of 46
Supplementary Materials for
Quantum imaging of biological organisms through spatial and polarization
entanglement
Yide Zhang
et al.
Corresponding author: Lihong V. Wang, lvw@caltech.edu
Sci. Adv.
10
, eadk1495 (2024)
DOI: 10.1126/sciadv.adk1495
This PDF file includes:
Notes S1 to S7
Figs. S1 to S17
References
Note
S
1
Sub
-
shot
-
noise signal retrieval in ICE
Each
round of ICE acquisition generates three images: the signal image
(
)
, the idler image
(
)
, and the coincidence image
(
)
.
(
)
and
(
)
contain photon counts from both SPDC
photon pairs (whose averaged value is denoted as
SPDC
) and stray light (whose averaged value is
denoted as
stray
). For simplicity, we assume the signal and idler detectors have the same
background light intensity and detection efficiency, denoted as
.
The imaging of an object here measures its transmittance
(
)
. Although the following derivation
applies to both
and
, we use
as an example. Classically, the transmittance is estimated as
0
(
)
=
(
)
0
(
)
,
(
S
1
)
where
0
denotes the signal image when the object is absent. In ICE
,
we estimate
0
using
b
(
)
, where
b
denotes a background region of the
image outside the target, and
...
denotes averaging over spatial locations.
By using the correlation between the SPDC photon pairs, two types of sub
-
shot
-
noise (SSN)
algorithms have been adopted to enhance the SNR of the transmittance measurements. The first
type relies on the ratio of the two images, where the object’s transmitta
nce is estimated as
(
34
,
71
)
1
(
)
=
(
)
(
)
(
)
b
(
)
.
(
S
2
)
The second type of SSN algorithms, termed optimized subtraction, suppresses the noise in
by
subtracting the variation of
(
18
,
72
)
:
SSN
(
)
=
(
)
(
)
Δ
(
)
,
(
S
3
)
where
(
)
is the unknown spatially varying multiplier, and
Δ
(
)
=
(
)
(
)
. The
ideal
(
)
is proportional to the transmittance
(
)
, the ground truth of which is unknown. To
estimate the
(
)
, one may use the approximated transmittance
̂
(
)
. For example,
̂
(
)
can be
acquired using
0
as in Eq. (S1) or as in
Re
f.
(
18
)
. The object’s transmittance estimated using the
second type of SSN algorithms is given by
2
(
)
=
(
)
b
(
)
̂
(
)
(
SPDC
stray
+
SPDC
)
2
Δ
(
)
b
(
)
.
(
S
4
)
Both Eqs. (
S2
) and (
S4
) achieve higher SNR than
Eq
. (
S1
), demonstrating the quantum advantage.
However, these methods require either prior knowledge of
(
)
or assumptions on the photon
distribution and minimal stray light intensity. Here, inspired by the two algorithms, we introduce
the covariance
-
over
-
variance (CoV) algorithm to further improve the SSN performance with fewer
assumptions.
The workflow of the CoV algorithm is shown in
Fig. S1
. We acquire the time
-
lapsed image stack
of
(
,
)
,
(
,
)
, and
(
,
)
. Following the basic framework of the optimized subtraction
(i.e., Eq. (
S3
)), instead of estimating
(
)
with approximated
(
)
, we derive the optimal
(
)
by
minimizing the variance of
SSN
(
,
)
:
Var
[
SSN
(
,
)
]
=
Var
[
(
,
)
]
+
2
(
)
Var
[
(
,
)
]
2
(
)
Cov
[
(
,
)
,
(
,
)
]
,
(
S
5
)
where
Var
and
Cov
denote the variance and covariance along the time sequence, respectively.
To minimize
Var
[
SSN
(
,
)
]
with regard to
(
)
:
Var
[
SSN
(
,
)
]
휕푘
(
)
=
2
(
)
Var
[
(
,
)
]
2
Cov
[
(
,
)
,
(
,
)
]
=
0
.
(
S
6
)
The optimized
(
)
is thus given by
(
)
=
Cov
[
(
,
)
,
(
,
)
]
Var
[
(
,
)
]
.
(
S
7
)
Since
SSN
(
)
=
(
)
according to Eq. (
S3
), combining Eqs. (
S1
), (
S3
), and (
S7
)
completes the CoV algorithm:
3
(
)
=
(
)
b
(
)
Cov
[
(
,
)
,
(
,
)
]
Var
[
(
,
)
]
Δ
(
)
b
(
)
.
(
S
8
)
From Eqs. (
S5
) and (
S7
), we can derive the minimized variance as
Var
[
SSN
(
,
)
]
=
Var
[
(
,
)
]
Cov
2
[
(
,
)
,
(
,
)
]
Var
[
(
,
)
]
=
Var
[
(
,
)
]
(
1
,
2
)
,
(
S
9
)
where
,
is the Pearson's correlation coefficient between
and
along the time sequence.
Note that the variance of the classical algorithm given by Eq. (
S1
) is
Var
[
(
,
)
]
. Since
,
2
0
, the
CoV
algorithm
guarantees SNR enhancement.
It is worth noting that the CoV algorithm requires repeated measurements over time. With a single
-
frame acquisition, we propose a similar algorithm to estimate
(
)
based on spatial repetitions,
named as the s
-
CoV algorithm.
The workflow of the s
-
CoV algorithm is shown in
Fig. S2
. From the single
-
frame image
(
)
(or
(
)
), we calculate the histogram and divide the pixel values into
bins. The selection of
depends on the experimental configuration and can be optimized through iteration. For the
-
th bin
(
=
1
,
2
,
...
,
), we select the pixels from the image whose values fall into this bin and form the
image subset
(
)
. The binary mask
(
)
used for segmentation (i.e.,
(
)
=
(
)
(
)
)
is then applied to
(
)
to get the image subset
(
)
. Following Eq. (
S7
), we can estimate the
subset of
(
)
(denoted as
,
) as
,
=
Cov
[
(
)
,
(
)
]
Var
[
(
)
]
,
(
S
10
)
where
Var
and
Cov
denote the variance and covariance along spatial locations, respectively.
The
same procedure is repeated for all
, and the resulting
(
)
is the summation of all
(
)
modified by
,
:
(
)
=
,
(
)
=
Cov
[
(
)
,
(
)
]
Var
[
(
)
]
(
)
.
(
S
11
)
Combining Eqs. (
S1
), (
S3
), and (
S11
) completes the s
-
CoV algorithm:
4
(
)
=
(
)
b
(
)
[
Cov
[
(
)
,
(
)
]
Var
[
(
)
]
(
)
]
Δ
(
)
b
(
)
.
(
S
12
)
A comparison of the workflows of the ratio, optimized subtraction, and CoV algorithms is shown