Minimal informationally complete measurements for probability representation of quantum dynamics

We start our consideration by introducing basic definitions and notations. Let $\mathcal{H}$ be a d-dimensional Hilbert space, where d is finite. We introduce $\mathfrak{B}\left(\mathcal{H}\right)$ as an algebra of bounded operators on $\mathcal{H}$. We also introduce the space of trace class operators $\mathfrak{T}\left(\mathcal{H}\right)$ and space of n × k-matrices over the field $\mathbb{F}$, which we refer to as ${\mathrm{M}\mathrm{a}\mathrm{t}}_{n{\times}k}\left(\mathbb{F}\right)$. Standardly, we use density operators $\rho \in \mathfrak{T}\left(\mathcal{H}\right)$ for the description of quantum states, where ρ ⩾ 0 and $\enspace \mathrm{T}\mathrm{r}\left(\rho \right)=1$. The convex space of quantum states is denoted as $\mathfrak{S}\left(\mathcal{H}\right)$. Extreme points of this space are called pure states, and we denote them as ${\mathfrak{S}}_{\text{pure}}\left(\mathcal{H}\right)$. An operator $B\in \mathfrak{B}\left(\mathcal{H}\right)$ is called an effect, if 0 ⩽ B ⩽ I_d, where we use I_n to denote n-dimensional identity operator. The space of all effects is denoted by $\mathfrak{E}\left(\mathcal{H}\right)$. The channel ${\Phi}:\mathfrak{S}\left({\mathcal{H}}^{\text{in}}\right)\to \mathfrak{S}\left({\mathcal{H}}^{\text{out}}\right)$ is a trace-preserving, completely-positive (CPTP) linear map between states on Hilbert spaces ${\mathcal{H}}^{\text{in}}$ and ${\mathcal{H}}^{\text{out}}$.

A set of effects E = {E₁, ..., E_m} with E_k ⩾ 0 and that satisfies the condition ∑_kE_k = I_d, is known as POVM (positive operator-valued measure). The Born rule implies that a given state ρ defines a probability distribution which we treat as a column-vector

$$\begin{equation}p={\left[\begin{matrix}\hfill {p}_{1}\dots {p}_{m}\hfill \end{matrix}\right]}^{\top },\quad {p}_{k}=\enspace \mathrm{T}\mathrm{r}\left(\rho {E}_{k}\right),\end{equation} \tag{ 1 }$$

where ⊤ the denotes a standard transposition. The POVM E is the MIC-POVM if it forms the basis of $\mathfrak{B}\left(\mathcal{H}\right)$. In this case, E contains d² elements, and every state ρ is fully described by the corresponding probability vector p. We denote a set of possible probability vectors with fixed MIC-POVM E and varying ρ as $\mathfrak{P}\left(E\right)$. We note that SIC-POVMs are a particular class of MIC-POVMs.

The elements of SIC-POVM E^sym have the following form:

$$\begin{equation}{E}_{k}^{\text{sym}}=\frac{1}{d}\vert {\psi }_{k}^{\text{sym}}\rangle \langle {\psi }_{k}^{\text{sym}}\vert ,\quad k=1,\dots ,{d}^{2}.\end{equation} \tag{ 2 }$$

Here vectors $\vert {\psi }_{k}^{\text{sym}}\rangle$ satisfy the following condition:

$$\begin{equation}\vert \langle {\psi }_{k}^{\text{sym}}\vert {\psi }_{l}^{\text{sym}}\rangle {\vert }^{2}=\frac{d{\delta }_{kl}+1}{d+1},\end{equation} \tag{ 3 }$$

and δ_kl is the Kronecker symbol.

As it is noted above, MIC-POVM and SIC-POVM measurements give probability distributions that fully describe quantum states. Therefore, in order to describe the dynamics of quantum systems, one has to find corresponding maps acting on these probability distributions. We note that stochastic maps are not sufficient for the description of quantum dynamics. As it is shown in reference [36], corresponding maps for a description of dynamics of quantum states, which are presented by a probability distribution, are quasistochastic or pseudostochastic. In line with references [35, 43, 44] in our work we prefer to use a term pseudostochastic to emphasize that we deal with classical probability distributions rather than quasi-probabilities.

We remind that a stochastic matrix $M\in {\mathrm{M}\mathrm{a}\mathrm{t}}_{n{\times}k}\left(\mathbb{R}\right)$ is a matrix, for which ${M}_{ij}{\geqslant}0,{\sum }_{i=1}^{n}{M}_{ij}=1$ for every j = 1, ..., k. It is called bistochastic, if n = k and also ∑_jM_ij = 1 for every i = 1, ..., n. Under bistochastic map, a fully chaotic state $\left[\begin{matrix}{c}\hfill 1/n\dots 1/n\hfill \end{matrix}\right]$ remains the same. For the description of quantum dynamics it is necessary to introduce pseudostochastic maps which can be presented as a matrix $M\in {\mathrm{M}\mathrm{a}\mathrm{t}}_{n{\times}k}\left(\mathbb{R}\right)$ with ${\sum }_{i=1}^{n}{M}_{ij}=1$ but without the restriction on the positivity of matrix elements. The square n × n matrix is pseudobistochastic if for any j ∈ {1, ..., n} one has ${\sum }_{j=1}^{n}{M}_{ij}=1$ as well (again, some elements of M_ij may be negative).

We consider a MIC-POVM $E={\left\{{E}_{k}\right\}}_{k=1}^{{d}^{2}}$ in the d-dimensional Hilbert space $\mathcal{H}$. There is a canonical duality between spaces $\mathfrak{B}\left(\mathcal{H}\right)$ and $\mathfrak{T}\left(\mathcal{H}\right)$, which is given by the bilinear form $\left(B,\rho \right){\mapsto}\enspace \mathrm{T}\mathrm{r}\left(\rho B\right)$ for $\rho \in \mathfrak{T}\left(\mathcal{H}\right)$ and $B\in \mathfrak{B}\left(\mathcal{H}\right)$. It means that any linear functional on $\mathfrak{B}\left(\mathcal{H}\right)$ can be represented as $\enspace \mathrm{T}\mathrm{r}\left(\rho \enspace \cdot \right)$, and any functional on $\mathfrak{T}\left(\mathcal{H}\right)$ can be represented as $\enspace \mathrm{T}\mathrm{r}\left(\cdot \enspace B\right)$.

