both cases, implying that the process is adiabatic. As $t\rightarrow\infty$,
the mixing angle goes to $\pi/2$ and $\pi/4$ in STIRAP and F-STIRAP
respectively. The coherence $\rho_{13}$ is zero in STIRAP while it is maximum,
0.5, in F-STIRAP.

In the previous sections, we analyzed the dressed (adiabatic) states in STIRAP
and the conditions for achieving adiabatic passage. As in STIRAP, two-photon
resonance is necessary in order to have adiabatic passage in F-STIRAP as well.
Apart from that, we saw that when $\Delta=0$, the variation in mixing angle
should be very slow compared to the rms Rabi frequency,
$|\Omega_{rms}(t)|\gg|\dot{\theta}(t)|$. The evolution of pulses, dressed
states, mixing angle and populations are shown in Fig. 26. The dressed
energies $\lambda_{+}$, $\lambda_{0}$ and $\lambda_{-}$ evolve similarly in
both the processes and there is no crossing of energy levels. The mixing angle
$\theta(t)$ evolve slowly in both cases indicating that $\dot{\theta}(t)$ is
very small compared to the difference between the dressed states $\lambda_{+}$
and $\lambda_{-}$. Note that the final value of $\theta(t)$ is $\pi/2(\approx
1.5)$ in STIRAP while it is $\pi/4(\approx 0.8)$ in F-STIRAP.

Fractional STIRAP can be understood as a generalized form of STIRAP. By
varying the constant mixing angle, it is possible to create any arbitrary
coherent superposition of the initial and final states. The final populations
and coherence are plotted the constant mixing angle in Fig. 27. For
$A=\pi/4\approx 0.8$, the coherence is maximum and for $A=\pi/2\approx 1.6$,
the coherence is zero and the final state population is 1. Any arbitrary
coherence between the initial and final states can be produced by carefully
choosing the angle $A$.

Figure 27: The plot of constant mixing angle $A$ vs populations and coherence.
If $A$=$\pi/4\approx 0.8$, the coherence is maximum and the populations
initial and final states are equal. If $A=\pi/2\approx 1.6$, the population is
completely transferred to the final state. This is equivalent to the ordinary
STIRAP.

###  5.4 Appilcation of F-STIRAP for Remote Detection

Figure 28: The schematic of using Fractional STIRAP technique to optimize the
signal for remote detection. First, fields $\Omega_{p}$ and $\Omega_{s}$
distribute the populations equally and maximize coherence between the
sublevels. Field $\Omega_{3}$ is then applied after some time, which generated
field $\Omega_{4}$ by coherent scattering with the system.

We showed that fractional-STIRAP is a robust and efficient technique to create
maximally coherent superposition states. In CARS, a coherent superposition
state is generated through driving by the pump and Stokes fields and the probe
pulse interacts with this superposition to generate the anti-Stokes signal. An
extension of fractional-STIRAP to the technique of CARS can be done by
applying a pump and Stokes first to create the coherence, followed by a probe
pulse at a later time. The schematic of this method is given in Fig. 28. This
scheme of creating maximum coherence can be combined with the semiclassical
theory we developed in section 3 to create control protocols that optimize the
signal used sensing and detection. We saw that chirping of pulses in STIRAP is
beneficial as it helps us to control the population flow to a desired state in
a four-level system. In the same way, chirping of pulses in fractional-STIRAP
can be used as a way to control the formation of coherent superposition
between a desired pair of states in a four-level system. Combining the
technique of chirping pulses in fractional-STIRAP with the semiclassical
theory for remote detection is expected to make considerable improvements to
the existing method for imaging, sensing and detection.

##  6 SUMMARY

In this chapter, we took a semiclassical approach to deal with light-matter
interactions and developed several methods to prepare quantum systems in a
predetermined state. The primary focus was to improve the existing methods of
detection and sensing by controlling the incident field parameters in order to
optimize the output signal. We learned that adiabatic passage regime of
interaction gives a robust way of preparing maximally coherent superpositions
of quantum states in a multilevel system. We presented ways of improving the
techniques of CARS and STIRAP by chirping the incident laser fields.

In the introductory section, the general theory of light-matter interaction
and Raman spectroscopy was discussed. In the second section, a theory of
quantum control method where the amplitudes and phases of all the incident
pulses in Coherent Anti-Stokes Raman Spectroscopy are carefully manipulated to
satisfy the conditions for adiabatic passage was developed. First, a large
one-photon detuning was assumed so that the two excited states can be
eliminated and the four-level system can be simplified to a “super-effective”
two-level system. This reveals the dynamics of energy levels and a control
scheme can be developed in order to maximize the vibrational coherence. The
amplitudes of probe and Stokes pulses should be equal and should be less than
the amplitude of pump by a factor of $\sqrt{2}$. The chirp rates of the pump
and Stokes needs to be opposite in sign before the central time and should be
equal after that. The probe should be chirped at a rate of equal to the
difference between pump and Stokes pulses all the time. This chirping scheme,
which we called C-CARS or Chirped-CARS, is a robust method to create a
maximally coherent superposition of the system via adiabatic passage while
suppressing the non-resonant background, overcoming one of the major
limitations of CARS spectroscopy. We also show that the selectivity of this
scheme can be increased by controlling the chirp parameter in the chirping
scheme.

In the third section, we developed a semiclassical theory which makes use of
the C-CARS scheme and presented a realistic model of the detection method by
taking methanol vapor as a surrogate system. The aim was to simulate the
optimized output signal from a cloud of molecules using both pulses and pulse
trains incident on the system. A detailed analysis was done to show the
advantages of using control pulses and pulse trains, and to understand the
effects of decoherence and propagation through atmosphere. A layer model of
molecular distribution was created where each layer is characterized by the
fractional density of the target molecules. A set of coupled Maxwell-Liouville
von Neumann equations was derived and numerically solved to find the output
from each layer. The results from each layer were applied to the subsequent
layers to generate the final output signal. When transform limited pulses are
propagated through a molecule distribution of 199 layers, equivalent to 0.5
meters from the center of the cloud, there is an amplification of 2 orders of
magnitude by the final scattering compared to the first scattering. To ensure
that the control pulses do not lose their phase values during the propagation
through multiple layers, a machine learning model was created to extract the
chirp parameters from numerical outputs. An exclusive look at this machine
learning technique, which is based on deep Convolutional Neural Networks
(CNN), was given in section 4. Two CNNs, one to classify the pulses based on
their phase values and another one to extract these values were created.
Primary results show that, the control scheme is efficient for hundred of
layers as the average change in chirp rate after each scattering is less than
0.001%.

In the final section, we discussed the theory of STIRAP process and the
conditions for adiabatic passage in STIRAP. We showed that by controlling the
sign of chirp rate of the incident pulses, the population can be exclusively
flown to a predetermined quantum state in a nearly degenerate system. Later,
we explained how a variation of STIRAP, namely Fractional STIRAP, can be used
to create a maximally coherent superposition of two quantum states in a
multilevel system. Later, we laid the groundwork for how the Fractional STIRAP
technique can be used to optimize the signal in detection and sensing methods.

## Acknowledgment

The authors gratefully acknowledge support from the Office of Naval Research
under awards N00014-20-1-2086 and N00014-22-1-2374. S.M. acknowledges the
Helmholtz Institute Mainz Visitor Program and J. Ch. the support from Johannes
Gutenberg University of Mainz.

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## Appendix A

Derivation of “super-effective" two-level Hamiltonian using adiabatic
elimination in density matrix frame

The Schrödinger Hamiltonian for the 4-level system in Fig. 1 is given by:

$\small{\textbf{H}=\hbar\begin{pmatrix}\omega_{1}&0&\mu_{31}E_{p}(t)&\mu_{41}E_{as}(t)\\\
0&\omega_{2}&\mu_{32}E_{s}(t)&\mu_{42}E_{pr}(t)\\\
\mu_{31}^{*}E_{p}^{*}(t)&\mu_{32}^{*}E_{s}^{*}(t)&\omega_{3}&0\\\
\mu_{41}^{*}E_{as}^{*}(t)&\mu_{42}^{*}E_{pr}^{*}(t)&0&\omega_{4}\\\
\end{pmatrix}}$ (63)

where the pulses $E_{q}(t)$ are given by:

$E_{q}(t)=\frac{1}{2}E_{q_{0}}(t)\left[e^{i\omega_{q}(t-t_{c})+i\frac{\alpha_{q}}{2}(t-t_{c})^{2}}+e^{-i\omega_{q}(t-t_{c})-i\frac{\alpha_{q}}{2}(t-t_{c})^{2}}\right]$
(64)

The Liouville-von Neumann equations are:

$\displaystyle\begin{aligned}
i\hbar\dot{\bm{\rho}}&=\left[\mathbf{H},\,\bm{\rho}\right]\\\
&=\mathbf{H}\bm{\rho}-\bm{\rho}\mathbf{H}\,.\end{aligned}$ (65)

Apply the following transformations:

$\displaystyle\begin{aligned} \rho_{11}=&\tilde{\rho}_{11}\\\
\rho_{12}=&\tilde{\rho}_{12}e^{-i(\omega_{1}-\omega_{2})(t-t_{c})}\\\
\rho_{13}=&\tilde{\rho}_{13}e^{i\omega_{p}(t-t_{c})}\\\
\rho_{14}=&\tilde{\rho}_{14}e^{i\omega_{as}(t-t_{c})}\\\
\rho_{22}=&\tilde{\rho}_{22}\\\
\rho_{23}=&\tilde{\rho}_{23}e^{-i(\omega_{2}-\omega_{1}-\omega_{p})(t-t_{c})}\\\
\rho_{24}=&\tilde{\rho}_{24}e^{-i(\omega_{2}-\omega_{1}-\omega_{as})(t-t_{c})}\\\
\rho_{33}=&\tilde{\rho}_{33}\\\
\rho_{34}=&\tilde{\rho}_{34}e^{-i(\omega_{p}-\omega_{as})(t-t_{c})}\\\
\rho_{44}=&\tilde{\rho}_{44}\end{aligned}$ (66)

After defining the rabi frequencies,
$\Omega_{q_{0}}(t)=\mu_{ij}E_{q_{0}}(t)/\hbar$, applying rotating wave
approximations, and removing the tilde, the equations can be written as:

$\displaystyle\begin{aligned}
i\dot{\rho}_{11}=&\tfrac{1}{2}\Omega_{p_{0}}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{31}+\tfrac{1}{2}\Omega_{as_{0}}(t)\rho_{41}-\tfrac{1}{2}\Omega_{p_{0}}^{*}(t)e^{-\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{13}-\tfrac{1}{2}\Omega_{as_{0}}^{*}(t)\rho_{41}\,\\\
i\dot{\rho}_{12}=&\tfrac{1}{2}\Omega_{p0}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{32}+\tfrac{1}{2}\Omega_{as0}(t)\rho_{42}-\tfrac{1}{2}\Omega_{s_{0}}^{*}(t)e^{i(\omega_{p}-\omega_{s}+\omega_{1}-\omega_{2})(t-t_{c})-\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{13}\\\
&-\tfrac{1}{2}\Omega_{pr_{0}}^{*}(t)e^{i(\omega_{as}-\omega_{pr}+\omega_{1}-\omega_{2})(t-t_{c})-\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{14}\\\
i\dot{\rho}_{13}=&(\omega_{1}+\omega_{p}-\omega_{3})\rho_{13}+\tfrac{1}{2}\Omega_{p_{0}}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{33}+\tfrac{1}{2}\Omega_{as_{0}}(t)\rho_{43}-\tfrac{1}{2}\Omega_{p_{0}}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{11}\\\
&-\tfrac{1}{2}\Omega_{s_{0}}^{*}(t)e^{i(\omega_{2}-\omega_{1}-\omega_{p}+\omega_{s})(t-t_{c})+\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{12}\\\
i\dot{\rho}_{14}=&(\omega_{1}+\omega_{4}-\omega_{as})\rho_{14}+\tfrac{1}{2}\Omega_{p_{0}}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{34}+\tfrac{1}{2}\Omega_{as_{0}}(t)\rho_{44}-\tfrac{1}{2}\Omega_{as_{0}}(t)\rho_{11}\\\
&-\tfrac{1}{2}\Omega_{pr_{0}}^{*}(t)e^{i(\omega_{2}-\omega_{1}-\omega_{pr}-\omega_{as})(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{12}\end{aligned}$

