This notebook explores the quantum Fisher information (QFI) for photonic states in quantum optics, using the Fubini–Study metric to quantify parameter sensitivity. Various one- and two-mode states are analyzed to illustrate their metrological potential and possible enhancements beyond the standard quantum limit. This analysis can be extended to do computational explorations of other photonic states in quantum metrology.
Install the Wolfram Quantum Framework:
In[]:=
PacletInstall["https://www.wolfr.am/DevWQCF",ForceVersionInstall->True]
Out[]=
PacletObject
Name: Wolfram/QuantumFramework
Version: 1.6.3

Load the functionalities:
In[]:=
<<Wolfram`QuantumFramework`​​<<Wolfram`QuantumFramework`SecondQuantization`​​<<Wolfram`QuantumFramework`QuantumOptimization`

Quantum Fisher Information

A classical statistical definition of Fisher information to estimate some parameter θ that has a probability distribution
p(x|θ)
is:
The Quantum Fisher Information (QFI) quantifies the maximum amount of information about an unknown parameter θ that can be extracted from a quantum measurement on a state ρ(θ). It sets a lower bound on the variance of any unbiased estimator of via the quantum Cramér-Rao Bound:
In quantum optics, a parameter that is typically chosen for estimation is the phase. Depending on the state used, there are limits that dictate bounds on how accurately this parameter can be estimated.
An interesting mathematical property of the Fisher information is its relation to the Fubini-Study metric tensor. If a quantum state is parametrized by a parameter θ, the following relation holds
where
FS
θ,θ
is the diagonal element of the tensor associated to the parameter θ, if the state only has one parameter the tensor is a scalar

Single mode states

Two state Fock superposition

Lets analyze the state
Pick random values of
n
and
m
:
In[]:=
{n,m}=RandomSample[Range[1,$FockSize-1],2];
Build the state :
In[]:=
ψ=QuantumStateCos[α]FockState[n]+
δ

Sin[α]FockState[m],"Parameters"->{α,δ}
Out[]=
cos(α)|5〉+
δ

sin(α)|10〉
The state has two parameters , so FubiniStudyMetricTensor returns a 2x2 matrix. We can get the Fisher information from the diagonal element of the tensor associated to the θ parameter (check documentation ):
Extracting the quantum Fisher information for both parameters:
In[]:=
QFI=4Simplify[Normal@Diagonal[FubiniStudyMetricTensor[ψ]["Matrix"]],Element[{α,δ},Reals]];
For α parameter the Quantum Fisher information is constant and its value is related to the maximal of Fisher information for two dimensional subspace (similar to a qubit in Bloch sphere). For the relative phase parameter δ the Fisher information has dependance on α, note that for
α=0orα=π/2
the superposition is lost and global phases cannot be measured
Define a qubit state:
In[]:=
qubitState=QuantumStateCos[α],
δ

Sin[α],"Parameters"->{α,δ};
Quantum Fisher information for the two level state:
In[]:=
qubitQFI=4Simplify[Normal@Diagonal[FubiniStudyMetricTensor[qubitState]["Matrix"]],Element[{α,δ},Reals]];
Check the Fisher information related to both parameters is the same:
In[]:=
QFI-qubitQFI
Out[]=
{0,0}
Plot the Crámer-Rao bound, also note the result does not depend on the
n
and m chosen:
In[]:=
Plot3DEvaluate
1
QFI
,{α,0,π/2},{δ,0,2π},

Out[]=
α parameter
δ parameter

Coherent state

In[]:=
SetFockSpaceSize[30];
Phase shifted coherent state:
In[]:=
ψ=QuantumStateCoherentState[][α
θ

],"Parameters"->{α,θ};
Norm of the state:
In[]:=
norm=CoherentState[][α]["Norm"];
Get the Quantum Fisher information:
In[]:=
QFI=Simplify[4FubiniStudyMetricTensor[ψ/norm]["Matrix"][[2,2]],θ∈Reals];
The Fisher information has quadratic dependence on |α|, or linear with |α
2
|
(SQL) :
In[]:=
Plot
QFI
,{α,0,3.},LabelStyle->13,AxesLabel->{"α","Fisher Information"}
Out[]=

Squeezed vacuum state

Squeezed vacuum with real squeeze parameter, phase shifted:
In[]:=
ψ=QuantumState[PhaseShiftOperator[θ]@SqueezeOperator[r][],"Parameters"->{r,θ}];
Quantum Fisher information:
In[]:=
QFI=4(FubiniStudyMetricTensor[ψ]["Matrix"][[2,2]]/.Conjugate[θ]->θ);
The resulting Quantum Fisher information has functional form
2
2
Sinh[2r]
, is expected to be a function of Sinh since
—
n
=
2
Sinh[r]
for a squeezed vacuum state:
In[]:=
data=Table[{rs,Re[QFI/.r->rs]},{rs,0.001,0.6,0.001}];​​NonlinearModelFit[data,A
F
(Sinh[Br])
,{A,B,F},r]["BestFitParameters"]
Out[]=
{A2.00002,B1.99999,F2.}
Plot the Fisher information and CR bound:
In[]:=
PlotQFI,
1
QFI
,{r,0,1},

Out[]=
ℱ
Q
CR Bound
Hence
for large
r
there is notable phase sensitivity, that is why squeezed states are used in many important experiments (LIGO for example). Makes sense because for an squeezed vacuum the mean number of photons is
2
Sinh[r]
, and Sinh[2r] behaves asymptotically as
2
Sinh[r]
so this is close to the Heisenberg
1
N
limit.

Two mode states

NOON state

In[]:=
SetFockSpaceSize[20];
A NOON state is defined as the superposition
:
In[]:=
NOONstate[N_,size_:$FockSize]:=QuantumState
1
2
FockState[{N,0},size]+FockState[{0,N},size]
Nθ

,"Parameters"->θ
NOON state with N = 2:
In[]:=
ψ=NOONstate[2]
Compute Quantum Fisher Information from the FubiniStudyMetricTensor, for different NOON states:
Consider N = 10:
Obviously these states are orthogonal:
The associated measurement probabilities are:

Entangled coherent states

Normalization constant for the whole state:
Normalized state:
Quantum Fisher information, using the Fubini Study Metric Tensor:
Analytical result of the Crámer-Rao bound:
There is a threshold value of |α| such the numerical error increases monotonically due to truncation effects:

Twin Fock State

Annihilation operators for two modes:
Define a twin Fock state for N = 4:
A twin Fock state is unaffected by this operation(no phase sensitivity):
Compute the QFI for different twin Fock states after beam-splitter evolution:
Resulting functional form, similar to NOON state but it has a reduced prefactor, here N is the total number of photons:
Spin Husimi Function for state |4,4〉
Helper definitions for 3D visualizations from this post:
Mapping the state after the beam-splitter operation:
Now for this state the spin Husimi function has ϕ dependence:

Fock-coherent state

A Fock-coherent state is defined as |N〉 ⊗ |α〉, where N is any Fock state number:
Two mode operators:
Coherent state norm:
Compute the Fisher information:

CITE THIS NOTEBOOK

Computing the quantum Fisher information via the Fubini–Study metric for photonic states​
by Bruno Tenorio​
Wolfram Community, STAFF PICKS, October 16, 2025
​https://community.wolfram.com/groups/-/m/t/3561108