Since MIC-POVM E forms a linear basis in $\mathfrak{B}\left(\mathcal{H}\right)$ one can construct a basis $e={\left\{{e}_{i}\right\}}_{i=1}^{{d}^{2}}$ in $\mathfrak{T}\left(\mathcal{H}\right)$, such that $\enspace \mathrm{T}\mathrm{r}\left({E}_{l}{e}_{k}\right)={\delta }_{l,k}$. This basis is usually referred to as a dual basis to E. Explicitly, the elements of this basis are as follows:

$$\begin{equation}{e}_{l}=\sum _{k=1}^{{d}^{2}}{\left({\mathbf{T}}^{-1}\right)}_{lk}{E}_{k},\quad {\mathbf{T}}_{nm}=\enspace \mathrm{T}\mathrm{r}\left({E}_{n}{E}_{m}\right).\end{equation} \tag{ 4 }$$

Then an arbitrary state $\rho \in \mathfrak{S}\left(\mathcal{H}\right)$ can be represented in the following form:

$$\begin{equation}\rho =\sum _{k=1}^{{d}^{2}}{p}_{k}{e}_{k},\quad {p}_{k}=\mathrm{T}\mathrm{r}\left(\rho {E}_{k}\right).\end{equation} \tag{ 5 }$$

We note that in the SIC-POVM case we have

$$\begin{equation}\begin{aligned}\hfill {\mathbf{T}}_{nm}& =\frac{d{\delta }_{nm}+1}{{d}^{2}\left(d+1\right)},\hfill \\ \hfill {\left({\mathbf{T}}^{-1}\right)}_{nm}& =d\left(d+1\right){\delta }_{nm}-1\hfill \end{aligned}\end{equation} \tag{ 6 }$$

Let s and p be two probability vectors corresponding to density operators ρ and σ. Then we can introduce a probability representation of the Hilbert–Schmidt product:

$$\begin{equation}\mathrm{T}\mathrm{r}\left(\rho \sigma \right)=\sum _{n,m=1}^{{d}^{2}}{p}_{n}{s}_{m}\mathrm{T}\mathrm{r}\left({e}_{n}{e}_{m}\right)={s}^{\top }{\mathbf{T}}^{-1}p,\end{equation} \tag{ 7 }$$

and an analog of the matrix–matrix multiplication:

$$\begin{align}\hfill {\left(s{\ast}p\right)}_{k}& \equiv \enspace \mathrm{T}\mathrm{r}\left(\sigma \rho {E}_{k}\right)\hfill \\ \hfill & =\sum _{n,m=1}^{{d}^{2}}\enspace \mathrm{T}\mathrm{r}\left({e}_{n}{e}_{m}{E}_{k}\right){s}_{n}{p}_{m}={p}^{\top }{{\Lambda}}^{\left(k\right)}s.\hfill \end{align} \tag{ 8 }$$

Here Λ^(k) is a d² × d² matrix with elements ${{\Lambda}}_{nm}^{\left(k\right)}=\enspace \mathrm{T}\mathrm{r}\left({e}_{n}{e}_{m}{E}_{k}\right)$.

One can check that operators e_k have a unit trace, but they are not necessarily positive. Therefore, not any probability vector p corresponds to a quantum state. The set of possible distributions $\mathfrak{P}\left(E\right)$ has a form

$$\begin{equation}\mathfrak{P}\left(E\right)=\left\{\enspace p\enspace \left\vert \enspace ,\sum _{k=1}^{{d}^{2}},{p}_{k}\right.=1,\enspace \sum _{k=1}^{{d}^{2}}{p}_{k}{e}_{k}{\geqslant}0\enspace \right\},\end{equation} \tag{ 9 }$$

which is referred to as qplex (see reference [34]). The set of distributions corresponding to pure states is as follows:

$$\begin{equation}{\mathfrak{P}}_{\text{pure}}\left(E\right)=\left\{\enspace p\enspace \left\vert \enspace ,\sum _{k=1}^{{d}^{2}},{p}_{k}\right.=1,\enspace p{\ast}p=p\enspace \right\}.\end{equation} \tag{ 10 }$$

The convex hull of this set is a set of distributions corresponding to all states $\mathfrak{P}\left(E\right)$. We show schematic diagram of relations between sets $\mathfrak{P}\left(E\right)$, ${\mathfrak{P}}_{\text{pure}}\left(E\right)$, and the full d²-dimensional simplex $\mathfrak{X}$ in figure 2.

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Since $\mathfrak{P}\left(E\right)$ does not occupy the full space of d²-dimensional simplex $\mathfrak{X}$ it is valuable to have a method for checking whether given distribution p belongs to $\mathfrak{P}\left(E\right)$. A straightforward way to cope with this task is to apply Equation (5) to reconstruct ρ and check whether ρ ⩾ 0. However, in the present work, we are interested in a method that does not require a transition to the standard formalism.

Consider a characteristic polynomial of a density operator ρ

$$\begin{equation}\chi \left(\lambda \right)=\prod _{k=1}^{d}\left(\lambda -{\lambda }_{k}\right),\end{equation} \tag{ 11 }$$

where ${\left\{{\lambda }_{n}\right\}}_{n=1}^{d}$ is the spectrum of ρ. Let us define a set ${\left\{{a}_{n}\right\}}_{n=1}^{d}$ with the following elements:

$$\begin{equation}{a}_{n}{:=}\enspace \mathrm{T}\mathrm{r}\left({\rho }^{n}\right)=\sum _{l}{\left({p}^{{\ast}n}\right)}_{l}=\sum _{k=1}^{d}{\lambda }_{k}^{n},\quad n=1,\dots ,d.\end{equation} \tag{ 12 }$$

In order to check that $p\in \mathfrak{P}\left(E\right)$, it is necessary and sufficient to check that λ_k ⩾ 0 for all k.

Using the Newton–Girard identities the characteristic polynomial can be rewritten in the form

$$\begin{equation}\chi \left(\lambda \right)=\sum _{m=0}^{d}{\left(-1\right)}^{m}{b}_{m}{\lambda }^{d-m},\end{equation} \tag{ 13 }$$

where b₀ = 1 and

$$\begin{equation}{b}_{n}=\frac{1}{n}\sum _{i=1}^{n}{\left(-1\right)}^{i-1}{b}_{n-i}{a}_{i},\quad n=1,\dots ,d.\end{equation} \tag{ 14 }$$

If starting from some

$$\begin{equation}{d}^{\prime }\in \left\{1,\dots ,d\right\},\quad \text{one}\enspace \text{has}\quad {b}_{{d}^{\prime }}={b}_{{d}^{\prime }+1}=\dots ={b}_{d}=0,\end{equation} \tag{ 15 }$$

then χ(λ) has d − d' + 1 zero roots. It is convenient to remove them from consideration by resetting

$$\begin{equation}\chi \left(\lambda \right){:=}\chi \left(\lambda \right)/{\lambda }^{d-{d}^{\prime }+1}.\end{equation} \tag{ 16 }$$

Otherwise we set d' := d.