$\displaystyle\begin{aligned}
i\dot{\rho_{22}}=&\tfrac{1}{2}\Omega_{s_{0}}(t)e^{i(\omega_{2}-\omega_{1}-\omega_{p}+\omega_{s})(t-t_{c})+\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{32}\\\
&+\tfrac{1}{2}\Omega_{pr_{0}}(t)e^{i(\omega_{2}-\omega_{1}-\omega_{as}+\omega_{pr})(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{42}\\\
&-\tfrac{1}{2}\Omega_{s_{0}}^{*}(t)e^{-i(\omega_{2}-\omega_{1}-\omega_{p}+\omega_{s})(t-t_{c})-\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{23}\\\
&-\tfrac{1}{2}\Omega_{pr_{0}}^{*}(t)e^{-i(\omega_{2}-\omega_{1}-\omega_{as}+\omega_{pr})(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{24}\\\
i\dot{\rho_{23}}=&-(\omega_{3}-\omega_{1}-\omega_{p})\rho_{23}+\tfrac{1}{2}\Omega_{s_{0}}(t)e^{i(\omega_{2}-\omega_{1}-\omega_{p}+\omega_{s})(t-t_{c})+\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{33}\\\
&+\tfrac{1}{2}\Omega_{pr_{0}}(t)e^{i(\omega_{2}-\omega_{1}-\omega_{as}+\omega_{pr})(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{43}\\\
&-\tfrac{1}{2}\Omega_{p_{0}}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{21}-\tfrac{1}{2}\Omega_{s_{0}}(t)e^{i(\omega_{2}-\omega_{1}-\omega_{p}+\omega_{s})(t-t_{c})+\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{22}\\\
i\dot{\rho_{24}}=&(-\omega_{4}+\omega_{1}+\omega_{as})\rho_{24}+\tfrac{1}{2}\Omega_{s_{0}}(t)e^{i(\omega_{2}-\omega_{1}-\omega_{p}+\omega_{s})(t-t_{c})+\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{34}\\\
&+\tfrac{1}{2}\Omega_{pr_{0}}(t)e^{i(\omega_{2}-\omega_{1}-\omega_{as}+\omega_{pr})(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{44}\\\
&-\tfrac{1}{2}\Omega_{as_{0}}(t)\rho_{21}-\tfrac{1}{2}\Omega_{pr_{0}}(t)e^{i(\omega_{2}-\omega_{1}-\omega_{as}+\omega_{pr})(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{22}\\\
i\dot{\rho_{33}}=&\tfrac{1}{2}\Omega_{p_{0}}^{*}(t)e^{-\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{13}+\tfrac{1}{2}\Omega_{s_{0}}^{*}(t)e^{-i(\omega_{2}-\omega_{1}-\omega_{p}+\omega_{s})(t-t_{c})-\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{23}\\\
&-\tfrac{1}{2}\Omega_{p_{0}}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{31}-\tfrac{1}{2}\Omega_{s_{0}}(t)e^{i(\omega_{2}-\omega_{1}-\omega_{p}+\omega_{s})(t-t_{c})+\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{32}\\\
i\dot{\rho_{34}}=&(\omega_{3}-\omega_{4}-\omega_{p}+\omega_{as})\rho_{34}+\tfrac{1}{2}\Omega_{p_{0}}^{*}(t)e^{-\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{14}\\\
&+\tfrac{1}{2}\Omega_{s_{0}}^{*}(t)e^{-i(\omega_{2}-\omega_{1}-\omega_{p}+\omega_{s})(t-t_{c})-\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{24}\\\
&-\tfrac{1}{2}\Omega_{as_{0}}(t)\rho_{31}-\tfrac{1}{2}\Omega_{pr_{0}}(t)e^{i(\omega_{2}-\omega_{1}-\omega_{as}+\omega_{pr})(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{32}\\\
i\dot{\rho_{44}}=&\tfrac{1}{2}\Omega_{as_{0}}^{*}(t)\rho_{14}+\tfrac{1}{2}\Omega_{pr_{0}}(t)e^{-i(\omega_{2}-\omega_{1}-\omega_{as}+\omega_{pr})(t-t_{c})-\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{24}\\\
&-\tfrac{1}{2}\Omega_{as_{0}}(t)\rho_{41}-\tfrac{1}{2}\Omega_{pr_{0}}(t)e^{i(\omega_{2}-\omega_{1}-\omega_{as}+\omega_{pr})(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{42}\end{aligned}$

Define the detunings:

$\displaystyle\begin{aligned}
\Delta_{s}&=\omega_{p}-(\omega_{3}-\omega_{1})\\\
\Delta_{as}&=\omega_{as}-(\omega_{4}-\omega_{1})\\\
\delta_{A}&=\omega_{p}-\omega_{s}-(\omega_{2}-\omega_{1})\\\
\delta_{B}&=\omega_{as}-\omega_{pr}-(\omega_{2}-\omega_{1})\,,\end{aligned}$
(67)

and rewrite the equations:

$\displaystyle\begin{aligned}
i\dot{\rho}_{11}=&\tfrac{1}{2}\Omega_{p0}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{31}+\tfrac{1}{2}\Omega_{as0}(t)\rho_{41}\\\
&-\tfrac{1}{2}\Omega_{p0}^{*}(t)e^{-\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{13}+\tfrac{1}{2}\Omega_{as0}^{*}(t)\rho_{14}\,,\\\
i\dot{\rho}_{12}=&\tfrac{1}{2}\Omega_{p0}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{32}+\tfrac{1}{2}\Omega_{as0}(t)\rho_{42}-\tfrac{1}{2}\Omega^{*}_{s0}(t)e^{i\delta_{A}(t-t_{c})-\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{13}\\\
&-\tfrac{1}{2}\Omega^{*}_{pr0}(t)e^{i\delta_{B}(t-t_{c})-\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{14}\,,\\\
i\dot{\rho}_{13}=&\Delta_{s}\rho_{13}+\tfrac{1}{2}\Omega_{p0}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{33}+\tfrac{1}{2}\Omega_{as0}(t)\rho_{43}-\tfrac{1}{2}\Omega_{p0}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{11}\\\
&-\tfrac{1}{2}\Omega_{s0}(t)e^{-i\delta_{B}(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{12}\,,\\\
i\dot{\rho}_{14}=&\Delta_{as}\rho_{14}+\tfrac{1}{2}\Omega_{p0}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{34}+\tfrac{1}{2}\Omega_{as0}(t)\rho_{44}-\tfrac{1}{2}\Omega_{a0}(t)\rho_{11}\\\
&-\tfrac{1}{2}\Omega_{pr0}(t)e^{-i\delta_{B}(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{12}\,,\\\
i\dot{\rho}_{22}=&\tfrac{1}{2}\Omega_{s_{0}}(t)e^{-i\delta_{A}(t-t_{c})+\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{32}+\tfrac{1}{2}\Omega_{pr_{0}}(t)e^{-i\delta_{B}(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{42}\\\
&-\tfrac{1}{2}\Omega_{s_{0}}^{*}(t)e^{i\delta_{A}(t-t_{c})-\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{23}-\tfrac{1}{2}\Omega_{pr0}^{*}(t)e^{i\delta_{B}(t-t_{c})-\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{24}\,,\\\
i\dot{\rho}_{23}=&\Delta_{s}\rho_{23}+\tfrac{1}{2}\Omega_{s0}(t)e^{-i\delta_{A}(t-t_{c})+\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{33}+\tfrac{1}{2}\Omega_{pr0}(t)e^{-i\delta_{B}(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{43}\\\
&-\tfrac{1}{2}\Omega_{p0}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{21}-\tfrac{1}{2}\Omega_{s0}(t)e^{-i\delta_{A}(t-t_{c})-\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{22}\,,\\\
i\dot{\rho}_{24}=&\Delta_{as}\rho_{24}+\tfrac{1}{2}\Omega_{s0}(t)e^{-i\delta_{A}(t-t_{c})+\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{34}+\tfrac{1}{2}\Omega_{pr0}(t)e^{-i\delta_{B}(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{44}\\\
&-\tfrac{1}{2}\Omega_{as0}(t)\rho_{21}-\tfrac{1}{2}\Omega_{pr0}(t)e^{-i\delta_{B}(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{22}\,,\\\
i\dot{\rho}_{33}=&\tfrac{1}{2}\Omega_{p_{0}}^{*}(t)e^{-\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{13}+\tfrac{1}{2}\Omega_{s_{0}}^{*}(t)e^{i\delta_{A}(t-t_{c})-\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{23}\\\
&-\tfrac{1}{2}\Omega_{p_{0}}(t)e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{31}-\tfrac{1}{2}\Omega_{s_{0}}(t)e^{-i\delta_{A}(t-t_{c})+\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{32}\,,\\\
i\dot{\rho}_{34}=&(\Delta_{as}-\Delta_{s})\rho_{34}+\tfrac{1}{2}\Omega^{*}_{p0}(t)e^{-\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{14}+\tfrac{1}{2}\Omega^{*}_{s0}(t)e^{i\delta_{A}(t-t_{c})-\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{24}\\\
&-\tfrac{1}{2}\Omega_{as0}(t)\rho_{31}-\tfrac{1}{2}\Omega_{pr0}(t)e^{-i\delta_{B}(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{32}\,.\\\
i\dot{\rho}_{44}=&\tfrac{1}{2}\Omega^{*}_{as0}(t)\rho_{14}+\tfrac{1}{2}\Omega^{*}_{pr0}(t)e^{i\delta_{B}(t-t_{c})-\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{24}\\\
&-\tfrac{1}{2}\Omega_{as0}(t)\rho_{41}-\tfrac{1}{2}\Omega_{pr0}(t)e^{-i\delta_{B}(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{42}\,,\\\
\end{aligned}$ (68)

After applying the conditions for adiabatic elimination:

$\displaystyle\begin{aligned} \dot{\rho_{33}}&=0\\\ \dot{\rho_{44}}&=0\\\
\dot{\rho_{34}}&=0\\\ \end{aligned}$ (69)

in the equations for $\dot{\rho_{33}}$, $\dot{\rho_{44}}$ and
$\dot{\rho_{34}}$, they can be written as:

$\displaystyle\begin{aligned}
\rho_{13}&=\frac{\Omega_{p_{0}}(t)}{2\Delta_{s}}e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{11}+\frac{\Omega_{s_{0}}(t)}{2\Delta_{s}}e^{-i\delta_{A}(t-t_{c})+\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{12}\\\
\rho_{14}&=\frac{\Omega_{as_{0}}(t)}{2\Delta_{as}}\rho_{11}+\frac{\Omega_{pr_{0}}(t)}{2\Delta_{as}}e^{-i\delta_{B}(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{12}\\\
\rho_{23}&=\frac{\Omega_{p_{0}}(t)}{2\Delta_{s}}e^{\frac{i}{2}\alpha_{p}(t-t_{c})^{2}}\rho_{21}+\frac{\Omega_{s_{0}}(t)}{2\Delta_{s}}e^{-i\delta_{A}(t-t_{c})+\frac{i}{2}\alpha_{s}(t-t_{c})^{2}}\rho_{22}\\\
\rho_{14}&=\frac{\Omega_{as_{0}}(t)}{2\Delta_{as}}\rho_{21}+\frac{\Omega_{pr_{0}}(t)}{2\Delta_{as}}e^{-i\delta_{B}(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{22}\end{aligned}$
(70)

Substituting these in the equations for $\dot{\rho_{11}}$, $\dot{\rho_{22}}$
and $\dot{\rho_{12}}$ will simplify the system of equations into the below set
of 3 equations which only includes the states $\ket{1}$ and $\ket{2}$:

