Conventions

Enforced by the Conventions exam.

Convention	qkd
Trust	an explicit trusted= flag, never inferred — Security
Unsupported combination	raises, naming the restriction; never approximated — refusals
$\hslash$	$1$
Quadratures	$\hat{x}=(\hat{a}+{\hat{a}}^{†})/\sqrt{2}$, $\hat{p}=(\hat{a}-{\hat{a}}^{†})/(i\sqrt{2})$, so $[\hat{x},\hat{p}]=i$
Vacuum	$\mathrm{\Delta }{x}_{\text{vac}}^2=\mathrm{\Delta }{p}_{\text{vac}}^2=\frac{1}{2}$, covariance $\frac{1}{2}\mathbb{1}$
Coherent state $\alpha$	mean $(\sqrt{2}\mathrm{Re}\alpha ,\sqrt{2}\mathrm{Im}\alpha )$
Ordering	xpxp: $\hat{\mathbf{r}}=({\hat{x}}_1,{\hat{p}}_1,\dots ,{\hat{x}}_N,{\hat{p}}_N{)}^{\mathrm{\top }}$, mode $m$ at indices $2m$, $2m+1$
Covariance	$V_{ij}=\frac{1}{2}⟨\{\mathrm{\Delta }{\hat{r}}_i,\mathrm{\Delta }{\hat{r}}_j\}⟩$

from qkd import gaussian as g

g.Vacuum(1).cov
# array([[0.5, 0. ],
#        [0. , 0.5]])

Ordering and the symplectic form

Symplectic form	$\mathrm{\Omega }={\mathbb{1}}_N\otimes \left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, block-diagonal in xpxp
Same $\mathrm{\Omega }$ in xxpp	the single global block $\left(\begin{array}{cc}0& {\mathbb{1}}_N\\ -{\mathbb{1}}_N& 0\end{array}\right)$
Gaussian unitary	$V\mapsto SV{S}^{\mathrm{\top }}$, $\overline{\mathbf{r}}\mapsto S\overline{\mathbf{r}}$, with $S\mathrm{\Omega }{S}^{\mathrm{\top }}=\mathrm{\Omega }$

The bona fide condition

$V+\frac{i}{2}\mathrm{\Omega }⪰0$, strictly stronger than symmetry plus positive-definiteness:

import numpy as np

V = np.diag([0.4, 0.4])
omega = np.array([[0.0, 1.0], [-1.0, 0.0]])

np.linalg.eigvalsh(V)                    # array([0.4, 0.4])  -- positive definite
np.linalg.eigvalsh(V + 0.5j * omega)     # array([-0.1,  0.9]) -- not a state

The negative eigenvalue	$\mathrm{\Delta }{x}^2\mathrm{\Delta }{p}^2\ge 1/4$ violated, $0.4\times 0.4=0.16$
Applied by	State.physical(), and the test harness after every channel and every symplectic map
Tolerance	absolute, not scaled by the state's own energy
Boundary	$V_{xx}{V}_{pp}=1/4$ accepted, out to $V_{pp}={10}^6$
Symplectic eigenvalues	${\nu }_k\ge 1/2$ (internal units), $=1/2$ for every mode iff the state is pure

from qkd import gaussian as g

g.Moments([0.0, 0.0], [[0.25e-6, 0.0], [0.0, 1e6]])   # accepted
g.Moments([0.0, 0.0], [[4.9e-97, 0.0], [0.0, 1e6]])
# ValueError: cov violates the bona fide condition V + i*Omega/2 >= 0

The SNU boundary

$V_{\text{SNU}}=2V_{\text{internal}}$, applied once, between the Gaussian layer and the key-rate layer.

Layer	Module	Units	Vacuum
Gaussian states, channels, sampling	qkd.gaussian	internal, $\hslash =1$	$1/2$
Number-basis states, Wigner, Husimi, negativity	qkd.fock	internal, $\hslash =1$	$1/2$
Excess-noise budget, key rate, security	qkd.budget, qkd.Link, qkd.Swap	SNU	$1$

$V_A$, $\xi$, $v_{el}$, ${\chi }_{\text{line}}$ and every published anchor are SNU; State.cov, State.spectrum() and State.purity() are internal. fock.State.shadow() needs no conversion.

from qkd import gaussian as g

st = g.Coherent(1.0, 0.0).thermal_loss(0, T=0.5, xi=0.01, ref="input")

st.cov[0, 0]        # 0.5025      internal
2.0 * st.cov[0, 0]  # 1.005       SNU: vacuum 1 plus T*xi = 0.005