Then we suggest using the Routh–Hurwitz criterion in order to verify that every root of the polynomial χ(λ) is nonnegative. Let

$$\begin{equation}\tilde {\chi }\left(\lambda \right)=\prod _{k=1}^{{d}^{\prime }}\left(\lambda +{\lambda }_{k}\right)={\left(-1\right)}^{{d}^{\prime }}\chi \left(-\lambda \right)=\sum _{m=0}^{{d}^{\prime }}{b}_{m}{\lambda }^{{d}^{\prime }-m}.\end{equation} \tag{ 17 }$$

One can see that nonnegative roots of χ(λ) imply nonpositive roots of $\tilde {\chi }\left(\lambda \right)$. The Routh–Hurwitz criterion states that every root of $\tilde {\chi }\left(\lambda \right)$ is negative if and only if the principal minors ${\left\{{{\Delta}}_{i}\right\}}_{i=1}^{{d}^{\prime }}$ of the Hurwitz matrix

$$\begin{equation}\mathcal{H}=\left[\begin{matrix}{cccccc}\hfill {b}_{1}\hfill & \hfill {b}_{3}\hfill & \hfill {b}_{5}\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {b}_{0}\hfill & \hfill {b}_{2}\hfill & \hfill {b}_{4}\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {b}_{1}\hfill & \hfill {b}_{3}\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\vdots}\hfill & \hfill {\vdots}\hfill & \hfill {\vdots}\hfill & \hfill \ddots \hfill & \hfill {\vdots}\hfill & \hfill {\vdots}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill {b}_{{d}^{\prime }-2}\hfill & \hfill {b}_{d}^{\prime }\hfill \end{matrix}\right]\end{equation} \tag{ 18 }$$

are positive:

$$\begin{equation}{{\Delta}}_{1}{ >}0,\enspace {{\Delta}}_{2}{ >}0,\enspace {{\Delta}}_{3}{ >}0,\dots ,{{\Delta}}_{{d}^{\prime }}{ >}0.\end{equation} \tag{ 19 }$$

These relations form the constructive way for checking whether $p\in \mathfrak{P}\left(E\right)$.

The use of MIC-POVM probability vectors allows one to employ a simpler description of tensor products. This is an important advantage in comparison to the SIC-POVM case [35]. Let ${\mathcal{H}}^{A}$ and ${\mathcal{H}}^{B}$ be Hilbert spaces with MIC-POVMs ${E}^{A}={\left\{{E}_{i}^{A}\right\}}_{i=1}^{{d}_{A}^{2}}$ and ${E}^{B}={\left\{{E}_{i}^{B}\right\}}_{i=1}^{{d}_{B}^{2}}$, correspondingly. We then take MIC-POVM

$$\begin{equation}{E}^{AB}={\left\{{E}_{i}^{A}\otimes {E}_{j}^{B}\right\}}_{i,j=1}^{{d}_{A}^{2},{d}_{B}^{2}}\equiv {E}^{A}\otimes {E}^{B}\end{equation} \tag{ 20 }$$

on the space ${\mathcal{H}}^{A}\otimes {\mathcal{H}}^{B}$. If e^A and e^B are dual bases for E^A and E^B, then e^AB = e^A ⊗ e^B is dual to E^AB. If ${\rho }^{A}\in \mathfrak{S}\left({\mathcal{H}}^{A}\right),{\rho }^{B}\in \mathfrak{S}\left({\mathcal{H}}^{B}\right)$ are states and p^A, p^B are corresponding probability vectors, then the probability vectors of ρ^A ⊗ ρ^B is as follows:

$$\begin{equation}{p}_{\left(kl\right)}^{AB}=\enspace \mathrm{T}\mathrm{r}\left(\left({\rho }^{A}\otimes {\rho }^{B}\right)\left({E}_{k}^{A}\otimes {E}_{l}^{B}\right)\right)={p}_{k}^{A}{p}_{l}^{B}.\end{equation} \tag{ 21 }$$

Here we use notation (α, β) with α ∈ {1, ..., d_A} and β ∈ {1, ..., d_B} to define a multiindex. One can think that (α, β) ≡ (d_A − 1)α + β according the the standard Kronecker product rules. In the vector form, we have p^AB = p^A ⊗ p^B.

The next step is to obtain the probability representation of quantum channels. Let ${\mathcal{H}}^{\text{in}}$, ${\mathcal{H}}^{\text{out}}$ be Hilbert spaces with MIC-POVMs Eⁱⁿ, E^out and ${\Phi}:\mathfrak{S}\left({\mathcal{H}}^{\text{in}}\right)\to \mathfrak{S}\left({\mathcal{H}}^{\text{out}}\right)$ be a quantum channel (CPTP map). Consider a state ${\rho }^{\text{in}}\in \mathfrak{S}\left({\mathcal{H}}^{\text{in}}\right)$ and let ${\rho }^{\text{out}}={\Phi}\left({\rho }^{\text{in}}\right)\in \mathfrak{S}\left({\mathcal{H}}^{\text{out}}\right)$. Denote the probability vectors corresponding to ρⁱⁿ and ρ^out as pⁱⁿ and p^out respectively. Then the channel Φ can be characterized with a matrix S such that

$$\begin{equation}{p}^{\text{out}}=\mathbf{S}{p}^{\text{in}},\quad {\mathbf{S}}_{lk}=\enspace \mathrm{T}\mathrm{r}\left({E}_{l}^{\text{out}}{\Phi}\left({e}_{k}^{\text{in}}\right)\right).\end{equation} \tag{ 22 }$$

This matrix is generally pseudostochastic (i.e. ∑_kS_kl = 1), but it is not necessarily stochastic. An action of the channel Φ on the state ρⁱⁿ can be written as follows:

$$\begin{equation}{\Phi}:{\rho }^{\text{in}}{\mapsto}\sum _{k,l=1}^{{d}_{\text{out}},{d}_{\text{in}}}{\mathbf{S}}_{kl}{e}_{k}^{\text{out}}\enspace \mathrm{T}\mathrm{r}\left({E}_{l}^{\text{in}}{\rho }^{\text{in}}\right).\end{equation} \tag{ 23 }$$

