Non-Gaussian states

qkd.fock holds one mode as a density matrix on levels $0\dots d-1$, with the conventions and (len(ps), len(xs)) grid layout of qkd.gaussian: the layers compare with no conversion. Scope.

from qkd import fock

Constructors

cutoff is keyword-only, $1\le d\le 512$. The cutoff is part of the state: the only check on a derived figure is convergence under at_cutoff(d).

Coherent(11.0, 0.0) at the default cutoff 40
Norm discarded	99.79 %
Reported Wigner negativity	2.5072, above every resource state here, on a state whose true negativity is zero
physical(), purity(), Gaussian shadow	True, $1.0$, bona fide
Artefact versus discarded population	1.4 to 6.7 orders of magnitude above it, so trunc_error() does not bound it

Parameter	Unit	Default	Description
Vacuum(*, cutoff)	levels	40	Number(0).
Number(n, *, cutoff)	—	required	$|n⟩$, $0\le n<d$. Zero truncation error.
Coherent(x, p, *, cutoff)	internal quadratures	required	$\beta =(x+ip)/\sqrt{2}$, as gaussian.Coherent(x, p). The Poissonian tail reaches every level, so discarded() is non-zero.
Squeezed(r, *, cutoff)	nepers	required	Squeezed vacuum, $x$ squeezed for $r>0$, as gaussian.Squeezed(r). True negativity zero at every $r$: the grid's calibration state.
Thermal(nbar, *, cutoff)	photons	required	$\overline{n}\ge 0$. Mixed and Gaussian, exercising both halves of the non-Gaussianity measure.
Cat(x, p, odd=False, *, cutoff)	internal quadratures	required	Even or odd superposition of the coherent states at $\pm (x,p)$. An odd cat at the origin raises: $|\beta ⟩-|-\beta ⟩$ vanishes.
Gkp(logical, delta, *, cutoff)	— / vacuum units	required, cutoff=None	Finite-energy grid state, logical $\in \{0,1\}$, $\delta \in [0.05,2]$; squeezing $-20{\log }_{10}\delta$ dB, so $\delta =0.3$ is 10.5 dB. None derives the cutoff as gkp_levels(delta); a pinned value is honoured; a derived cutoff that cannot hold the state raises.
Density(rho)	—	required	Square complex density matrix; its dimension is the cutoff and discarded() is 0. Validated for shape, finiteness, Hermiticity to ${10}^{-9}$, unit trace and positive semidefiniteness — the number-basis bona fide condition.

Module constants

Name	Value	Bounds
CUTOFF	40	Default cutoff for every constructor except Gkp. Not safe for a large displacement.
MAX_LEVELS	512	The native core's cap, so a derived cutoff clamps before the core refuses.
GKP_SCALE, GKP_TAIL	8.0, 1e-6	The numerator of gkp_levels, and the discarded norm a derived-cutoff grid state may reach before its constructor raises. A pinned cutoff is never refused.
gkp_levels(delta)	ceil(GKP_SCALE / delta**2), capped at MAX_LEVELS	Levels needed for a discarded population below GKP_TAIL track $7.1/{\delta }^2$ over $\delta \in [0.15,0.6]$: 66, 89, 128 and 200 at $\delta$ = 0.35, 0.3, 0.25, 0.2. $\delta \lesssim 0.12$ is refused — no cutoff within the cap converges.
EDGE	3	Default levels for tail(), and the depth trunc_error() sums over.
STEP	0.06	Default phase-space sample spacing for window().

Truncation

Method	Catches
discarded()	norm thrown away at construction — a state born too big for its cutoff
tail(levels=3)	population in the top few levels — a state born inside its cutoff, then pushed against the ceiling by a displacement or a channel
trunc_error()	the worse of the two, as one number to threshold on

A truncated matrix is renormalised, so it passes every physicality check. Neither instrument bounds the kernel: both measure population, and the displaced-parity kernel oscillates, so the truncated tail contributes to $\iint |W|$ with alternating sign.

Operation	Truncation cost
rotate(theta)	none — diagonal in the number basis
loss(eta)	none — the Kraus operators only move population down, so it is trace-preserving inside the cutoff and shrinks the tail
displace(x, p)	yes — the truncated displacement is not unitary: it loses norm and pushes population up. The loss is folded into discarded(), not renormalised away

Phase space

$W$ and $Q$ as in the Gaussian layer, in $dxdp$. A general density matrix has no closed form, so $W$ uses the Cahill–Glauber displaced-parity kernel on a rescaled three-term Laguerre recursion.

from qkd import fock

grid = [-1e-9, 0.0, 1e-9]
vac = fock.Vacuum(cutoff=24)

vac.wigner(grid, grid)[1, 1]   # 0.3183098861837907  == 1/pi
vac.husimi(grid, grid)[1, 1]   # 0.15915494309189535 == 1/(2 pi)

$W_{|1⟩}(0,0)=-1/\pi$.