$\displaystyle\begin{aligned}
i\dot{\rho}_{11}=&\left(\frac{\Omega_{p0}(t)\Omega^{*}_{s0}(t)}{4\Delta_{s}}e^{i\delta_{A}(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}+\frac{\Omega^{*}_{pr0}(t)\Omega_{as0}(t)}{4\Delta_{as}}e^{i\delta_{B}(t-t_{c})-\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\right)\rho_{21}\\\
&-\left(\frac{\Omega^{*}_{p0}(t)\Omega_{s0}(t)}{4\Delta_{s}}e^{-i\delta_{A}(t-t_{c})+\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\right.\\\
&+\left.\frac{\Omega_{pr0}(t)\Omega^{*}_{as0}(t)}{4\Delta_{as}}e^{-i\delta_{B}(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\right)\rho_{12}\\\
i\dot{\rho}_{12}=&\left(\frac{\Omega_{p0}(t)\Omega^{*}_{p0}(t)}{4\Delta_{s}}\rho_{12}+\frac{\Omega_{p0}(t)\Omega^{*}_{s0}(t)}{4\Delta_{s}}e^{i\delta_{A}(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{22}\right)\\\
&+\left(\frac{\Omega_{as0}(t)\Omega^{*}_{as0}(t)}{4\Delta_{as}}\rho_{12}+\frac{\Omega^{*}_{pr0}(t)\Omega_{as0}(t)}{4\Delta_{as}}e^{i\delta_{B}(t-t_{c})-\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{22}\right)\\\
&-\left(\frac{\Omega^{*}_{s0}(t)\Omega_{p0}(t)}{4\Delta_{s}}e^{i\delta_{A}(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{11}+\frac{\Omega_{s0}(t)\Omega^{*}_{s0}(t)}{4\Delta_{s}}\rho_{12}\right)\\\
&-\left(\frac{\Omega^{*}_{pr0}(t)\Omega_{as0}(t)}{4\Delta_{as}}e^{i\delta_{B}(t-t_{c})-\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\rho_{11}+\frac{\Omega_{pr0}(t)\Omega^{*}_{pr0}(t)}{4\Delta_{as}}\rho_{12}\right)\\\
i\dot{\rho}_{22}=&\left(\frac{\Omega_{p0}(t)\Omega^{*}_{p0}(t)}{4\Delta_{s}}e^{-i\delta_{A}(t-t_{c})+\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\right.\\\
&\left.+\frac{\Omega_{pr0}(t)\Omega^{*}_{as0}(t)}{4\Delta_{as}}e^{-i\delta_{B}(t-t_{c})+\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\right)\rho_{12}\\\
&-\left(\frac{\Omega^{*}_{s0}(t)\Omega_{p0}(t)}{4\Delta_{s}}e^{i\delta_{A}(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\right.\\\
&+\left.\frac{\Omega^{*}_{pr0}(t)\Omega_{as0}(t)}{4\Delta_{as}}e^{i\delta_{B}(t-t_{c})-\frac{i}{2}\alpha_{pr}(t-t_{c})^{2}}\right)\rho_{21}\\\
\end{aligned}$ (71)

In CARS, the two-photon detunings are generally equal,
$\delta_{A}=\delta_{B}=\delta$. In the C-CARS scheme, the chirp rate of probe
pulse is given by: $\alpha_{pr}=\alpha_{s}-\alpha_{p}$. Applying these
conditions, the the above equations can be simplified to:

$\displaystyle\begin{aligned}
i\dot{\rho}_{11}=&\left(\frac{\Omega_{p0}(t)\Omega^{*}_{s0}(t)}{4\Delta_{s}}+\frac{\Omega^{*}_{pr0}(t)\Omega_{as0}(t)}{4\Delta_{as}}\right)e^{i\delta(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{21}\\\
&-\left(\frac{\Omega^{*}_{p0}(t)\Omega_{s0}(t)}{4\Delta_{s}}+\frac{\Omega_{pr0}(t)\Omega^{*}_{as0}(t)}{4\Delta_{as}}\right)e^{-i\delta(t-t_{c})+\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{12}\\\
i\dot{\rho}_{12}=&\left(\frac{|\Omega_{p0}(t)|^{2}}{4\Delta_{s}}+\frac{|\Omega_{as0}(t)|^{2}}{4\Delta_{as}}\right)\rho_{12}\\\
&+\left(\frac{\Omega_{p0}(t)\Omega^{*}_{s0}(t)}{4\Delta_{s}}+\frac{\Omega^{*}_{pr0}(t)\Omega_{as0}(t)}{4\Delta_{as}}\right)e^{i\delta(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{22}\\\
&-\left(\frac{|\Omega_{s0}(t)|^{2}}{4\Delta_{s}}+\frac{|\Omega_{pr0}(t)|^{2}}{4\Delta_{as}}\right)\rho_{12}\\\
&-\left(\frac{\Omega_{p0}(t)\Omega^{*}_{s0}(t)}{4\Delta_{s}}+\frac{\Omega^{*}_{pr0}(t)\Omega_{as0}(t)}{4\Delta_{as}}\right)e^{i\delta(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{11}\\\
i\dot{\rho}_{22}=&\left(\frac{\Omega^{*}_{p0}(t)\Omega_{s0}(t)}{4\Delta_{s}}+\frac{\Omega_{pr0}(t)\Omega^{*}_{as0}(t)}{4\Delta_{as}}\right)e^{-i\delta(t-t_{c})+\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{12}\\\
&-\left(\frac{\Omega_{p0}(t)\Omega^{*}_{s0}(t)}{4\Delta_{s}}+\frac{\Omega^{*}_{pr0}(t)\Omega_{as0}(t)}{4\Delta_{as}}\right)e^{i\delta(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{21}\\\
\end{aligned}$ (72)

Now, define the new Rabi frequencies:

$\Omega_{1}(t)=\frac{|\Omega_{p0}(t)|^{2}}{4\Delta_{s}}+\frac{|\Omega_{as0}(t)|^{2}}{4\Delta_{as}},\
\ \ \ \
\Omega_{2}(t)=\frac{|\Omega_{s0}(t)|^{2}}{4\Delta_{s}}+\frac{|\Omega_{pr0}(t)|^{2}}{4\Delta_{as}}\,,$
(73)

and

$\Omega_{3}(t)=\frac{\Omega_{p0}(t)\Omega^{*}_{s0}(t)}{4\Delta_{s}}+\frac{\Omega^{*}_{pr0}(t)\Omega_{as0}(t)}{4\Delta_{as}}\,.$
(74)

to receive:

$\displaystyle\begin{aligned}
i\dot{\rho}_{11}=&\Omega_{3}(t)e^{i\delta(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{21}-\Omega^{*}_{3}(t)e^{-i\delta(t-t_{c})+\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{12}\,\\\
i\dot{\rho}_{12}=&\Omega_{1}(t)\rho_{12}+\Omega_{3}(t)e^{i\delta(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{22}\\\
&-\Omega_{3}(t)e^{i\delta(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{11}-\Omega_{2}(t)\rho_{12}\\\
i\dot{\rho}_{22}=&\Omega^{*}_{3}(t)e^{-i\delta(t-t_{c})+\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{12}-\Omega_{3}(t)e^{i\delta(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\rho_{21}\,\\\
\end{aligned}$ (75)

The Hamiltonian can now be written, in the interaction representation as:

$\small{\textbf{H}=\hbar\begin{pmatrix}\Omega_{1}(t)&\Omega_{3}(t)e^{i\delta(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\\\
\Omega^{*}_{3}(t)e^{-i\delta(t-t_{c})+\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}&\Omega_{2}(t)\\\
\end{pmatrix}}$ (76)

To derive the Hamiltonian in the field-interaction representation, the density
matrix elements are transformed as below:

$\displaystyle\begin{aligned} \rho_{11}=&\tilde{\rho}_{11}\\\
\rho_{12}=&\tilde{\rho}_{12}e^{i\delta(t-t_{c})-\frac{i}{2}(\alpha_{s}-\alpha_{p})(t-t_{c})^{2}}\\\
\rho_{22}=&\tilde{\rho}_{22}\end{aligned}$ (77)

Applying these transformations in the Liouville von-Neumann equations (65)
will give:

$\displaystyle\begin{aligned}
i\dot{\rho_{11}}=&\Omega_{3}(t)\rho_{21}-\Omega^{*}_{3}(t)\rho_{12}\\\
i\dot{\rho_{12}}=&\Omega_{1}(t)\rho_{12}+\Omega_{3}(t)\rho_{22}-\Omega_{3}(t)\rho_{11}-\left[\Omega_{2}(t)-\delta+(\alpha_{s}-\alpha_{p})(t-t_{c})\right]\rho_{12}\\\
i\dot{\rho_{22}}=&\Omega^{*}_{3}(t)\rho_{12}-\Omega_{3}(t)\rho_{21}\end{aligned}$
(78)

The Hamiltonian is now given by:

$\small{\textbf{H}=\hbar\begin{pmatrix}\Omega_{1}(t)&\Omega_{3}(t)\\\
\Omega^{*}_{3}(t)&\Omega_{2}(t)-(\delta-(\alpha_{s}-\alpha_{p})(t-t_{c}))\\\
\end{pmatrix}}$ (79)

The final “super-effective" Hamiltonian can be found by adding
$\frac{1}{2}(\delta-(\alpha_{s}-\alpha_{p})(t-t_{c})-\Omega_{1}(t)-\Omega_{2}(t))$
to both the diagonal:

$\mathbf{H}_{se}(t)=\frac{\hbar}{2}\begin{pmatrix}\small\delta-(\alpha_{s}-\alpha_{p})(t-t_{c})+\Omega_{1}(t)-\Omega_{2}(t)&2\Omega_{3}(t)\\\
2\Omega^{*}_{3}(t)&-\delta+(\alpha_{s}-\alpha_{p})(t-t_{c})-\Omega_{1}(t)+\Omega_{2}(t)\\\
\end{pmatrix}\ $ (80)

## Appendix B

Derivation of coupled Maxwell - Liouville von Neumann Equations

Maxwell’s equations, with no free currents and charges, read

$\displaystyle\nabla\cdot(\epsilon_{0}E+P)=0$ (81) $\displaystyle\nabla\times
E=-\partial B/\partial t$ (82) $\displaystyle\nabla\times
B=\mu_{0}\partial(\epsilon_{0}E+P)/\partial t$ (83) $\displaystyle\nabla\cdot
B=0$ (84)

From Eqs.(82,83) we obtain the wave equation

$\nabla^{2}E-\epsilon_{0}\mu_{0}\frac{\partial^{2}E}{\partial^{2}t}=\nabla(\nabla\cdot
E)+\mu_{0}\frac{\partial^{2}P}{\partial^{2}t}.$ (85)

It follows from Eq.(81) that $\nabla\cdot E=\nabla\cdot P/\epsilon_{0}$ in a
space free from charges. In a plane wave limit, when the wave length is much
less than the beam radius and neglecting any diffraction effects in transverse
direction, fields propagate in the $\hat{z}$ direction and have polarization
in the XY plane. Then $\nabla\cdot P$ may be set to zero and the wave equation
reads

$\displaystyle\left(\partialderivative{z}+\frac{1}{c}\partialderivative{t}\right)\left(-\partialderivative{z}+\frac{1}{c}\partialderivative{t}\right)E=-\mu_{0}\partialderivative[2]{P}{t}$
(86)

Assuming the field is $E(z,t)=\frac{1}{2}(E_{0}(z,t)e^{-i[\omega
t-kz-\phi(z,t)]}+c.c)$ and polarization is
$P(z,t)=\frac{1}{2}(P_{0}(z,t)e^{-i[\omega t-kz-\phi(z,t)]}+c.c)$ and
considering $E_{0}(z,t)$ and $\phi(z,t)$ as slowly varying functions of
position and time, we write

$\displaystyle-\partialderivative{E(z,t)}{z}=-\frac{1}{2}(e^{-i\omega
t}e^{ikz}e^{i\phi(z,t)}\partialderivative{E_{0}(z,t)}{z}+ikE_{0}(z,t)e^{-i\omega
t}e^{ikz}e^{i\phi(z,t)}$ (87)
$\displaystyle+i\partialderivative{\phi(z,t)}{z}E_{0}(z,t)e^{-i\omega
t}e^{ikz}e^{i\phi(z,t)}+c.c.)$
$\displaystyle\frac{1}{c}\partialderivative{E(z,t)}{t}=\frac{1}{2c}(e^{-i\omega
t}e^{ikz}e^{i\phi(z,t)}\partialderivative{E_{0}(z,t)}{t}-i\omega
E_{0}(z,t)e^{-i\omega t}e^{ikz}e^{i\phi(z,t)}$ (88)
$\displaystyle+i\partialderivative{\phi(z,t)}{t}E_{0}(z,t)e^{-i\omega
t}e^{ikz}e^{i\phi(z,t)}+c.c.)$

Then

$\displaystyle\left(-\partialderivative{z}+\frac{1}{c}\partialderivative{t}\right)E$
$\displaystyle=$ $\displaystyle-\frac{ik}{2}(E_{0}(z,t)e^{-i\omega
t}e^{ikz}e^{i\phi(z,t)}-c.c.)$ (89)
$\displaystyle-\frac{i\omega}{2c}(E_{0}(z,t)e^{-i\omega
t}e^{ikz}e^{i\phi(z,t)}-c.c.)$ $\displaystyle=$ $\displaystyle-
ik(E_{0}(z,t)e^{-i(\omega t-kz-\phi(z,t))}-c.c.)$ $\displaystyle=$
$\displaystyle-2ik\Im{E}.$