In turn, an action of the dual channel ${{\Phi}}^{{\ast}}:\mathfrak{T}\left({\mathcal{H}}^{\text{out}}\right)\to \mathfrak{T}\left({\mathcal{H}}^{\text{in}}\right)$ is then given by

$$\begin{equation}{{\Phi}}^{{\ast}}:{M}^{\text{out}}{\mapsto}\sum _{k,l=1}^{{d}_{\text{in}},{d}_{\text{out}}}{\mathbf{S}}_{kl}{E}_{k}^{\text{in}}\enspace \mathrm{T}\mathrm{r}\left({e}_{l}^{\text{out}}{M}^{\text{out}}\right).\end{equation} \tag{ 24 }$$

In the case of Kraus representation where the operation of the channel is defined in the form

$$\begin{equation}{\Phi}\left(\rho \right)=\sum _{n}{V}_{n}\rho {V}_{n}^{{\dagger}},\end{equation} \tag{ 25 }$$

the corresponding elements of the pseudostochastic matrix are

$$\begin{equation}{\mathbf{S}}_{lk}=\sum _{n}\enspace \mathrm{T}\mathrm{r}\left({E}_{l}^{\text{out}}{V}_{n}{e}_{k}^{\text{in}}{V}_{n}^{{\dagger}}\right).\end{equation} \tag{ 26 }$$

The representation of tensor products for quantum channels can be used similarly to section 2.3. If

$$\begin{equation}{{\Phi}}^{A}:\mathfrak{S}\left({\mathcal{H}}^{\text{in},A}\right)\to \mathfrak{S}\left({\mathcal{H}}^{\text{out},A}\right)\end{equation} \tag{ 27 }$$

and

$$\begin{equation}{{\Phi}}^{B}:\mathfrak{S}\left({\mathcal{H}}^{\text{in},B}\right)\to \mathfrak{S}\left({\mathcal{H}}^{\text{out},B}\right)\end{equation} \tag{ 28 }$$

are quantum channels with corresponding pseudostochastic matrices S^A and S^B, then

$$\begin{align}\hfill {\mathbf{S}}_{\left(kl\right)\left(nm\right)}^{AB}& =\mathrm{T}\mathrm{r}\left(\left\{{E}_{k}^{\text{out},A}\otimes {E}_{l}^{\text{out},B}\right\}\left\{{{\Phi}}^{A}\otimes {{\Phi}}^{B}\right\}\left\{{e}_{n}^{\text{in},A}\otimes {e}_{m}^{\text{in},B}\right\}\right)\hfill \\ \hfill & =\enspace \mathrm{T}\mathrm{r}\left({E}_{k}^{\text{out},A}{{\Phi}}^{A}\left({e}_{n}^{\text{in},A}\right)\otimes {E}_{l}^{\text{out},B}{{\Phi}}^{B}\left({e}_{m}^{\text{in},B}\right)\right)\hfill \\ \hfill & ={\mathbf{S}}_{\left(k,n\right)}^{A}\otimes {\mathbf{S}}_{\left(l,m\right)}^{B}.\hfill \end{align} \tag{ 29 }$$

Thus, the tensor product of two quantum channels maps to the tensor product of two corresponding matrices.

The channel of a partial trace

$$\begin{equation}{\mathrm{T}\mathrm{r}}_{{\mathcal{H}}^{B}}\left(\cdot \right):{\rho }^{AB}{\mapsto}{\mathrm{T}\mathrm{r}}_{{\mathcal{H}}^{B}}\left({\rho }^{AB}\right)\end{equation} \tag{ 30 }$$

taking an input ${\rho }^{AB}\in \mathfrak{S}\left({\mathcal{H}}^{A}\otimes {\mathcal{H}}^{B}\right)$ then corresponds to the matrix in the form:

$$\begin{align}\hfill {\mathbf{S}}_{l\left(nm\right)}& =\enspace \mathrm{T}\mathrm{r}\left({E}_{l}^{A}{\mathrm{T}\mathrm{r}}_{B}\left({e}_{n}^{A}\otimes {e}_{m}^{B}\right)\right)\hfill \\ \hfill & =\enspace \mathrm{T}\mathrm{r}\left({E}_{l}^{A}{e}_{n}^{A}\right)={\delta }_{\mathrm{ln}}.\hfill \end{align} \tag{ 31 }$$

As in the case of states, not any pseudostochastic matrix S corresponds to a (physical) quantum channel Φ. In order to formulate a criterion, we use the Choi–Jamiołkowski duality [45, 46].

Let ${\Phi}:\mathfrak{S}\left({\mathcal{H}}^{\text{in}}\right)\to \mathfrak{S}\left({\mathcal{H}}^{\text{out}}\right)$ be a trace-preserving map. By fixing the orthonormal basis ${\left\{\vert n\rangle \right\}}_{n=1}^{{d}_{\text{in}}}$ in ${\mathcal{H}}^{\text{in}}$ (${d}_{\text{in}}=\mathrm{dim}{\mathcal{H}}^{\text{in}}$), we define a state $\sigma \in \mathfrak{S}\left({\mathcal{H}}^{\prime \mathrm{i}\mathrm{n}}\otimes {\mathcal{H}}^{\text{in}}\right)$ with ${\mathcal{H}}^{\prime \mathrm{i}\mathrm{n}}={\mathcal{H}}^{\text{in}}$ of the following form:

$$\begin{equation}\sigma =\frac{1}{{d}_{\text{in}}}\sum _{n,m=1}^{{d}_{\text{in}}}\vert n\rangle \langle m\vert \otimes \vert n\rangle \langle m\vert .\end{equation} \tag{ 32 }$$

One can see that it is a density matrix of the pure state

$$\begin{equation}\vert \phi \rangle =\frac{1}{\sqrt{{d}_{\text{in}}}}\sum _{k=1}^{{d}_{\text{in}}}\vert k\rangle \otimes \vert k\rangle .\end{equation} \tag{ 33 }$$