Wigner negativity as the witness

Theorem	Statement
Spekkens 2008	negativity of a quasi-probability representation and the failure of generalised non-contextuality are the same statement
Mari & Eisert 2012; Veitch et al. 2013	Wigner-positive states, under Gaussian operations and homodyne readout, are efficiently classically simulable

The computed quantity is the Kenfack–Życzkowski indicator,

$$\delta (\rho )=\iint |W(x,p)|dxdp-1,$$

zero iff $W\ge 0$ everywhere. Its logarithm is the resource monotone of Albarelli et al. 2018; qkd reports the volume.

The anchor

$\delta (|1⟩)=4/\sqrt{e}-2=0.4261226\dots$ in closed form.

from qkd import fock

fock.Number(1, cutoff=12).negativity().value   # 0.4262150
fock.Vacuum(cutoff=12).negativity().value      # -9.1e-07, i.e. zero

Four significant figures; the residual is the default grid's quadrature error — validation quotes the same state on a stated grid. A number state discards nothing at any $d>n$. The volume grows monotonically from $|0⟩$ to $|4⟩$.

The grid is part of the answer

negativity() returns a Volume, not a bare float: the integral runs over a window nobody chose explicitly.

Member	Description
value	$\iint |W|-1$. float(vol) returns it.
trunc_error	The state's truncation error.
cutoff	The cutoff the state was held at.
points	(len(xs), len(ps)).
span	(x_lo, x_hi, p_lo, p_hi).
step	(dx, dp).

window(state, reach=None, step=0.06)
reach=None	five standard deviations of the state's second moments plus its displacement, on the wider quadrature, held to $\sqrt{2\text{cutoff}}+5$
Five	a Gaussian leaves $1.1\times {10}^{-6}$ of its mass outside a $5\sigma$ square and $1.4\times {10}^{-5}$ outside $4.5\sigma$, the latter already the size of a truncated state's spurious volume
The cap	a state too big for its cutoff has the renormalised matrix's moments: Coherent(11, 0) at cutoff 40 asks for reach 25.65, an ${857}^2$ grid. Levels $0..d-1$ cannot reach past the top Hermite function's turning point $\sqrt{2d-1}$ plus a Gaussian tail — the bound gkp() sizes its quadrature with. Capping moves volumes by at most $1.2\times {10}^{-4}$
step	resolves the kink in $|W|$ at the zero contour; too coarse a step reports a smaller volume

State	trunc_error	Volume, true value $0$
Squeezed(0.8), cutoff 25	$2.2\times {10}^{-5}$	$4.5\times {10}^{-3}$
the same at cutoff 40	—	$2.7\times {10}^{-4}$
the same at cutoff 60	—	$4.1\times {10}^{-6}$

Refining the grid does not move those; raising the cutoff does, monotonically. A volume is evidence of negativity when it stops changing as the cutoff grows. Volume over trunc_error is not a screen: a wholly spurious $2.505$ scores $2.5$, a wholly spurious $4.9\times {10}^{-3}$ scores $1400$, a mostly physical GKP volume $27$.

Two witnesses

non_gaussianity() is the relative-entropy non-Gaussianity $S({\rho }_G)-S(\rho )$, ${\rho }_G$ the Gaussian state with $\rho$'s first and second moments (Genoni & Paris 2010) — shadow(). It returns a Divergence: value, trunc_error, cutoff, and float().

Truncation moves both halves, so a Gaussian state can score large and no cheap bound separates the columns:

At cutoff 40	Artefact	Resource
non_gaussianity(), bits	Squeezed(2.0) $2.284$, Coherent(11.0, 0.0) $5.324$	odd cat $2.16$, Number(10) $4.83$

Negativity implies non-Gaussianity; the converse fails. An equal mixture of two coherent states is non-Gaussian with a strictly positive Wigner function: non_gaussianity() sees it, negativity() does not. Loss separates them at a calculable point:

from qkd import fock

cat = fock.Cat(3.0, 0.0, True, cutoff=40)

for eta in (1.0, 0.7, 0.51, 0.5, 0.49):
    st = cat.loss(eta)
    print(eta, st.negativity().value, float(st.non_gaussianity()))
# 1.00   0.600926   2.5555
# 0.70   0.024912   1.3117
# 0.51   0.000048   1.0931
# 0.50  -0.000000   1.0798
# 0.49  -0.000000   1.0662

Negativity reaches zero at $\eta =\frac{1}{2}$ and stays there; non-Gaussianity falls smoothly through it. The same $\eta =\frac{1}{2}$ kills $|1⟩$'s negativity, where the parity $⟨(-1{)}^n⟩=1-2\eta$ crosses zero. Negativity is the resource, non-Gaussianity a symptom.

Parity

parity() returns $⟨(-1{)}^{\hat{n}}⟩=\pi W(0,0)$ — no grid, and the quantity an experiment measures. A negative value certifies negativity at the origin: sufficient, not necessary. The odd cat sits at $-1$, the even cat at $+1$.