By substituting Eq.(89) to Eq.(86) and using $\omega/c=k$ and later assuming
real fields we arrive at

$\displaystyle-
ik\left(\partialderivative{z}+\frac{1}{c}\partialderivative{t}\right)(E_{0}(z,t)e^{-i(\omega
t-kz-\phi(z,t))}+c.c.)$ (91) $\displaystyle=$ $\displaystyle-
ik\partialderivative{E_{0}(z,t)}{z}e^{-i(\omega
t-kz-\phi(z,t))}+ik\partialderivative{E_{0}^{*}(z,t)}{z}e^{i(\omega
t-kz-\phi(z,t))}$
$\displaystyle-\frac{ik}{c}\partialderivative{E_{0}(z,t)}{t}e^{-i(\omega
t-kz-\phi(z,t))}+\frac{ik}{c}\partialderivative{E_{0}^{*}(z,t)}{t}e^{i(\omega
t-kz-\phi(z,t))}$ $\displaystyle=$ $\displaystyle-
ik\left[\left(\partialderivative{E_{0}(z,t)}{z}+\frac{1}{c}\partialderivative{E_{0}(z,t)}{t}\right)e^{-i(\omega
t-kz-\phi(z,t))}\right]$
$\displaystyle+ik\left[\left(\partialderivative{E_{0}^{*}(z,t)}{z}+\frac{1}{c}\partialderivative{E_{0}^{*}(z,t)}{t}\right)e^{i(\omega
t-kz-\phi(z,t))}\right]$ $\displaystyle=$
$\displaystyle-2k\left(\partialderivative{E_{0}(z,t)}{z}+\frac{1}{c}\partialderivative{E_{0}(z,t)}{t}\right)\frac{1}{2i}\left(-e^{-i(\omega
t-kz-\phi(z,t))}+e^{i(\omega t-kz-\phi(z,t))}\right)$ $\displaystyle=$
$\displaystyle-2k\left(\partialderivative{E_{0}(z,t)}{z}+\frac{1}{c}\partialderivative{E_{0}(z,t)}{t}\right)\sin{(\omega
t-kz-\phi(z,t))}$ $\displaystyle=$
$\displaystyle-\mu_{0}\partialderivative[2]{t}P(z,t)$

For $P(z,t)=\frac{1}{2}(P_{0}(z,t)e^{-i[\omega t-kz-\phi(z,t)]}+c.c)$,

$\displaystyle\partialderivative[2]{t}P(z,t)=-\omega^{2}\left(\frac{1}{2}(P_{0}(z,t)e^{-i[\omega
t-kz-\phi(z,t)]}+c.c)\right)=-\omega^{2}\real\left[P(z,t)\right]$ (92)

Substituting these in Eq.(91) gives

$\displaystyle-2k(\partialderivative{E_{0}(z,t)}{z}+\frac{1}{c}\partialderivative{E_{0}(z,t)}{t})\sin{(\omega
t-kz-\phi(z,t))}=$
$\displaystyle\mu_{0}\omega^{2}\left(Re\left[P_{0}(z,t)\right]\cos{(\omega
t-kz)}+Im\left[P_{0}(z,t)\right]\sin{(\omega t-kz-\phi(z,t))}\right),$ (93)

leading to

$\displaystyle-2k(\partialderivative{E_{0}(z,t)}{z}+\frac{1}{c}\partialderivative{E_{0}(z,t)}{t})=$
$\displaystyle\mu_{0}\omega^{2}Im\left[P_{0}(z,t)\right].$ (94)

In quantum theory, a measurable quantity is the expectation value, which for
macroscopic polarization is an expectation value of the electric dipole moment
operator $\hat{\mu}$, $\langle
P(z,t)\rangle=N_{s}Tr\\{\langle\rho(z,t)\cdot\mu\rangle\\}$, where $N_{s}$ is
the atomic density of the target molecules. Applied to the four-level system
of CARS, it contains four components corresponding to each of four
transitions:

$\displaystyle P_{p}(z,t)$
$\displaystyle=2N_{s}\real\left[\mu_{13}\rho_{13}(z,t)e^{i(\omega_{p}t-k_{p}z-\phi(z,t))}\right]$
(95) $\displaystyle P_{s}(z,t)$
$\displaystyle=2N_{s}\real\left[\mu_{23}\rho_{23}(z,t)e^{i(\omega_{s}t-k_{s}z-\phi(z,t))}\right]$
$\displaystyle P_{pr}(z,t)$
$\displaystyle=2N_{s}\real\left[\mu_{24}\rho_{24}(z,t)e^{i(\omega_{pr}t-k_{pr}z-\phi(z,t))}\right]$
$\displaystyle P_{as}(z,t)$
$\displaystyle=2N_{s}\real\left[\mu_{14}\rho_{14}(z,t)e^{i(\omega_{as}t-k_{as}z-\phi(z,t))}\right],$

giving $P_{0p}(z,t)=N_{s}\mu_{13}\rho_{13}(z,t)$,
$P_{0s}(z,t)=N_{s}\mu_{23}\rho_{23}(z,t)$,
$P_{0pr}(z,t)=N_{s}\mu_{24}\rho_{24}(z,t)$, and
$P_{0as}(z,t)=N_{s}\mu_{14}\rho_{14}(z,t)$.

For four components of propagating fields in CARS, the Eq.(94) reads as
follows

$\displaystyle\partialderivative{E_{p}}{z}+\frac{1}{c}\partialderivative{E_{p}}{t}=-N_{s}\frac{\mu_{0}\mu_{13}\omega_{p}^{2}}{k_{p}}\Im{\rho_{13}(z,t)}$
(96)
$\displaystyle\partialderivative{E_{s}}{z}+\frac{1}{c}\partialderivative{E_{s}}{t}=-N_{s}\frac{\mu_{0}\mu_{23}\omega_{s}^{2}}{k_{s}}\Im{\rho_{23}(z,t)}$
$\displaystyle\partialderivative{E_{pr}}{z}+\frac{1}{c}\partialderivative{E_{pr}}{t}=-N_{s}\frac{\mu_{0}\mu_{24}\omega_{pr}^{2}}{k_{pr}}\Im{\rho_{24}(z,t)}$
$\displaystyle\partialderivative{E_{as}}{z}+\frac{1}{c}\partialderivative{E_{as}}{t}=-N_{s}\frac{\mu_{0}\mu_{14}\omega_{as}^{2}}{k_{as}}\Im{\rho_{14}(z,t)}.$

If $\bar{t}=(t-\frac{z}{c})$, then
$\derivative{t}{z}=(\derivative{\bar{t}}{z}+\frac{1}{c}),$ which leads to
$\partialderivative{z}=\partialderivative{t}\partialderivative{t}{z}=\frac{1}{c}\partialderivative{t}$.
Taking into account that $k_{q}=\omega_{q}/c$, and $c\omega_{q}\hbar=E_{q}$,
where $q=p,s,pr,as$, the Eq. (96) becomes

$\frac{1}{c}\partialderivative{E_{q}}{t}=-N_{s}\mu_{0}\mu_{ij}\frac{E_{q}(t)}{\hbar}\Im{\rho_{ij}}$
(97)

We find the density matrix elements $\rho_{ij}$ from the Liouville von Neumann
equation $i\hbar\dot{\rho}=[H,\rho]$ and using the above Hamiltonian in
Eq.(63). We start by opening the commutator and applying the substitutions

$\displaystyle\begin{aligned}
\rho_{12}&=\tilde{\rho}_{12}e^{i(\alpha_{p}-\alpha_{s})t^{2}/2}\\\
\rho_{13}&=\tilde{\rho}_{13}e^{i(\Delta_{s}t+\alpha_{p}t^{2}/2)}\\\
\rho_{14}&=\tilde{\rho}_{14}e^{i\Delta_{as}t}\\\
\rho_{23}&=\tilde{\rho}_{23}e^{i(\Delta_{s}t+\alpha_{s}t^{2}/2)}\\\
\rho_{24}&=\tilde{\rho}_{24}e^{i(\Delta_{as}t+\alpha_{pr}t^{2}/2)}\\\
\rho_{34}&=\tilde{\rho}_{34}e^{i(\Delta_{as}-\Delta_{s})t-i\alpha_{p}t^{2}/2}\end{aligned}$

Next, we apply the rotating wave approximation and use the control condition
on the chirp parameters $\alpha_{s}-\alpha_{p}=\alpha_{pr}$, and arrive at

$\displaystyle\dot{\rho}_{11}$
$\displaystyle=-\tfrac{i}{2}\Omega_{p0}(t)\tilde{\rho}_{31}+\tfrac{i}{2}\Omega^{*}_{p0}(t)\tilde{\rho}_{13}-\tfrac{i}{2}\Omega_{as0}(t)\tilde{\rho}_{41}+\tfrac{i}{2}\Omega^{*}_{as0}(t)\tilde{\rho}_{14}$
$\displaystyle\dot{\rho}_{22}$
$\displaystyle=-\tfrac{i}{2}\Omega_{s0}(t)\tilde{\rho}_{32}+\tfrac{i}{2}\Omega^{*}_{s0}(t)\tilde{\rho}_{23}-\tfrac{i}{2}\Omega_{pr0}(t)\tilde{\rho}_{42}+\tfrac{i}{2}\Omega^{*}_{pr0}(t)\tilde{\rho}_{24}$
$\displaystyle\dot{\rho}_{33}$
$\displaystyle=\tfrac{i}{2}\Omega_{p0}(t)\tilde{\rho}_{31}-\tfrac{i}{2}\Omega^{*}_{p0}(t)\tilde{\rho}_{13}+\tfrac{i}{2}\Omega_{s0}(t)\tilde{\rho}_{32}-\tfrac{i}{2}\Omega^{*}_{s0}(t)\tilde{\rho}_{23}$
$\displaystyle\dot{\rho}_{44}$
$\displaystyle=\tfrac{i}{2}\Omega_{as0}(t)\tilde{\rho}_{41}-\tfrac{i}{2}\Omega^{*}_{as0}(t)\tilde{\rho}_{14}+\tfrac{i}{2}\Omega_{pr0}(t)\tilde{\rho}_{42}-\tfrac{i}{2}\Omega^{*}_{pr0}(t)\tilde{\rho}_{24}$
$\displaystyle\dot{\tilde{\rho}}_{12}$
$\displaystyle=i\alpha_{pr}t\tilde{\rho}_{12}-\tfrac{i}{2}\Omega_{p0}(t)\tilde{\rho}_{32}-\tfrac{i}{2}\Omega_{as0}(t)\tilde{\rho}_{42}+\tfrac{i}{2}\Omega^{*}_{s0}(t)\tilde{\rho}_{13}+\tfrac{i}{2}\Omega^{*}_{pr0}(t)\tilde{\rho}_{14}$
$\displaystyle\dot{\tilde{\rho}}_{13}$
$\displaystyle=-i(\Delta_{s}+\alpha_{p}t)\tilde{\rho}_{13}-\tfrac{i}{2}\Omega_{p0}(t)(\rho_{33}-\rho_{11})-\tfrac{i}{2}\Omega_{as0}(t)\tilde{\rho}_{43}+\tfrac{i}{2}\Omega_{s0}(t)\tilde{\rho}_{12}$
$\displaystyle\dot{\tilde{\rho}}_{14}$
$\displaystyle=-i\Delta_{as}\tilde{\rho}_{14}-\tfrac{i}{2}\Omega_{p0}(t)\tilde{\rho}_{34}-\tfrac{i}{2}\Omega_{as0}(t)(\rho_{44}-\rho_{11})+\tfrac{i}{2}\Omega_{pr0}(t)\tilde{\rho}_{12}$
$\displaystyle\dot{\tilde{\rho}}_{23}$
$\displaystyle=-i(\Delta_{s}+\alpha_{s}t)\tilde{\rho}_{23}-\tfrac{i}{2}\Omega_{s0}(t)(\rho_{33}-\rho_{22})-\tfrac{i}{2}\Omega_{pr0}(t)\tilde{\rho}_{43}+\tfrac{i}{2}\Omega_{p0}(t)\tilde{\rho}_{21}$
$\displaystyle\dot{\tilde{\rho}}_{24}$
$\displaystyle=-i(\Delta_{as}+\alpha_{pr}t)\tilde{\rho}_{24}-\tfrac{i}{2}\Omega_{pr0}(t)(\rho_{44}-\rho_{22})-\tfrac{i}{2}\Omega_{s0}(t)\tilde{\rho}_{34}+\tfrac{i}{2}\Omega_{as0}(t)\tilde{\rho}_{21}$
$\displaystyle\dot{\tilde{\rho}}_{34}$
$\displaystyle=i(\Delta_{s}-\Delta_{as}+\alpha_{p}t)\tilde{\rho}_{34}-\tfrac{i}{2}\Omega^{*}_{p0}(t)\tilde{\rho}_{14}-\tfrac{i}{2}\Omega^{*}_{s0}(t)\tilde{\rho}_{24}+\tfrac{i}{2}\Omega_{as0}(t)/2\tilde{\rho}_{31}$
$\displaystyle+\tfrac{i}{2}\Omega_{pr0}(t)\tilde{\rho}_{32}.$