We then call Choi state an operator ρ_Φ

$$\begin{equation}{\rho }_{{\Phi}}=\left(\mathrm{I}\mathrm{d}\otimes {\Phi}\right)\left(\sigma \right)=\frac{1}{{d}_{\text{in}}}\sum _{n,m=1}^{{d}_{\text{in}}}\vert n\rangle \langle m\vert \otimes {\Phi}\left(\vert n\rangle \langle m\vert \right),\end{equation} \tag{ 34 }$$

where Id is an identical map. The Choi–Jamiołkowski isomorphism says that the operator ρ_Φ is a quantum state if and only if Φ is a quantum channel. One can reconstruct an action of Φ on an arbitrary input using the following formula:

$$\begin{equation}{\Phi}\left(\rho \right)={\mathrm{T}\mathrm{r}}_{{\mathcal{H}}^{\prime \mathrm{i}\mathrm{n}}}\left(\left({\rho }^{\text{in}\top }\otimes {\mathbf{I}}_{{d}_{\text{in}}}\right){\rho }_{{\Phi}}\right).\end{equation} \tag{ 35 }$$

The Choi–Jamiołkowski isomorphism can be naturally formulated in the probability representation. Let S be a matrix corresponding to a trace-preserving map Φ, and s be a vector corresponding to σ. We assume that s is obtained with MIC-POVM Eⁱⁿ ⊗ Eⁱⁿ. Then the Choi probability vector has the form

$$\begin{equation}{p}_{\mathbf{S}}=\left({\mathbf{I}}_{{d}_{\text{in}}^{2}}\otimes \mathbf{S}\right)s.\end{equation} \tag{ 36 }$$

One can see that S corresponds to the quantum channel only in case ${p}_{\mathbf{S}}\in \mathfrak{P}\left({E}^{\text{in}}\otimes {E}^{\text{out}}\right)$. In order to reconstruct S via the vector p_S, one can use the following relation:

$$\begin{align}\hfill {\mathbf{S}}_{lk}& =\enspace \mathrm{T}\mathrm{r}\left({E}_{l}^{\mathrm{o}\mathrm{u}\mathrm{t}}{\Phi}\left({e}_{k}^{\mathrm{i}\mathrm{n}}\right)\right)=\enspace \mathrm{T}\mathrm{r}\left({\rho }_{{\Phi}}\left({e}_{k}^{\text{in}\top }\otimes {E}_{l}^{\mathrm{o}\mathrm{u}\mathrm{t}}\right)\right)\hfill \\ \hfill & =\sum _{n,m=1}^{{d}_{\text{in}}^{2}}{p}_{\mathbf{S}\left(nm\right)}\enspace \mathrm{T}\mathrm{r}\left({e}_{n}^{\mathrm{i}\mathrm{n}}{e}_{k}^{\text{in}\top }\right)\enspace \mathrm{T}\mathrm{r}\left({e}_{m}^{\mathrm{o}\mathrm{u}\mathrm{t}}{E}_{l}^{\mathrm{o}\mathrm{u}\mathrm{t}}\right)\hfill \\ \hfill & =\sum _{n=1}^{{d}_{\text{in}}^{2}}\enspace \mathrm{T}\mathrm{r}\left({e}_{n}^{\mathrm{i}\mathrm{n}}{e}_{k}^{\text{in}\top }\right){p}_{\mathbf{S}\left(n,l\right)}.\hfill \end{align} \tag{ 37 }$$

It is useful to define s in terms of probability vectors without the notion of the Hilbert space ${\mathcal{H}}^{\text{in}}$ and the state σ. Consider random pure state $\vert {\psi }_{1}\rangle \langle {\psi }_{1}\vert \in \mathfrak{S}\left({\mathcal{H}}^{\text{in}}\right)$ and denote its probability vector as p⁽¹¹⁾. Let us construct as a set of orthonormal probability vectors ${\left\{{p}^{\left(kk\right)}\right\}}_{k=1}^{{d}_{\text{in}}}$ using the following equations for each k = 2, ..., d_in:

$$\begin{equation}\begin{aligned}\hfill & \sum _{l=1}^{{d}_{\text{in}}}{p}_{l}^{\left(kk\right)}=1;\hfill \\ \hfill & {p}^{\left(kk\right)}\;{\ast}\; {p}^{\left(kk\right)}={p}^{\left(kk\right)};\hfill \\ \hfill & \sum _{l=1}^{{d}_{\text{in}}}{\left({p}^{\left(kk\right)}{\ast} {p}^{\left(nn\right)}\right)}_{l}=0,\quad n=1,\dots ,k-1,\hfill \end{aligned}\end{equation} \tag{ 38 }$$

where we consider orthonormality with respect to the Hilbert–Schmidt product 7. One can think about p^(kk) as a probability vector of state |ψ_k⟩⟨ψ_k| taken from an orthonormal basis constructed from |ψ₁⟩⟨ψ₁|. Of course, Equation (38) has infinite number of solutions.

Then the vectors p^(nm) with n ≠ m, corresponding to states |ψ_n⟩⟨ψ_m|, can be obtained using straightforward multiplicative relations. For example, vector p⁽¹²⁾ can be obtained as a solution of the following equations:

$$\begin{equation}\begin{aligned}\hfill {p}^{\left(12\right)}{\ast} {p}^{\left(kk\right)}& =0,\enspace \enspace k\ne 2;\quad \hfill & \hfill {p}^{\left(kk\right)}{\ast} {p}^{\left(12\right)}& =0,\enspace \enspace k\ne 1;\hfill \\ \hfill {p}^{\left(12\right)}{\ast} {p}^{\left(11\right)}& =0;\quad \hfill & \hfill {p}^{\left(11\right)}{\ast} {p}^{\left(12\right)}& ={p}^{12};\hfill \\ \hfill {p}^{\left(12\right)}{\ast} {p}^{\left(12\right)}& =0;\quad \hfill & \hfill {p}^{\left(12\right)}{\ast} {p}^{\left(12\right)}& =0;\hfill \\ \hfill {p}^{\left(12\right)}{\ast} {p}^{\left(22\right)}& ={p}^{12};\quad \hfill & \hfill {p}^{\left(22\right)}{\ast} {p}^{\left(12\right)}& =0.\hfill \end{aligned}\end{equation} \tag{ 39 }$$

By finding p^(nm) for all n, m = 1, ..., d_in, we obtain the Choi distribution in the form

$$\begin{equation}{p}_{\mathbf{S}}=\frac{1}{{d}_{\text{in}}}\sum _{n,m=1}^{{d}_{\text{in}}}{p}^{\left(nm\right)}\otimes \mathbf{S}{p}^{\left(nm\right)}.\end{equation} \tag{ 40 }$$

