
  * Ferenczi and Lütkenhaus [2012] A. Ferenczi and N. Lütkenhaus.  Symmetries in quantum key distribution and the connection between optimal attacks and optimal cloning.  _Phys. Rev. A_ , 85:052310, 2012.  doi: 10.1103/PhysRevA.85.052310.
  * Borwein and Wolkowicz [1981] J. M. Borwein and H. Wolkowicz.  Regularizing the abstract convex program.  _J. Math. Anal. Appl._ , 83(2):495–530, 1981\.  ISSN 0022-247X.  doi: 10.1017/S1446788700017250.
  * Sremac et al. [2021] S. Sremac, H. J. Woerdeman, and H. Wolkowicz.  Error bounds and singularity degree in semidefinite programming.  _SIAM J. Optim._ , 31(1):812–836, 2021.  ISSN 1052-6234.  doi: 10.1137/19M1289327.
  * Rockafellar [1970] R. T. Rockafellar.  _Convex analysis_.  Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J., 1970.  doi: 10.1515/9781400873173.
  * Luenberger and Ye [2008] D. G. Luenberger and Y. Ye.  _Linear and Nonlinear Programming_ , volume 116 of _International series in operations research & management science_.  Springer, Boston, USA, 2008.  ISBN 9781441945044.  doi: 10.1007/978-0-387-74503-9.
  * Nocedal and Wright [2006] J. Nocedal and S. J. Wright.  _Numerical optimization_.  Springer Series in Operations Research and Financial Engineering. Springer, New York, NY, USA, second edition, 2006.  ISBN 978-0387-30303-1; 0-387-30303-0.  doi: 10.1007/978-0-387-40065-5.
  * Dedieu and Shub [2000] J. P. Dedieu and M. Shub.  Newton’s method for overdetermined systems of equations.  _Math. Comp._ , 69(231):1099–1115, 2000.  ISSN 0025-5718.  doi: 10.1090/S0025-5718-99-01115-1.
  * Dennis Jr. and Schnabel [1996] J. E. Dennis Jr. and R. B. Schnabel.  _Numerical methods for unconstrained optimization and nonlinear equations_ , volume 16 of _Classics in Applied Mathematics_.  Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.  ISBN 0-89871-364-1.  doi: 10.1137/1.9781611971200.  Corrected reprint of the 1983 original.
  * Monteiro and Todd [2000] R. D. C. Monteiro and M. J. Todd.  Path-following methods.  In _Handbook of Semidefinite Programming_ , volume 27 of _International Series in Operations Research & Management Science_, pages 267–306. Springer, Boston, MA, 2000.  doi: 10.1007/978-1-4615-4381-7˙10.
  * Demmel [1997] J. W. Demmel.  _Applied numerical linear algebra_.  Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997.  ISBN 0-89871-389-7.  doi: 10.1137/1.9781611971446.
  * Lin et al. [2019] J. Lin, T. Upadhyaya, and N. Lütkenhaus.  Asymptotic security analysis of discrete-modulated continuous-variable quantum key distribution.  _Phys. Rev. X_ , 9:041064, 2019.  doi: 10.1103/PhysRevX.9.041064.
  * Fawzi et al. [2019] H. Fawzi, J. Saunderson, and P. A. Parrilo.  Semidefinite approximations of the matrix logarithm.  _Foundations of Computational Mathematics_ , 19:259–296, 2019.  doi: 10.1007/s10208-018-9385-0.  Package cvxquad at https://github.com/hfawzi/cvxquad.
  * Boyd and Vandenberghe [2004] S. Boyd and L. Vandenberghe.  _Convex Optimization_.  Cambridge University Press, Cambridge, UK, 2004.  doi: 10.1017/CBO9780511804441.
  * Luenberger [1969] D. G. Luenberger.  _Optimization by Vector Space Methods_.  John Wiley & Sons, New York, USA, 1969.
  * Dennis Jr. and Wolkowicz [1993] J. E. Dennis Jr. and H. Wolkowicz.  Sizing and least-change secant methods.  _SIAM J. Numer. Anal._ , 30(5):1291–1314, 1993\.  ISSN 0036-1429.  doi: 10.1137/0730067.
  * Lo et al. [2012] H.-K. Lo, M. Curty, and B. Qi.  Measurement-device-independent quantum key distribution.  _Phys. Rev. Lett._ , 108:130503, 2012.  doi: 10.1103/PhysRevLett.108.130503.
  * Cao et al. [2015] Z. Cao, Z. Zhang, H.-K. Lo, and X. Ma.  Discrete-phase-randomized coherent state source and its application in quantum key distribution.  _New J. Phys._ , 17:053014, 2015.  doi: 10.1088/1367-2630/17/5/053014.
  * Lucamarini et al. [2018] M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields.  Overcoming the rate-distance limit of quantum key distribution without quantum repeaters.  _Nature_ , 557:400–403, 2018.  doi: 10.1038/s41586-018-0066-6.
  * Curty et al. [2019] M. Curty, K. Azuma, and H.-K. Lo.  Simple security proof of twin-field type quantum key distribution protocol.  _npj. Quantum Inf._ , 5:64, 2019.  doi: 10.1038/s41534-019-0175-6.