After performing adiabatic elimination of the excited states assuming that
$\dot{\rho}_{13},\dot{\rho}_{14},\dot{\rho}_{23},\dot{\rho}_{24},\dot{\rho}_{34}\approx
0,\;\rho_{34}\approx 0,\;\rho_{33},\rho_{44}\ll\rho_{11},\rho_{22}$ and
$\dot{\rho}_{33},\dot{\rho}_{44}\approx 0$, the density matrix elements
$\rho_{13},\rho_{23},\rho_{14},\rho_{24}$ read in terms of
$\rho_{11},\rho_{22}$ and $\rho_{12}$ as follows

$\displaystyle\rho_{13}$
$\displaystyle=\dfrac{1}{2(\Delta_{s}+\alpha_{p}t)}\Omega_{p0}(t)\rho_{11}+\dfrac{1}{2(\Delta_{s}+\alpha_{p}t)}\Omega_{s0}(t)\rho_{12}$
(98) $\displaystyle\rho_{23}$
$\displaystyle=\dfrac{1}{2(\Delta_{s}+\alpha_{s}t)}\Omega_{s0}(t)\rho_{22}+\dfrac{1}{2(\Delta_{s}+\alpha_{s}t)}\Omega_{p0}(t)\rho_{21}$
$\displaystyle\rho_{14}$
$\displaystyle=\dfrac{1}{2\Delta_{as}}\Omega_{as0}(t)\rho_{11}+\dfrac{1}{2\Delta_{as}}\Omega_{pr0}(t)\rho_{12}$
$\displaystyle\rho_{24}$
$\displaystyle=\dfrac{1}{2(\Delta_{as}+\alpha_{pr}t)}\Omega_{pr0}(t)\rho_{22}+\dfrac{1}{2(\Delta_{as}+\alpha_{pr}t)}\Omega_{as0}(t)\rho_{21}$

Substituting Eq.(98) in Eq.(97) and rewriting the equations in terms of Rabi
frequencies provide the following Maxwell’s equations:

$\displaystyle\partialderivative{\Omega_{p0}}{t}$
$\displaystyle=c\partialderivative{\Omega_{p0}}{z}=-\dfrac{\eta}{2(\Delta_{s}+\alpha_{p}t)}\kappa_{13}\omega_{p}\Omega_{s0}(t)\imaginary[\rho_{12}]$
(99) $\displaystyle\partialderivative{\Omega_{s0}}{t}$
$\displaystyle=c\partialderivative{\Omega_{s0}}{z}=\dfrac{\eta}{2(\Delta_{s}+\alpha_{s}t)}\kappa_{23}\omega_{s}\Omega_{p0}(t)\imaginary[\rho_{12}]$
$\displaystyle\partialderivative{\Omega_{pr0}}{t}$
$\displaystyle=c\partialderivative{\Omega_{pr0}}{z}=\dfrac{\eta}{2(\Delta_{as}+\alpha_{pr}t)}\kappa_{24}\omega_{pr}\Omega_{as0}(t)\imaginary[\rho_{12}]$
$\displaystyle\partialderivative{\Omega_{as0}}{t}$
$\displaystyle=c\partialderivative{\Omega_{as0}}{z}=-\dfrac{\eta}{2(\Delta_{as})}\kappa_{14}\omega_{as}\Omega_{pr0}(t)\imaginary[\rho_{12}].$


# Entropic Bounds as Uncertainty Measure of Unitary Operators

Jesni Shamsul Shaari  Faculty of Science, International Islamic University
Malaysia (IIUM), Jalan Sultan Ahmad Shah, Bandar Indera Mahkota, 25200
Kuantan, Pahang, Malaysia  Rinie N. M. Nasir  Faculty of Science,
International Islamic University Malaysia (IIUM), Jalan Sultan Ahmad Shah,
Bandar Indera Mahkota, 25200 Kuantan, Pahang, Malaysia  Stefano Mancini
School of Science & Technology, University of Camerino, I-62032 Camerino,
Italy  INFN Sezione di Perugia, I-06123 Perugia, Italy

###### Abstract

We reformulate the notion of uncertainty of pairs of unitary operators within
the context of guessing games and derive an entropic uncertainty relation for
a pair of such operators. We show how distinguishable operators are compatible
while maximal incompatibility of unitary operators can be connected to bases
for some subspace of operators which are mutually unbiased.

###### pacs:

03.65.Aa, 03.67.-a

##  I Introduction

The notion of uncertainty, or incompatibility has almost become synonymous
with quantum theory. The most immediate case would be the uncertainty for a
pair of noncommuting observables captured in the famous inequality scho  for
observables $A$ and $B$ given by $\Delta A\Delta B\geq\langle\psi|[AB-
BA]|\psi\rangle|/2$ with the uncertainties in observables $O$ given as
standard deviations, $\Delta
O=\sqrt{|\langle\psi|O^{2}|\psi\rangle|-|\langle\psi|O|\psi\rangle|^{2}}$, for
a quantum state $|\psi\rangle$.

This has to some extent motivated the study of uncertainty of pairs of unitary
operators which obey a specific commutation relation maasar . A generalisation
to include arbitrary pairs of unitary operators with tighter bounds was later
reported bagchi . In these works, the uncertainty of an operator, $U$, was
defined in an analogous way to the uncertainties of observables, in terms of
variances, $\Delta U=\sqrt{1-|\langle\psi|U|\psi\rangle|^{2}}$. The
uncertainty defined this way can be more precisely understood as the
uncertainty in state $|\psi\rangle$ associated to $U$ in the context of the
overlap between the state $|\psi\rangle$ and the evolved state
$U|\psi\rangle$. Extending this to a pair of operators, $U,V$, an uncertainty
relation would supposedly reflect the distinguishability of two distinct
unitary state evolutions bagchi .

However, notwithstanding the motivations mentioned in the Refs. maasar ;
bagchi  of defining $\Delta U$ as above, the meaning of uncertainty of unitary
operators in reference to such a definition can be somewhat ambiguous. We
shall refer to a simple example for our argument, the Pauli unitary operators,
$\sigma_{x}$ and $\sigma_{z}$ operating on the two dimensional Hilbert space,
$\mathcal{H}_{2}$. The uncertainty relation
$\Delta\sigma_{x}^{2}+\Delta\sigma_{z}^{2}\geq 1$ then was reinterpreted in
maasar  as an uncertainty relation for the Mach-Zender interferometer.
However, in terms of unitary operators, it can be a little unclear what this
greater than zero value would imply as such a pair of unitary operators
actually can result in distinguishable states for a specific input. Namely,
for input $|0\rangle$ (in terms of the computational basis), for which it is
easy to note that $|\langle 0|\sigma_{x}\sigma_{z}|0\rangle|=0$. Thus the
distinguishability of the evolved states is perfect. This is hardly surprising
as the Pauli operators are actually orthogonal and can be distinguished
perfectly.

Therefore, we shall consider in this work a more ‘operational’ notion for the
uncertainty of unitary operators. Intuitively, we would like some definition
such that the notion would reflect the uncertainty one can have when testing
such an operator. Testing itself can be understood as a means to ascertain the
nature of action the operator has on some quantum state. In other words, if
one can test perfectly such an operator, then the uncertainty would be zero.
Extending to the case of testing a pair of operators, we require our
formulation to give zero uncertainty ONLY when the tester has zero uncertainty
with respect to both the operators while perfectly distinguishing the
operators. The framework of testing operators is well established in the
context of Process Positive Operator Value Measurement (PPOVM) ziman  or
quantum tester bisio ; sedlak . This can be understood as analogous to what
POVMs are to quantum states.

As for the notion of uncertainty, we will resort to the most obvious Shannon
entropy, defined as $H(x)=-\sum p(x)\log{p(x)}$ for $x$ being the outcome of
the tester occurring with probability $p(x)$. We note that our work is
essentially motivated by the use of entropic uncertainty relations for
observables of finite dimensional Hilbert spaces D ; maassen ; step ; coles .
The entropic uncertainty relation for such observables was first proposed by D
and later refined by maassen  to eventually replace the older relations based
on standard deviations. It is more often described in terms of guessing games;
i.e. one party, say Alice, provides another, say Bob with some arbitrary state
for which the latter would measure in either one of two measurement basis. Bob
then informs Alice of his choice of measurement and she then guesses the
result of his measurement. Her guesses for both measurements cannot be free of
uncertainty if the measurements are incompatible. Inspired by that, we will
formulate the notion of uncertainty of unitary operators pairs in our work,
within the context of some guessing game. The main results of our work
highlighted as:

  1. 1.

uncertainty of unitary operators simply reflects the distinguishability of the
operators. The incompatibility of observables can be connected to the
distinguishability of unitary operators as follows: if unitary operators $U$
and $V$ are (not perfectly) distinguishable, then the measurement operators
defined by $\\{U^{\dagger}|\chi_{i}\rangle\langle\chi_{i}|U\\}$ and
$\\{V^{\dagger}|\chi_{i}\rangle\langle\chi_{i}|V\\}$ are incompatible for any
projective measurement, $\\{|\chi_{i}\rangle\langle\chi_{i}|\\}$. Unless
stated otherwise, we assume throughout the text, projective measurements such
that $\langle\chi_{i}|\chi_{j}\rangle=\delta_{ij}$ and
$\\{|\chi_{i}\rangle\\}$ is a complete basis for the $d$ dimensional Hilbert
space.

  2. 2.

if the uncertainty saturates the maximal entropic bound (i.e. $\log{d}$) for
some pair $U$ and $V$, then the measurement operators defined by
$\\{U^{\dagger}|\chi_{i}\rangle\langle\chi_{i}|U\\}$ and
$\\{V^{\dagger}|\chi_{i}\rangle\langle\chi_{i}|V\\}$ are maximally
incompatible. In the special case where each of these operators comes from two
distinct unitary bases and the maximal entropic bound is reached for any pairs
thus formed, then the bases are mutually unbiased scott ; jssarxiv ; arxiv .

We should note the difference (though related) work of Ref.annals  where an
entropic relation was defined for a pair of unitary testers. In the mentioned
reference, an uncertainty relation using Shannon entropy was defined for a
tester pair in an analogous way to that of a pair observables. In simple
terms, it reflected the uncertainty when considering a pair of testers for
some (common) unitary tested. This is markedly different from the current work
which defines uncertainty of results when a pair of unitary operators are
tested by a common tester. As such it complements Ref.annals  and, we shall
see later how some of the resulting propositions relate to those of Ref.annals
.

The outline of the work is as follows. We begin with a brief review of
testers, mutually unbiased bases (MUB) and mutually unbiased unitary bases
(MUUB). We then commit to a thorough study of uncertainty of pairs of
operators using testers with a (pure) $d$-dimensional input state and
projective measurements. This would include propositions related to the
distinguishability of the operators and another which connects maximal
incompatibility to MUUB. We further present a short section on using testers
with maximally entangled input states as well as those using POVMs for
measurement purposes before concluding the work.