We also would like to mention a special case where ${E}^{\text{in}}={\left\{\vert {\psi }_{i}^{\text{sym}}\rangle \langle {\psi }_{i}^{\text{sym}}\vert \right\}}_{i=1}^{{d}_{\text{in}}}$ is a SIC-POVM. Let ${\bar{E}}^{\text{in}}={\left\{\vert {\bar{\psi }}_{i}^{\text{sym}}\rangle \langle {\bar{\psi }}_{i}^{\text{sym}}\vert \right\}}_{i=1}^{{d}_{\text{in}}}$ with

$$\begin{equation}\vert {\bar{\psi }}_{i}^{\text{sym}}\rangle =\sum _{i=1}^{{d}_{\text{in}}}\vert i\rangle \bar{\langle i\vert {\psi }_{i}^{\text{sym}}\rangle }=\sum _{i=1}^{{d}_{\text{in}}}\vert i\rangle \langle {\psi }_{i}^{\text{sym}}\vert i\rangle ,\end{equation} \tag{ 41 }$$

where $\bar{x}$ stands for complex conjugate of x, and ${\left\{\vert n\rangle \right\}}_{n=1}^{{d}_{\text{in}}}$ is a computational basis as usual. Then the probability vector of the state σ = |ϕ⟩⟨ϕ| (see Equation (35)) takes the following form with respect to the MIC-POVM ${\bar{E}}^{\text{in}}\otimes {E}^{\text{in}}$:

$$\begin{align}\hfill {s}_{\left(nm\right)}& =\enspace \mathrm{T}\mathrm{r}\left(\vert \phi \rangle \langle \phi \vert \left({\bar{E}}_{n}^{\mathrm{i}\mathrm{n}}\otimes {E}_{m}^{\mathrm{i}\mathrm{n}}\right)\right)\hfill \\ \hfill & =\frac{1}{{d}_{\text{in}}^{2}}\sum _{k,l}\langle k\vert {\bar{\psi }}_{n}^{\mathrm{s}\mathrm{y}\mathrm{m}}\rangle \langle {\bar{\psi }}_{n}^{\mathrm{s}\mathrm{y}\mathrm{m}}\vert l\rangle \langle k\vert {\psi }_{m}^{\mathrm{s}\mathrm{y}\mathrm{m}}\rangle \langle {\psi }_{m}^{\mathrm{s}\mathrm{y}\mathrm{m}}\vert l\rangle \hfill \\ \hfill & =\frac{1}{{d}_{\text{in}}}{\left\vert \langle {\psi }_{m}\vert {\psi }_{n}\rangle \right\vert }^{2}=\frac{{d}_{\text{in}}{\delta }_{nm}+1}{{d}_{\text{in}}^{2}\left({d}_{\text{in}}+1\right)}.\hfill \end{align} \tag{ 42 }$$

It then can be substituted to Equation (36) in order to obtain a Choi probability vector and verify that it corresponds to valid quantum state.

Here we consider a MIC-POVM-based probability representation of an arbitrary measurement with the finite number of outcomes. In the general case it is given by a POVM $M={\left\{{M}_{i}\right\}}_{i=1}^{m}$ with some finite m. Note that M may not belong to MIC class. According to the Born rule the probability to obtain ith outcome for an input state ρ is given by

$$\begin{equation}{q}_{i}=\enspace \mathrm{T}\mathrm{r}\left(\rho {M}_{i}\right),\quad i=1,\dots ,m\end{equation} \tag{ 43 }$$

(here we assume that M and ρ are defined with respect to the same d-dimensional Hilbert space $\mathcal{H}$). Taking ρ in the probability representation given by Equation (5), we obtain the following expression for the probability vector:

$$\begin{equation}\begin{aligned}\hfill q& ={\left[\begin{matrix}{c}\hfill {q}_{1}\dots {q}_{m}\hfill \end{matrix}\right]}^{\top },\hfill \\ \hfill q& =\mathbf{M}p,\quad {\mathbf{M}}_{ij}=\enspace \mathrm{T}\mathrm{r}\left({M}_{i}{e}_{j}\right).\hfill \end{aligned}\end{equation} \tag{ 44 }$$

One can see that M is m × d² pseudostochastic matrix because of normalization condition ${\sum }_{i=1}^{m}{M}_{i}={\mathbf{I}}_{d}$. We note that given matrix M, the effects of the POVM in the standard formalism are given by

$$\begin{equation}{M}_{k}=\sum _{l=1}^{{d}^{2}}\enspace \mathrm{T}\mathrm{r}\left({M}_{k}{e}_{l}\right){E}_{l}=\sum _{l=1}^{{d}^{2}}{\mathbf{M}}_{kl}{E}_{l}.\end{equation} \tag{ 45 }$$

Next, we consider a problem of the verification that a given m × d² pseudostochastic matrix M corresponds to some valid POVM M with m outcomes. An idea behind such a test is very similar to the case of states, which is considered in section 2.2, with the main difference that we swap the basis E and the dual basis e.

Consider two operators $X,Y\in \mathfrak{T}\left(\mathcal{H}\right)$. Using a dual basis ${\left\{{e}_{i}\right\}}_{i=1}^{d}$ one can represent them with row-vectors $\lambda =\left[\begin{matrix}{c}\hfill {\lambda }_{1}\dots {\lambda }_{d}\hfill \end{matrix}\right]$ and $\mu =\left[\begin{matrix}{c}\hfill {\mu }_{1}\dots {\mu }_{d}\hfill \end{matrix}\right]$ according to the following expressions:

$$\begin{equation}\begin{aligned}\hfill X& =\sum _{i=1}^{{d}^{2}}{\lambda }_{i}{E}_{i},\quad \hfill & \hfill {\lambda }_{i}& =\enspace \mathrm{T}\mathrm{r}\left({e}_{i}X\right),\quad i=1,\dots ,{d}^{2};\hfill \\ \hfill Y& =\sum _{j=1}^{{d}^{2}}{\mu }_{j}{E}_{j}\quad \hfill & \hfill {\mu }_{j}& =\enspace \mathrm{T}\mathrm{r}\left({e}_{j}Y\right),\quad j=1,\dots ,{d}^{2}.\hfill \end{aligned}\end{equation} \tag{ 46 }$$

Note that the trace operation takes the form:

$$\begin{equation}\enspace \mathrm{T}\mathrm{r}\left(X\right)=\sum _{i=1}^{{d}^{2}}{\lambda }_{i}\enspace \mathrm{T}\mathrm{r}\left({E}_{i}\right)=\lambda \kappa \end{equation} \tag{ 47 }$$

with $\kappa ={\left[\begin{matrix}{c}\hfill \mathrm{T}\mathrm{r}\left({E}_{1}\right)\dots \mathrm{T}\mathrm{r}\left({E}_{{d}^{2}}\right)\hfill \end{matrix}\right]}^{\top }$.