The Gaussian shadow

moments() returns $(⟨x⟩,⟨p⟩)$ and the $2\times 2$ covariance; shadow() is gaussian.Moments(mean, cov) over it.

from qkd import fock

fock.Coherent(1.0, 0.0).shadow().cov
# array([[0.5, 0. ],
#        [0. , 0.5]])

What the shadow is used for
Hudson as a live test	a coherent, squeezed or thermal state built in both layers must give pointwise-equal Wigner grids
Non-Gaussian states	the shadow stays non-negative where the state it shadows does not
$S({\rho }_G)\ge S(\rho )$	a Gaussian maximises entropy at fixed second moments, so a negative non_gaussianity() is numerical noise

Scope

No protocol consumes this layer: Gaussian-modulation CV-QKD is Gaussian end to end, the discrete-modulation analysis works from a covariance matrix, and the click family is written in photons.

Limit
Single mode	no two-mode gates, no tensor network, no MPS — Architecture
GKP is the finite-energy approximation	a Gaussian-enveloped comb of Gaussian peaks in the position wavefunction, at $x_s=2s\sqrt{\pi }$ for $|0_L⟩$ and $(2s+1)\sqrt{\pi }$ for $|1_L⟩$; not the ideal grid state
Operations	pure loss, rotation and displacement only. No thermal-loss channel, no cubic-phase gate, no Kerr evolution, no photon-number-resolving measurement

Read surface

from qkd import fock

odd = fock.Cat(3.0, 0.0, True, cutoff=30)
odd.parity()                 # -1.0 exactly
odd.negativity().value       # 0.600926  at cutoff 30
odd.trunc_error()            # 9.1e-13 -- how much the cutoff cost

gkp = fock.Gkp(0, 0.3)       # cutoff derived: gkp_levels(0.3) == 89
gkp.cutoff                   # 89
gkp.negativity().value       # 0.926206  at cutoff 89
gkp.trunc_error()            # 9.4e-08

thermal = fock.Thermal(0.5, cutoff=30)
thermal.negativity().value   # -9.8e-07 -- Gaussian, so Hudson applies
float(thermal.non_gaussianity())   # 4.6e-13  -- and it is Gaussian too

The cat reads $0.600926$ at cutoff 30 and at at_cutoff(45). The GKP state:

Cutoff	negativity().value	trunc_error()
60 (the former fixed default)	0.933283	$2.6\times {10}^{-5}$
80	0.926781	$4.2\times {10}^{-7}$
89 (gkp_levels(0.3), the derived default)	0.926206	$9.4\times {10}^{-8}$
110	0.925878	$2.2\times {10}^{-9}$

trunc_error improves four orders of magnitude while the volume moves in the third decimal: at 60 the volume's error was $\approx 7\times {10}^{-3}$, about 280× the trunc_error. The derived 89 cuts the overstatement from 0.8 % to 0.04 %.

Method	Returns
cutoff	Levels this state is held on. A property
at_cutoff(d)	The same state on $d$ levels — the convergence check, and the only guard on a Volume
matrix()	The density matrix
populations()	$P(n)={\rho }_{nn}$
eigenvalues()	Spectrum, ascending
photons()	Mean photon number
purity()	A Purity: value is $\mathrm{Tr}({\rho }^2)$ of the truncated state, with discarded and the certified lower bound least
entropy()	Von Neumann entropy in bits
parity()	$⟨(-1{)}^{\hat{n}}⟩=\pi W(0,0)$
physical(tol=1e-9)	Hermitian, unit trace and positive semidefinite
moments(), shadow()	The Gaussian shadow, as a (mean, cov) pair of arrays or as a gaussian.State
wigner(xs, ps), husimi(xs, ps)	Phase-space grids
negativity(xs=None, ps=None)	A Volume; both axes default to window(self)
non_gaussianity()	A Divergence: $S({\rho }_G)-S(\rho )$ in bits, with its truncation error and cutoff
discarded(), tail(levels=3), trunc_error()	The truncation instrumentation

Array dtypes, shapes and aliasing: Bulk returns are numpy arrays. Volume, Divergence and Purity are frozen records carrying __float__.

purity() is the truncated state's purity
Mechanism	the constructors renormalise whatever fits under the cutoff, rescaling every retained element by $1/\text{kept}$ and $\mathrm{Tr}({\rho }^2)$ by $1/{\text{kept}}^2$
Thermal(200) at cutoff 40	keeps 18 % of its norm and reports $0.025083$ against the true $1/(1+2\times 200)=0.002494$ — ten times high, on a state that passes every physical check
Sign	not one-signed: a state whose discarded tail is purer than its retained head reads too low
least	the one certified statement, $\mathrm{Tr}(P\rho P{)}^2\le \mathrm{Tr}({\rho }^2)$ for the truncating projector $P$, and $\mathrm{Tr}(P\rho P{)}^2={\text{kept}}^2\times$value
Above	nothing bounds it usefully — the discarded tail could be pure. entropy() and eigenvalues() carry the same distortion with no bound at all

References

• Spekkens 2008 — PRL 101, 020401 (2008)
• Genoni and Paris 2010 — PRA 82, 052341 (2010)
• Mari & Eisert 2012 — PRL 109, 230503 (2012)
• Veitch et al. 2013 — NJP 15, 013037 (2013)
• Albarelli et al. 2018 — PRA 98, 052350 (2018)