##  II MUBs, MUUBs and Testers

Given the measurement bases
$\mathbb{V}=\\{|\mathbb{V}_{i}\rangle\langle\mathbb{V}_{i}|\\}$ and
$\mathbb{W}=\\{|\mathbb{W}_{i}\rangle\langle\mathbb{W}_{i}|\\}$, with
$\\{|\mathbb{V}_{i}\rangle\\}$ and $\\{|\mathbb{W}_{j}\rangle\\}$ being
distinct bases for $\mathcal{H}_{d}$, the notion of incompatibility of
measurements is captured through the well known uncertainty relation maassen ,

$\displaystyle
H(\mathbb{V})+H(\mathbb{W})\geq-\log{\max_{i,j}|\langle\mathbb{V}_{i}|\mathbb{W}_{j}\rangle}|^{2}$
(1)

with $H(\cdot)$ the Shannon entropy of the outcomes. The bound is saturated if
and only if $\\{|\mathbb{V}_{i}\rangle\\}$ and $\\{|\mathbb{W}_{j}\rangle\\}$
are mutually unbiased to one another step ; coles ; i.e,

$\displaystyle|\langle\mathbb{W}_{i}|\mathbb{V}_{j}\rangle|=1/\sqrt{d},~{}\forall
i,j,=0,\ldots,d-1.$ (2)

These bases, referred to as mutually unbiased bases (MUB), are described by
having equiprobable transition between states from one basis to the other durt
.

An analogous structure to MUB for the space, $\mathcal{L}(\mathcal{H}_{d})$,
i.e. the space of linear operators on $\mathcal{H}_{d}$ was studied in
ref.scott ; jssarxiv ; arxiv  as mutually unbiased unitary basis (MUUB). The
definition carries similarly, the notion of inner product (in this case, the
Hilbert-Schmidt inner-product); two distinct orthogonal basis,
$\\{P_{0},...,P_{D-1}\\}$ and $\\{Q_{0},...,Q_{D-1}\\}$ comprising of unitary
transformations for some $D$-dimensional subspace of the space
$\mathcal{L}(\mathcal{H}_{d})$ are sets of MUUB if

$\displaystyle\left|\text{Tr}(P_{i}^{\dagger}Q_{j})\right|^{2}=\kappa~{}~{},~{}~{}\forall
i,j,=0,\ldots,D-1$ (3)

for some constant, $\kappa$. This constant takes on value $1$ and $d$ for the
$d^{2}$ and $d$ dimensional subspaces of $\mathcal{L}(\mathcal{H}_{d})$
respectively jssarxiv ; arxiv .

The discrimination of operators can be done using testers. Physically, a
tester consist of some known quantum state which acts as a probe that
undergoes the action of some (unknown) operator and subsequently, measurements
are made to determine how the state has evolved; thus one may infer properties
of the operator. Testers can then be described as a pair,
$T=(\Psi,\\{F_{i}\\})$ where $\Psi$ is the known quantum state and
$\\{F_{i}\\}$ is a POVM for the measurement purpose. In principle, $\Psi$ can
be any quantum state, although it should preferably be pure; thus avoiding
further uncertainties in determining an operator. We note that this is the
same description used in Ref.annals  and is sufficient for our purpose. We
refer the interested reader to Ref.ziman  for a more precise definition in
terms of PPOVM.

##  III Entropic Bounds for Pairs of Unitary Testing

In what follows, we shall make use of testers with a pure $d$-dimensional
input state, $|\psi\rangle\in\mathcal{H}_{d}$ and measurement operators
$\\{|\chi_{i}\rangle\langle\chi_{i}|\\}$ assumed to be orthogonal projectors.
As noted earlier, we formulate the notion of uncertainty of unitary operators
in terms of testers within the framework of a guessing game as follows:

Consider a pair of unitary operators $V,W\in SU(d)$. Let Alice prepare a
tester, $\mathcal{T}=(|\psi\rangle,\\{|\chi_{i}\rangle\langle\chi_{i}|\\})$,
to distinguish between the two operators and submits the tester to Bob. Bob
then would randomly test either $V$ or $W$ and then informs Alice of the
operator he used. Alice then needs to guess Bob’s results.

Assuming the unitary operators were chosen with equal probability, and the
tester used as above, Alice’s average uncertainty of Bob’s result would be
given as

$\displaystyle
H(\mathcal{T}|V)+H(\mathcal{T}|W)=-\sum_{i}|\langle\chi_{i}|V|\psi\rangle|^{2}\log{|\langle\chi_{i}|V|\psi\rangle|^{2}}$
(4)
$\displaystyle-\sum_{i}|\langle\chi_{i}|W|\psi\rangle|^{2}\log{|\langle\chi_{i}|W|\psi\rangle|^{2}}$

The right hand side of the above equation can be understood equivalently as
the average uncertainty of making measurements using the measurement operators
$\mathcal{V}=\\{V^{\dagger}|\chi_{i}\rangle\langle\chi_{i}|V\\}$ and
$\mathcal{W}=\\{W^{\dagger}|\chi_{i}\rangle\langle\chi_{i}|W\\}$ given the
state $|\psi\rangle$. Hence

$\displaystyle
H(\mathcal{T}|V)+H(\mathcal{T}|W)=H(\mathcal{V}||\psi\rangle)+H(\mathcal{W}||\psi\rangle)$
(5)

and we can immediately apply the well-known entropic bound for measurement
operators (independent of $|\psi\rangle$)and thus we write an inequality for
eq.(5) as

$\displaystyle
H(\\{|\chi_{i}\rangle\langle\chi_{i}|\\}|V)+H(\\{|\chi_{i}\rangle\langle\chi_{i}|\\}|W)\geq-\log{\max_{i,j}{|\langle\chi_{i}|WV^{\dagger}|\chi_{j}\rangle|^{2}}}.$
(6)

We see that this relation is dependent on the measurement operator of the
tester (for any possible input of the tester, thus we write
$\\{|\chi_{i}\rangle\langle\chi_{i}|\\}$ for notation rather than
$\mathcal{T}$). This is as it should be, i.e. it reflects the physical fact
that any uncertainty is borne out of the measurements and not due to the
deterministic nature of unitary evolutions. Hence an uncertainty relation in
the form of entropic inequality makes sense only when referred to the tester
involved.

The entropic relation reflect Alice’s uncertainty in guessing Bob’s
measurement results. Incompatibility of unitary operators is thus interpreted
as Alice’s inability to prepare a tester to distinguish between the operators
which would give zero uncertainty for both $V$ and $W$. Notwithstanding this,
the relation has another clear physical meaning. It is in fact, the
uncertainty relation for a common measurement operator type used by the two
parties independently, whose reference frames are related by a certain unitary
relation. Let Alice and Bob independently measure the observable,
$\\{|\chi_{i}\rangle\langle\chi_{i}|\\}$, in their respective locality for a
given state sent by Alice to Bob (or identical states by a third party to
both). If their reference frames are rotated to one another, such that any
state prepared by Alice is viewed by Bob as rotated by the operator
$WV^{\dagger}$, then the entropic uncertainty relation describes the
uncertainty of their measurement outcomes.

For a given measurement operator, $\\{|\chi_{i}\rangle\langle\chi_{i}|\\}$,
one can obviously derive an infinite number of testers by having different
inputs. However, only a specific subset of those, if exists, may achieve the
minimal entropic value,
$-\log{\max_{i,j}{|\langle\chi_{i}|WV^{\dagger}|\chi_{j}\rangle|^{2}}}$. We
refer to such testers as saturating testers.

Note that for any unitary operator pair $V$ and $W$, one can always have zero
uncertainty, i.e. when one considers a tester such that the measurement
$\\{|\chi_{i}\rangle\langle\chi_{i}|\\}$ projects onto the eigenvectors of
$WV^{\dagger}$ for some specific input. This is quite obvious and we refer to
testers with such measurement operators as trivial testers for the unitary
operators pair $V$ and $W$. These trivial testers unfortunately have no real
operational use. It tells us nothing about the difference between $V$ and $W$,
beyond being a passive operation for some input of the tester. In what
follows, we shall exclude trivial testers from our considerations.

###  III.1 The Minimal Bound; Perfect Discrimination

######  Proposition 1.

Consider a pair of unitary operators, $V$ and $W$. Excluding trivial testers,
the uncertainty relation gives the trivial (zero) bound, i.e.

$\displaystyle
H(\\{|\chi_{i}\rangle\langle\chi_{i}|\\}|V)+H(\\{|\chi_{i}\rangle\langle\chi_{i}|\\}|W))=0$
(7)

for some $\\{|\chi_{i}\rangle\langle\chi_{i}|\\}$ if and only if they are
perfectly distinguishable

###### Proof.

A pair of unitary $V$ and $W$ being perfectly distinguishable implies the
overlap between $V|\chi_{j}\rangle$ and $W|\chi_{j}\rangle$ for some
$W|\chi_{j}\rangle$ is given by
$|\langle\chi_{j}|WV^{\dagger}|\chi_{j}\rangle|=0$. Hence, there exists a
$\\{|\chi_{i}\rangle\langle\chi_{i}|\\}$, such that
$|\langle\chi_{j}|WV^{\dagger}|\chi_{i}\rangle|=1$. This results in
$H(\\{|\chi_{i}\rangle\langle\chi_{i}|\\}|V)+H(\\{|\chi_{i}\rangle\langle\chi_{i}|\\}|W)=0$.

Proving the other way, it is obvious that a pair of unitary operators, $V$ and
$W$, which gives the minimal (zero) bound implies
$H(\\{|\chi_{i}\rangle\langle\chi_{i}|\\}|V)=H(\\{|\chi_{i}\rangle\langle\chi_{i}|\\}|W)=0$
and $|\langle\chi_{i}|WV^{\dagger}|\chi_{j}\rangle|=1$ for some $i\neq j$
(having excluded the trivial tester), thus
$|\langle\chi_{i}|V^{\dagger}W|\chi_{i}\rangle|=0$ which implies $V$ and $W$
can be perfectly distinguished.

∎

This of course means that any two unitary operators, $V$ and $W$ such that
$\operatorname{Tr}{(V^{\dagger}W)}=0$; i.e orthogonal, and thus
distinguishable would saturate the minimal bound for some
$\\{|\chi_{i}\rangle\langle\chi_{i}|\\}$. Such pairs would imply the existence
of some measurement operator, $\\{|\chi_{i}\rangle\langle\chi_{j}|\\}$, such
that $\\{V^{\dagger}|\chi_{i}\rangle\langle\chi_{i}|V\\}$ and
$\\{W^{\dagger}|\chi_{i}\rangle\langle\chi_{i}|W\\}$ are compatible.

As an example, the unitary operators $P,Q\in SU(d)$ (as defined in Ref.maasar
with $j,k=-[d/2],...,[(d-1)/2]$) such that $PQ=QPe^{2i\pi/d}$ given as

$\displaystyle P=\sum_{j}e^{i2\pi j/d}|a_{j}\rangle\langle
a_{j}|~{},~{}Q=\sum_{k}e^{-i2\pi k/d}|b_{k}\rangle\langle b_{k}|$ (8)

with the two orthonormal basis $\\{|a_{j}\rangle\\}$ and $\\{|b_{k}\rangle\\}$
related based by the Discrete Fourier transform,

$\displaystyle|b_{k}\rangle=\sum_{j}e^{i2\pi jk/d}|a_{j}\rangle$ (9)

are orthogonal (it can easily be shown that
$\operatorname{Tr}{(PQ^{\dagger})}=0$), therefore distinguishable and
$H(P)+H(Q)=0$. If we consider a basis of unitary operators (for some space of
operators), the orthogonality of the operators imply that they are all in fact
compatible, in the sense discussed above. This is notwithstanding the fact
that they may be non-commuting. Commutation (or otherwise) of operators has no
bearing on its compatibility within our notion of uncertainty of unitary
operators and is only meaningful in the context of operators as observables.
Owing to the different frameworks subscribed when considering the roles of the
operators, the Pauli operators, say, for $SU(2)$ which are non-commuting are
thus non-compatible in the context of observables though compatible as unitary
operators.

Let us consider how this can be related to the trivial entropic bound of
ref.annals . Let $T_{1}$ and $T_{2}$ be two testers and if they both satisfy
eq.(7), then, rewriting $U_{1}$ for $V$ and $U_{2}$ for $W$ we have,

$\displaystyle H(T_{1}|U_{1})+H(T_{1}|U_{2})+H(T_{2}|U_{1})+H(T_{2}|U_{2})=0$
(10)

which implies

$\displaystyle H(T_{1}|U_{i})+H(T_{2}|U_{i})=0,~{}i=1,2.$ (11)

By further considering the notion of complete set of testers annals  and
assuming the above to also hold for the set of operators
$\\{U_{1},U_{2},...,U_{d}\\}$, the proposition 5 related to trivial bounds of
ref.annals  follows.