Then we can introduce a 'multiplication' of vectors μ and λ, denoted by ⊛, as follows:

$$\begin{align}\hfill {\left(\lambda {\circledast}\mu \right)}_{k}& \equiv \enspace \mathrm{T}\mathrm{r}\left(XY{e}_{k}\right)\hfill \\ \hfill & =\sum _{n,m=1}^{{d}^{2}}\enspace \mathrm{T}\mathrm{r}\left({E}_{n}{E}_{m}{e}_{k}\right){\lambda }_{n}{\mu }_{m}=\lambda {\tilde {{\Lambda}}}^{\left(k\right)}{\mu }^{\top },\hfill \end{align} \tag{ 48 }$$

where ${\tilde {{\Lambda}}}^{\left(k\right)}$ is d² × d² matrix with elements

$$\begin{equation}{\tilde {{\Lambda}}}_{nm}^{\left(k\right)}=\enspace \mathrm{T}\mathrm{r}\left({E}_{n}{E}_{m}{e}_{k}\right).\end{equation} \tag{ 49 }$$

Now we are ready to describe the verification algorithm. The normalization condition ${\sum }_{i=1}^{m}{M}_{i}={\mathbf{I}}_{d}$ follows from the fact that M is pseudostochastic. So the only remaining issue is to check the semi-positivity condition M_i ⩾ 0. We note that in the case of states we derived expressions for $\enspace \mathrm{T}\mathrm{r}\left(\rho \right)$, ..., $\mathrm{T}\mathrm{r}\left({\rho }^{d}\right)$ from the probability representation of ρ and then substitute them into Routh–Hurwitz-like criterion. Here we act in a similar manner. For each ith row of the matrix M set ${\lambda }^{\left(i\right)}{:=}\left[\begin{matrix}{c}\hfill {\mathbf{M}}_{i,1} \dots {\mathbf{M}}_{i,{d}^{2}}\hfill \end{matrix}\right]$ and compute

$$\begin{equation}{a}_{j}^{\left(i\right)}{:=}{\lambda }^{\left(i\right){\circledast}j}\kappa =\enspace \mathrm{T}\mathrm{r}\left({M}_{i}^{j}\right).\end{equation} \tag{ 50 }$$

Then proceed with same steps as in the case of states replacing ${\left\{{a}_{n}\right\}}_{n=1}^{d}$ with ${\left\{{a}_{i}\right\}}_{n=1}^{d}$. If the positivity condition is fulfilled for all i = 1, ..., m, then S corresponds to valid 'physical' quantum measurement.

Finally, we consider the case a measurement given by some Hermitian operator $O={O}^{{\dagger}}\in \mathfrak{T}\left(\mathcal{H}\right)$, also known as an observable. To obtain its probability representation we first take it spectral decomposition in the form

$$\begin{equation}O=\sum _{i=1}^{m}{x}_{i}{{\Pi}}_{i},\end{equation} \tag{ 51 }$$

where ${\left\{{x}_{i}\right\}}_{i=1}^{m}$ are physical quantities which can be observed, and ${\left\{{{\Pi}}_{i}\right\}}_{i=1}^{m}$ are complete set of orthogonal (self-adjoint) projectors: ${\sum }_{i=1}^{m}{{\Pi}}_{i}={\mathbf{I}}_{d}$, Π_iΠ_j = δ_ij. One can consider a POVM M_O with effects ${\left\{{{\Pi}}_{i}\right\}}_{i=1}^{m}$ and its corresponding pseudostochastic matrix M keeping in mind that each ith outcome corresponds the quantity x_i. However, it is important to note if one is interested in mean value for some state ρ is given by $\langle O\rangle =\enspace \mathrm{T}\mathrm{r}\left(O\rho \right)$, then one can consider a row-vector

$$\begin{equation}{\mathbf{O}}^{\text{mean}}{:=}\left[\begin{matrix}\hfill {x}_{1} \dots {x}_{m}\hfill \end{matrix}\right]\mathbf{M}\end{equation} \tag{ 52 }$$

and compute mean value as ⟨O⟩ = O_meanp, where p is a probability vector of ρ.

Finally, we note that the rules of the tensor product remain the same as in the case of quantum channels: pseudostochastic matrix of measurements on several physical subsystems is given by a tensor product of pseudostochastic measurements on each of subsystems.

Up to this point the MIC-POVM, which determines the probability representation, was fixed. Here we consider a question of how to make a transition between representations determined by different MIC-POVMs. Let E and F be two MIC-POVMs defined with respect to the same d-dimensional Hilbert space $\mathcal{H}$, and let e and f be their corresponding dual bases. We use superscript [E] or [F] to emphasize that given probability vector or pseudostochastic matrix of a channel/measurement is given in E- or F-based representation.

Consider elements of a pseudostochastic matrix of the measurement in E given in F-based representation:

$$\begin{equation}{\left({\mathbf{M}}_{E}^{\left[F\right]}\right)}_{m,n}=\enspace \mathrm{T}\mathrm{r}\left({E}_{m}{f}_{n}\right),\quad m,n=1,\dots ,{d}^{2}.\end{equation} \tag{ 53 }$$

One can see the pseudostochastic matrix of the measurement in F given in E-based representation is given by ${\mathbf{M}}_{F}^{\left[E\right]}={\left({\mathbf{M}}_{E}^{\left[F\right]}\right)}^{-1}$

Then we come to the following relations:

$$\begin{equation}{p}^{\left[E\right]}={\mathbf{M}}_{E}^{\left[F\right]}{p}^{\left[F\right]},\end{equation} \tag{ 54 }$$

$$\begin{equation}{\mathbf{S}}^{\left[E\right]}={\mathbf{M}}_{E}^{\left[F\right]}{\mathbf{S}}^{\left[F\right]}{\mathbf{M}}_{F}^{\left[E\right]},\end{equation} \tag{ 55 }$$

$$\begin{equation}{\mathbf{M}}^{\left[E\right]}={\mathbf{M}}^{\left[F\right]}{\mathbf{M}}_{F}^{\left[E\right]},\end{equation} \tag{ 56 }$$

where p^[⋅], S^[⋅], and M^[⋅] corresponds to some state, channel, and POVM, correspondingly.