Where the factor surfaces	Form
thermal_loss	multiplies the SNU xi argument by $0.5$ on the way in
purity()	$\mu =1/(2^n\sqrt{detV})$ internally versus $1/\sqrt{det{V}_{\text{SNU}}}$
entropy()	converts ${\nu }_{\text{SNU}}=2\nu$ before applying $G$
Bona fide, in SNU	$V+i\mathrm{\Omega }⪰0$, and $\mathrm{\Delta }{x}^2\mathrm{\Delta }{p}^2\ge 1$

Reference states

State	Internal ($\hslash =1$, vacuum $\frac{1}{2}$)	SNU (vacuum $1$)
Vacuum(n)	$\frac{1}{2}{\mathbb{1}}_{2n}$	${\mathbb{1}}_{2n}$
Coherent(x, p)	$\frac{1}{2}{\mathbb{1}}_2$, mean $(x,p)$	${\mathbb{1}}_2$
Thermal(nbar)	$(\overline{n}+\frac{1}{2}){\mathbb{1}}_2$	$(2\overline{n}+1){\mathbb{1}}_2$
Squeezed(r)	$\frac{1}{2}\mathrm{diag}(e^{-2r},e^{+2r})$	$\mathrm{diag}(e^{-2r},e^{+2r})$
Epr(r)	$\frac{1}{2}\left(\begin{array}{cc}\cosh 2r{\mathbb{1}}_2& \sinh 2r{\sigma }_z\\ \sinh 2r{\sigma }_z& \cosh 2r{\mathbb{1}}_2\end{array}\right)$	$\left(\begin{array}{cc}V{\mathbb{1}}_2& \sqrt{V^2-1}{\sigma }_z\\ \sqrt{V^2-1}{\sigma }_z& V{\mathbb{1}}_2\end{array}\right)$, $V=\cosh 2r$

${\sigma }_z=\mathrm{diag}(1,-1)$. The key-rate layer builds the SNU Epr at $V=V_A+1$: Gaussian-modulated coherent states are equivalent to Alice holding half of a two-mode squeezed vacuum.

Excess noise carries a plane

${\xi }_{\text{Bob}}=T{\xi }_{\text{Alice}}$.

qkd's plane	the channel input (Alice's side) everywhere, as Lodewyck, Fossier and Leverrier
Never defaulted	thermal_loss and q.Channel(xi=...) require ref= — see the ref= plane

Bulk returns are numpy arrays

Bulk returns cross the PyO3 boundary as numpy.ndarray, so numpy is a hard runtime dependency. None aliases its source.

Call	dtype	Shape	Buffer
gaussian.State.mean	float64	(2n,)	copied from the state
gaussian.State.cov	float64	(2n, 2n)	copied from the state, reshaped from flat
gaussian.State.homodyne(...)	float64	(shots,)	built for the call
gaussian.State.heterodyne(...)	float64	(shots, 2)	built for the call
fock.State.populations()	float64	(cutoff,)	built for the call
fock.State.eigenvalues()	float64	(cutoff,)	built for the call
fock.State.wigner(xs, ps), .husimi(xs, ps)	float64	(len(ps), len(xs))	built for the call
fock.State.matrix()	complex128	(cutoff, cutoff)	copied from the state
_core.SimOut.frames_x, .frames_p	float64	(n_used,)	copied from the run
_core.cpu_words(...), _core.gpu_words(...)	uint32	(4n,)	built for the call

cpu_words and gpu_words are Threefry-4×32-20 words; uint32 arithmetic wraps at $2^{32}$. A dtype is the representation, not the precision computed at — see dispatch on precision, not device.

Where the literature disagrees

Point	qkd	Alternative in the literature
Vacuum variance	$1/2$ internally, $1$ in the QKD layer	$\hslash =2$ conventions put it at $1$ throughout
Quadrature ordering	xpxp	xxpp
Plane of $\xi$	channel input	channel output (Laudenbach 2018, Eq. (4.8)); divide by $T$ on import
Heterodyne 3 dB penalty	inside ${\chi }_{\text{het}}$ (Lodewyck/Fossier)	$T\to T/2$, $\xi \to \xi /2$ per quadrature (Laudenbach Eq. (5.10))
Entropy function	$G(x)=(x+1){\log }_2(x+1)-x{\log }_{2}x$ at $(\nu -1)/2$	$g(\nu )$ applied to $\nu$ directly; identical, $g(\nu )=G((\nu -1)/2)$