###  III.2 The Maximal Bound; MUUBs

Given the measurement of our tester results in any one of $d$ possible
outcomes, the maximal bound, $\log{d}$, can be achieved for a pair of unitary
operators $V$ and $W$ if $WV^{\dagger}|\chi_{j}\rangle$ and $|\chi_{k}\rangle$
are mutually unbiased for all $j,k$.

However, when this holds true for unitary operators pairwise taken from two
distinct bases, it has some interesting implications on the inner-product of
the operators. We summarise it in the following proposition:

######  Proposition 2.

Let $\mathfrak{V}=\\{V_{i}\\}$ and $\mathfrak{W}=\\{W_{j}\\}$ be two distinct
bases for some $d$-dimensional subspace for, $\mathcal{L}(\mathcal{H}_{d})$,
the space of linear operators acting on a $d$-dimensional Hilbert space.
$\mathfrak{V}$ and $\mathfrak{W}$ are mutually unbiased, i.e.
$|\operatorname{Tr}{(V_{n}^{\dagger}W_{m})}|=\sqrt{d}$ if for any $m,n$, the
unitary pair $V_{n}$ and $W_{m}$ saturates the maximal bound for some tester,
$T_{m,n}$.

###### Proof.

Let
$H(\\{|\chi_{i}\rangle\langle\chi_{i}|\\}|V_{n})+H(\\{|\chi_{i}\rangle\langle\chi_{i}|\\}|W_{m}))=-\log{1/d}$,
i.e. $V_{n}$ and $W_{m}$ saturate the maximal bound for any $m,n$ given some
tester, $T_{m,n}$. This condition is met if and only if
$\\{V_{n}^{\dagger}|\chi_{i}\rangle\\}$ and
$\\{W_{m}^{\dagger}|\chi_{i}\rangle\\}$ are mutually unbiased; i.e. $\forall
i,j,|\langle\chi_{i}|W_{m}V_{n}^{\dagger}|\chi_{j}\rangle|=1/\sqrt{d}$. The
trace of the operator $W_{m}V_{n}^{\dagger}$ can be written as

$\displaystyle\operatorname{Tr}{(W_{m}V_{n}^{\dagger})}=\sum_{i}\langle\chi_{i}|W_{m}V_{n}^{\dagger}|\chi_{i}\rangle$
(12)

and

$\displaystyle\sum_{i}|\langle\chi_{i}|W_{m}V_{n}^{\dagger}|\chi_{i}\rangle|\geq|\sum_{i}\langle\chi_{i}|W_{m}V_{n}^{\dagger}|\chi_{i}\rangle|=|\operatorname{Tr}{(W_{m}V_{n}^{\dagger})}|\Rightarrow|\operatorname{Tr}{(W_{m}V_{n}^{\dagger})}|\leq\sqrt{d}.$
(13)

We will now demonstrate how $|\operatorname{Tr}{(W_{m}V_{n}^{\dagger})}|$ is
in fact necessarily equal to $\sqrt{d}$. Let us write
$W_{m}=\sum\alpha_{i}V_{i}$ and thus
$|\operatorname{Tr}{(W_{m}V_{n}^{\dagger})}|^{2}=|\alpha_{n}|^{2}d^{2}$.
However note that

$\displaystyle|\operatorname{Tr}{(W_{m}W_{m}^{\dagger})}|=d(\sum_{i}|\alpha_{i}|^{2})=d\Rightarrow\sum_{i}|\alpha_{i}|^{2}=1.$
(14)

Therefore,

$\displaystyle\sum_{k}|\operatorname{Tr}{(W_{m}V_{k}^{\dagger})}|^{2}=d^{2}\sum|\alpha_{k}|^{2}=d^{2}$
(15)

From (13) earlier, we have
$|\operatorname{Tr}{(W_{m}V_{n}^{\dagger})}|\leq\sqrt{d}$, for which squaring
both sides of the inequality gives
$|\operatorname{Tr}{(W_{m}V_{n}^{\dagger})}|^{2}\leq d$ and
$\sum_{k}|\operatorname{Tr}{(W_{m}V_{k}^{\dagger})}|^{2}\leq d^{2}$. The
requirement of equation (15) implies $\forall
k,|\operatorname{Tr}{(W_{m}V_{k}^{\dagger})}|^{2}=d$ or
$|\operatorname{Tr}{(W_{m}V_{n}^{\dagger})}|=\sqrt{d}$. ∎

The proposition in a nutshell describes how any operator pair formed with
elements each taken from two unitary operator basis, which are maximally
incompatible implies that the two operator bases are in fact mutually
unbiased.

Our notation of tester, $T_{m,n}$, in the above proposition reflects the
possibility of having testers of differing inputs dependent on the different
values of $m,n$, despite the common measurement operators. Let us consider how
we can define the tester. Let
$W_{m}V_{n}^{\dagger}|\chi_{i}\rangle=|\zeta_{i}^{(m,n)}\rangle$ where
$|\langle\chi_{i}|\zeta_{i}^{(m,n)}\rangle|=1/\sqrt{d}$. We further denote
$|\psi_{i}^{(n)}\rangle=V_{n}^{\dagger}|\chi_{i}\rangle$, implying
$V_{n}|\psi_{i}^{(n)}\rangle=|\chi_{i}\rangle$. Hence,
$W_{m}|\psi_{i}^{(n)}\rangle=W_{m}V_{n}^{\dagger}|\chi_{i}\rangle=|\zeta_{i}^{(m,n)}\rangle$.
Thus, if we pick a tester,
$T_{m,n}=(|\psi_{i}^{(n)}\rangle,\\{|\chi_{i}\rangle\langle\chi_{i}|\\})$, the
saturation of the maximal entropic bound,
$H(T_{m,n}|W_{m})+H(T_{m,n}|V_{n})=\log{d}$, holds with the uncertainties in
testing $V_{n}$ and $W_{m}$ being zero and $\log{d}$ respectively.

Using a similar argument to the above, we can obviously consider another
tester, say, $T_{m,n}^{\prime}$, such that, its input is
$|\psi_{i}^{(m)}\rangle=W_{m}^{\dagger}|\chi_{i}\rangle$ and thus the
uncertainty in testing $V_{n}$ would be maximal and $W_{m}$ would be zero
instead. We can therefore write

$\displaystyle
H(T_{m,n}|W_{m})+H(T_{m,n}^{\prime}|W_{m})=H(T_{m,n}|V_{n})+H(T_{m,n}^{\prime}|V_{n})=\log{d}.$
(16)

Considering the case where the above holds for any pair of testers, $T_{m,n}$
and $T_{m,n}^{\prime}$, each coming from two different complete set of
testers, proposition 6 of Ref.annals  follows immediately.

###  III.3 The case for SU(2)

We consider here a special case for operators of SU(2), some specific unitary
operators acting on the 2-dimensional Hilbert space. Its simple structure
would allow us to explicitly see the features of the uncertainty for selected
pairs of operators using testers as described earlier.

We consider two pairs of operators; the identity, $I$ with the Pauli operator,
$\sigma_{y}$, and the identity with $(I-i\sigma_{y})/\sqrt{2}$. The projective
measurement operators in both cases are given by measurement
$\\{|\chi_{i}\rangle\langle\chi_{i}|\\}$ with

$\displaystyle|\chi_{1}\rangle=\cos{\theta}|0\rangle+e^{i\phi}\sin{\theta}|1\rangle~{},~{}|\chi_{2}\rangle=-\sin{\theta}|0\rangle+e^{i\phi}\cos{\theta}|1\rangle$
(17)

with $\theta,\phi\in[0,\pi]$. Now, the overlaps between
$\sigma_{y}|\chi_{i}\rangle$ and $|\chi_{j}\rangle$ for $i,j=1,2$ are given
as,

$\displaystyle|\langle\chi_{i}|I\sigma_{y}|\chi_{j}\rangle|^{2}=\begin{cases}\sin^{2}2\theta\sin^{2}\phi&\text{,
$i=j$}\\\
\cos^{4}\theta+2\cos^{2}\theta\cos^{2}2\phi\sin^{2}\theta+\sin^{4}\theta&\text{,
$i\neq j$}\\\ \end{cases}$ (18)

and that for $I$ and $\Omega=(I-i\sigma_{y})/\sqrt{2}$

$\displaystyle|\langle\chi_{i}|I((I-i\sigma_{y})/\sqrt{2})^{\dagger}|\chi_{j}\rangle|^{2}=\begin{cases}[1+\sin^{2}(2\theta)\sin^{2}(\phi)]/2&\text{,
$i=j$}\\\ [1-\sin^{2}(2\theta)\sin^{2}(\phi)]/2&\text{, $i\neq j$}\\\
\end{cases}$ (19)

We plot below in Fig. 1 (a) and Fig. 2 (a) respectively, the graphs for
$\max_{i,j}{|\langle\chi_{i}|I\sigma_{y}|\chi_{j}\rangle|^{2}}$ and
$\max_{i,j}{|\langle\chi_{i}|(I-i\sigma_{y})^{\dagger}/\sqrt{2}|\chi_{j}\rangle|^{2}}$
as functions of $\theta,\phi$. Additionally, Fig. 1 (b) and Fig. 2 (b)
respectively illustrates $|\langle\chi_{i}|I\sigma_{y}|\chi_{i}\rangle|^{2}$
and
$|\langle\chi_{i}|(I-i\sigma_{y})^{\dagger}/\sqrt{2}|\chi_{i}\rangle|^{2}$,
highlighting the overlap between the states evolved from the different
operators in each pair.

(a)  (b)

Figure 1:  The plots (a) for
$\max_{i,j}{|\langle\chi_{i}|I\sigma_{y}|\chi_{j}\rangle|^{2}}$ and (b) for
$|\langle\chi_{i}|I\sigma_{y}|\chi_{i}\rangle|^{2}$ for any $i=1,2$, are
against the angles $\theta$ and $\phi$.

(a)  (b)

Figure 2: The plots (a) for
$\max_{i,j}{|\langle\chi_{i}|(I-i\sigma_{y})^{\dagger}/\sqrt{2}|\chi_{j}\rangle|^{2}}$
and (b) for
$|\langle\chi_{i}|(I-i\sigma_{y})^{\dagger}/\sqrt{2}|\chi_{i}\rangle|^{2}$ for
any $i=1,2$, are against the angles $\theta$ and $\phi$. In both the plots
here, one can note the minimal value as $0.5$. The graphs are essentially
identical as
$\max_{i,j}{|\langle\chi_{i}|(I-i\sigma_{y})^{\dagger}/\sqrt{2}|\chi_{j}\rangle|^{2}}=|\langle\chi_{i}|(I-i\sigma_{y})^{\dagger}/\sqrt{2}|\chi_{i}\rangle|^{2}$.

The graph of Fig1(a) demonstrates that
$\max_{i,j}{|\langle\chi_{i}|I\sigma_{y}|\chi_{j}\rangle|^{2}}$, achieves the
maximal value 1 (hence zero uncertainty) at the ‘edges’ as well as at the
centres. The edges correspond to projectors that project onto the states
$\cos{\theta}|0\rangle+\sin{\theta}|1\rangle,\forall\theta$, i.e. those lying
on the equator of the Poincare sphere. The edges also include the case for
$\theta=0,\pi$ for any possible value of $\phi$; which are really the states
of the computational basis. In these cases, one can in fact use any of these
states as the input state to form the saturating tester.

We can observe that
$\max_{i,j}{|\langle\chi_{i}|I\sigma_{y}|\chi_{j}\rangle|^{2}}$, decreases in
value (increases in uncertainty) up to 0.5 before taking a turn to increase to
1 at the centres. This increase is contributed by the scenario where
$\max_{i,j}{|\langle\chi_{i}|I\sigma_{y}|\chi_{j}\rangle|^{2}}=|\langle\chi_{i}|I\sigma_{y}|\chi_{i}\rangle|^{2}$
for any $i=1,2$; as depicted in Fig.1 (b), and reflects the distinguishability
of the states $\sigma_{y}|\chi_{i}\rangle$ and $I|\chi_{i}\rangle$. The
maximum value at the centre correspond to the eigenvectors of $I\sigma_{y}$
(trivial tester) and we exclude this from our considerations.

In the context of Bob’s rotated frame with respect to Alice’s, this trivial
bound can be interpreted as Alice measuring in the operators
$\\{|\chi_{i}\rangle\langle\chi_{i}|\\}$ while Bob measures
$\\{\sigma_{y}|\chi_{i}\rangle\langle\chi_{i}|\sigma_{y}\\}$. But as
$|\langle\chi_{j}|\sigma_{y}|\chi_{i}\rangle|=1$ for some $j\neq i$, thus
$|\langle\chi_{i}|\sigma_{y}|\chi_{i}\rangle|=0$ for all other $i$ and given
the fact that we are only dealing with qubits here,
$\\{\sigma_{y}|\chi_{i}\rangle\langle\chi_{i}|\sigma_{y}\\}=\\{|\chi_{i}\rangle\langle\chi_{i}|\\}$.

The graph of Fig. 2(a) on the other hand demonstrates that, for any state,
$|\chi_{i}\rangle$, lying on the equator of the Poincare sphere,
$\max_{i,j}{|\langle\chi_{i}|(I-i\sigma_{y})^{\dagger}/\sqrt{2}|\chi_{j}\rangle|^{2}}=1/2$.
The overlap between $(I-i\sigma_{y})^{\dagger}/\sqrt{2}|\chi_{i}\rangle$ and
$I|\chi_{i}\rangle$ for any $i=1,2$ as shown in Fig. 2(b) has its its minimal
value of $1/2$ for those states and becomes less distinguishable otherwise.
The graphs are essentially identical due to the fact that the contribution for
$\max_{i,j}{|\langle\chi_{i}|(I-i\sigma_{y})^{\dagger}/\sqrt{2}|\chi_{j}\rangle|^{2}}$
comes solely from
$|\langle\chi_{i}|(I-i\sigma_{y})^{\dagger}/\sqrt{2}|\chi_{i}\rangle|^{2}$ for
any $i=1,2$. We also note in this case, a peak at the centre which corresponds
to the zero uncertainty scenario happens for the case of trivial testers. It
is instructive to note that, similar graphs can be drawn for the operator
pairs $\sigma_{y}$ and $(I-i\sigma_{y})/\sqrt{2}$, $\sigma_{y}$ and
$(I+i\sigma_{y})/\sqrt{2}$ as well as $I$ and $(I+i\sigma_{y})/\sqrt{2}$,
demonstrating the connection between the saturation of the maximal bound to
MUUBs. The bases $\\{I,\sigma_{y}\\}$ and
$\\{(I-i\sigma_{y})/\sqrt{2},(I+i\sigma_{y})/\sqrt{2}\\}$ for the subspace of
operators of SU(2) are in fact mutually unbiased.

##  IV Generalisations

An immediate extension of the earlier sections would be the case where the
tester uses a bipartite maximally entangled state (MES),
$|\Psi\rangle\in\mathcal{H}_{d}\otimes\mathcal{H}_{d}$, as input and the
measurement operators, $\\{|\nu_{i}\rangle\langle\nu_{i}|\\}$, would project
onto some MES as well. This is to avoid any contribution of uncertainty due to
measurements that project an unitarily evolved maximally entangled input onto
non maximally entangled states111a measurement which projects a MES onto
separable states would result in maximal uncertainty independently of the
unitary operators involved. This does not modify the preceeding results much
beyond the following:

  1. 1.

Trivial testers do not exist. This is due to the fact that the operator
$WV^{\dagger}\otimes I_{d}$ for any $W,V\in SU(d)$ is an eigenoperator for a
MES only if $WV^{\dagger}$ is some multiple of $I_{d}$ and only thus can
$|\langle\nu_{i}|WV^{\dagger}\otimes I_{d}|\nu_{i}\rangle|$ be equal to unity.

  2. 2.

Proposition 2 remains except that $\mathfrak{V}=\\{V_{i}\\}$ and
$\mathfrak{W}=\\{W_{j}\\}$ are now two distinct bases for the
$d^{2}$-dimensional space of operators and
$|\operatorname{Tr}{(W_{m}V_{n}^{\dagger})}|=1$, for any
$W_{m}\in\mathfrak{W},V_{n}\in\mathfrak{V}$ inline with the MUUB definition
for the space.

As these are rather straightforward, the explicit demonstration is referred to
only in the Appendix section for the interested reader.

Another immediate direction of generalisation would be the straightforward
case for using testers with POVMs for measurement purposes. Similar to the
earlier section, we will see how it reduces to uncertainty relations for
measurements by two parties with rotated POVMs. Let $\rho$ be the input and
$\\{M_{k}\\}$ as the POVM for tester $T$. The average uncertainty now gives,

$\displaystyle H(V|T)+H(W|T)=-\sum_{k}\text{Tr}[M_{k}V^{\dagger}\rho
V]\log\text{Tr}[M_{k}V^{\dagger}\rho V]$
$\displaystyle-\sum_{k}\text{Tr}[M_{k}W^{\dagger}\rho
W]\log\text{Tr}[M_{k}W^{\dagger}\rho W]$
$\displaystyle=-\sum_{k}\text{Tr}[VM_{k}V^{\dagger}\rho]\log\text{Tr}[VM_{k}V^{\dagger}\rho]$
$\displaystyle-\sum_{k}\text{Tr}[WM_{k}W^{\dagger}\rho]\log\text{Tr}[WM_{k}W^{\dagger}\rho]$
(20)

Hence, for any $\rho$, this becomes the entropic bound for POVMs sankhya ,
$\mathcal{M}_{V}=\\{M_{i}^{(V)}~{}|~{}M_{i}^{(V)}=VM_{i}V^{\dagger}\\}$ and
$\mathcal{M}_{W}=\\{M_{j}^{(W)}~{}|~{}M_{j}^{(W)}=WM_{j}W^{\dagger}\\}$,

$\displaystyle
H(\mathcal{M}_{V})+H(\mathcal{M}_{W})\geq-2\log{\max_{i,j}\|\sqrt{M_{i}^{(V)}}\sqrt{M_{j}^{(W)}}}\|.$
(21)

where $\|\cdot\|$ is the operator norm.

##  V Conclusion

The notion of uncertainty pairs of unitary operators in terms of variances
derived quite straightforwardly from those related to operator observables is
certainly not without its charm. Aiming to sideline the possible ambiguity in
such a formulation, especially when dealing with pairs that deterministically
evolved states which are completely distinguishable, we have revised such
notion of uncertainty to one based on entropy using testers. Formulated within
the framework of a guessing game, the uncertainty in testing a unitary
operator pair, $V$ and $W$, reflects the inability of one party to prepare a
tester from which conclusive statements an be made for both the operators.

We immediately see how operators which are completely distinguishable; i.e.
orthogonal are in fact compatible in the sense that it saturates the minimal
entropic bound. It essentially implies the existence of some measurement
operator for which the evolution of its eigenbasis under two compatible
unitary operators are in fact perfectly distinguishable. Hence, any two
operators from any common basis for some subspace of the space of operators
are compatible. This also means that the (generalised) Pauli operators are
actually compatible irrespective of the matter of commutation of operators. It
is thus instructive to determine what a pair of unitary operator represents;
i.e. deterministic evolution or as observables of a quantum system in order to
discuss notions of incompatibility.

On the other extreme, we observe that operators derived from distinct bases
for some subspace of the space of operators saturating the maximal bound are
in fact mutually unbiased to one another. This is true for both the $d$ as
well as the $d^{2}$ dimensional subspaces $\mathcal{L}(\mathcal{H}_{d})$.

The framework we have described in terms of the uncertainty relation with
testers provides a more obvious and practical meaning to the idea of
‘compatibility’ of the operators. It rightfully delegates the focus of
uncertainty to measurement in the testers themselves and not the unitary
operation which are deterministic. Its physical implication can also be
captured in terms of independent measurements made by the parties Alice and
Bob for which their measurement operators are connected by some unitary
relation. In a nutshell, incompatibility of unitary operators should be viewed
in terms of its action on measurement operators, and any uncertainties hence,
are derived from subsequent measurements.

It’s worth noting the relation the current work has to Ref.annals , where the
latter considers the entropic uncertainty relation involved when a unitary is
tested by one of two choices of testers. Our propositions in the current work,
as we have demonstrated, can actually be understood as primitives for some of
the results of Ref.annals .

##  VI Acknowledgement

J. S. S. and R. M. N would like to acknowledge financial support under the
project FRGS19-030-0639 from the Ministry of Higher Education’s Fundamental
Research Grant Scheme and the University’s Research Management Centre (RMC)
for their support and facilities provided. J. S. S would also like to extend a
very special thanks to Su’aidah (RMC, Kuantan) for her kind assistance and
support. S. M. acknowledges the financial support of the Future and Emerging
Technologies (FET) programme, within the Horizon-2020 Programme of the
European Commission, under the FET-Open grant agreement QUARTET, number
862644.

##  VII Appendix

######  Proposition 3.

Consider $\mathfrak{S}=\\{S_{i}\\}$ and $\mathfrak{R}=\\{R_{j}\\}$ be two
distinct bases for the $d^{2}$-dimensional space of operators acting on the
$d$-dimensional Hilbert space. A pair of unitary operators, $S_{p}$ and
$R_{q}$ saturating the maximal bound for any $p,q$, implies that
$\mathfrak{S}$ and $\mathfrak{R}$ are mutually unbiased, i.e.
$|\text{Tr}(S_{p}R_{q}^{\dagger})|=1$.

###### Proof.

Let $H(S_{p})+H(R_{q})=-\log{1/d^{2}}$, i.e. $S_{p}$ and $R_{q}$ saturates the
maximal bound for any $p,q$. saturating the maximal bound. This condition is
met if and only if $\\{R_{q}^{\dagger}\otimes\mathcal{I}|\nu_{i}\rangle\\}$
and $\\{S_{p}\otimes\mathcal{I}|\nu_{j}\rangle\\}$ are mutually unbiased; i.e.
$\forall
i,j,|\langle\nu_{i}|S_{p}R_{q}^{\dagger}\otimes\mathcal{I}|\nu_{j}\rangle|=1/d$.
Note that,

$\displaystyle\operatorname{Tr}{(S_{p}R_{q}^{\dagger}\otimes\mathcal{I})}$
$\displaystyle=\operatorname{Tr}{(S_{p}R_{q}^{\dagger})}\operatorname{Tr}{(\mathcal{I})}$
$\displaystyle=d\operatorname{Tr}{(S_{p}R_{q}^{\dagger})}$ (22)

Now, consider that the trace of the operator
$S_{p}R_{q}^{\dagger}\otimes\mathcal{I}$ can be written as

$\displaystyle\sum_{i}\langle\nu_{i}|S_{p}R_{q}^{\dagger}\otimes\mathcal{I}|\nu_{i}\rangle=d\operatorname{Tr}{(S_{p}R_{q}^{\dagger})}$
(23)

and

$\displaystyle\sum_{i}|\langle\nu_{i}|S_{p}R_{q}^{\dagger}\otimes\mathcal{I}|\nu_{i}\rangle|\geq|\sum_{i}\langle\nu_{i}|S_{p}R_{q}^{\dagger}\otimes\mathcal{I}|\nu_{i}\rangle|=|\operatorname{Tr}{(S_{p}R_{q}^{\dagger}\otimes\mathcal{I}})|=d|\operatorname{Tr}{(S_{p}R_{q}^{\dagger})}|\Rightarrow|\operatorname{Tr}{(S_{p}R_{q}^{\dagger})}|\leq
1.$ (24)

We will now demonstrate how $|\operatorname{Tr}{(S_{p}R_{q}^{\dagger})}|$ is
in fact necessarily equal to $1$.

Let us write $S_{p}=\sum\alpha_{i}R_{i}$ and thus
$|\operatorname{Tr}{(S_{p}R_{q}^{\dagger})}|^{2}=d^{2}|\alpha_{q}|^{2}$. Note
that

$\displaystyle|\operatorname{Tr}{(S_{p}S_{p}^{\dagger})}|=d(\sum_{i}|\alpha_{i}|^{2})=d\Rightarrow\sum_{i}|\alpha_{i}|^{2}=1.$
(25)

Therefore,

$\displaystyle\sum_{k}|\operatorname{Tr}{(S_{p}R_{k}^{\dagger})}|^{2}=d^{2}\sum|\alpha_{k}|^{2}=d^{2}$
(26)

From (24) earlier, we have $|\operatorname{Tr}{(S_{p}R_{q}^{\dagger})}|\leq
1$, for which squaring both sides of the inequality gives
$|\operatorname{Tr}{(S_{p}R_{q}^{\dagger})}|^{2}\leq 1$ and
$\sum_{k}|\operatorname{Tr}{(S_{p}R_{k}^{\dagger})}|^{2}\leq d^{2}$. The
requirement of equation (26) implies $\forall
k,|\operatorname{Tr}{(S_{p}R_{k}^{\dagger})}|^{2}=1$. ∎

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