Consider a d-dimensional dimensional Hilbert space $\mathcal{H}$. The evolution of a quantum state under the Hamiltonian $H={H}^{{\dagger}}\in \mathfrak{T}\left(\mathcal{H}\right)$ is described by the Liouville–von Neumann equation

$$\begin{equation}\dot {\rho }\left(t\right)=-\frac{{\imath}}{\hslash }\left[H,\enspace \rho \left(t\right)\right],\end{equation} \tag{ 57 }$$

where $\left[\cdot ,\enspace \cdot \right]$ denotes commutator. In what follows we use dimensionless units and set ℏ ≡ 1.

Using Equation (5), then left multiplying by E_l and taking trace, the equation takes the form

$$\begin{equation}{\dot {p}}_{l}\left(t\right)=-{\imath}\sum _{k=1}^{{d}^{2}}\enspace \mathrm{T}\mathrm{r}\left({E}_{l}\left[H,\enspace {e}_{k}\right]\right){p}_{k}\left(t\right).\end{equation} \tag{ 58 }$$

Therefore, the Liouville–von Neumann equation takes the form of the ordinary linear differential equation with generator H

$$\begin{equation}\dot {p}\left(t\right)=\mathbf{H}p\left(t\right),\quad {\mathbf{H}}_{l,k}={\imath}\enspace \mathrm{T}\mathrm{r}\left(H\left[{E}_{l},\enspace {e}_{k}\right]\right).\end{equation} \tag{ 59 }$$

We note that such a form of the equation for the unitary dynamics on the probability language has been obtained in reference [38]. In the following section we present the generalization of this results for the case of the GKSL equation.

The d² × d² matrix H is a probabilistic representation of the Hamiltonian, which has following properties.

• (a)  
  The matrix H is real: $\mathbf{H}\in {\mathrm{M}\mathrm{a}\mathrm{t}}_{{d}^{2}{\times}{d}^{2}}\left(\mathbb{R}\right)$.
• (b)  
  The sum of each column is zero: ${\sum }_{l=1}^{{d}^{2}}{\mathbf{H}}_{l,k}=0$.The first property follows from the fact that $\left[{E}_{l},\enspace {e}_{k}\right]$ is Hermitian, and second fact come from the normalization condition on MIC-POVM effects E_i. It is worth to note that if E is a SIC-POVM, then H becomes antisymmetric (H^⊤ = −H), and thus all its diagonal elements are zero, and all rows also sum to zero (see reference [35] for more details).The solution to Equation (59) can be presented in the form

  $$\begin{equation}p\left(t\right)=\mathbf{U}\left(t\right){p}_{0}\quad \mathbf{U}\left(t\right)={e}^{\mathbf{H}t},\end{equation} \tag{ 60 }$$

  where p₀ is a probability vector at t = 0. Note that U(t) is pseudostochastic. The unitarity of the evolution governed by the Liouville–von Neumann equation implies preserving the Hilbert–Schmidt product between two arbitrary vectors during their evolution according to Equation (59). Taking into account the probability representation of the Hilbert–Schmidt product is given by Equation (7) we obtain the identity

  $$\begin{equation}{\mathbf{T}}^{-1}=\mathbf{U}{\left(t\right)}^{\top }{\mathbf{T}}^{-1}\mathbf{U}\left(t\right).\end{equation} \tag{ 61 }$$

  Considering small times t = δt ≪ 1 and expanding exponent of the evolution operator in Equation (60) into Taylor series we arrive at the 3rd property of the MIC-POVM-based representation of a Hamiltonian.
• (a)  
  The following identity holds:

  $$\begin{equation}{\mathbf{H}}^{\top }{\mathbf{T}}^{-1}+{\mathbf{T}}^{-1}\mathbf{H}=0.\end{equation} \tag{ 62 }$$

Consider a matrix $\tilde {\mathbf{H}}{:=}{\mathbf{T}}^{-1}\mathbf{H}$. One can see that since T is symmetric, $\tilde {\mathbf{H}}$ is antisymmetric: ${\tilde {\mathbf{H}}}^{\top }+\tilde {\mathbf{H}}=0$. Combining this fact with the property 2 one has ${\sum }_{i=1}^{{d}^{2}}{\left(\mathbf{T}\tilde {\mathbf{H}}\right)}_{ij}=0$. Antisymmetric matrix $\tilde {\mathbf{H}}$ possessing these properties can be defined with (d² − 2)(d² − 1)/2 independent parameters. Note that for d > 2 this quantity is larger than d² − 1—the number of independent parameters required to define physical properties of a Hamiltonian (the term −1 comes from the fact that energy is always defined up to some constant). So the properties 2 and 3 are insufficient to determine a set of possible probability representations of Hamiltonians. In what follows we consider a necessary and sufficient condition on matrix $\mathbf{H}\in {\mathrm{M}\mathrm{a}\mathrm{t}}_{{d}^{2}{\times}{d}^{2}}\left(\mathbb{R}\right)$ to be a probability representation of some Hamiltonian H.

In order to proceed, we first need to introduce the 'vectorised' notation for operators. If $A\in \mathfrak{T}\left(\mathcal{H}\right)$ is an operator, then it can be written in a form

$$\begin{equation}A=\sum _{n,m=1}^{d}{A}_{nm}\vert n\rangle \langle m\vert ,\quad {A}_{nm}=\langle n\vert A\vert m\rangle .\end{equation} \tag{ 63 }$$

Then the bra- and ket-representations of A will be denoted as

$$\begin{equation}\vert A\rangle \rangle =\sum _{n,m}{A}_{nm}\vert n\rangle \otimes \vert m\rangle ,\enspace \enspace \langle \langle A\vert =\sum _{n,m}\bar{{A}_{nm}}\langle n\vert \otimes \langle m\vert .\end{equation} \tag{ 64 }$$

These representations may be understood as raising or lowering index. The inner product between such vectors yields $\langle \langle A\vert \vert B\rangle \rangle =\enspace \mathrm{T}\mathrm{r}\left({A}^{{\dagger}}B\right)$. We also note that B = UAV^† transforms into $\vert B\rangle \rangle =U\otimes \bar{V}\vert A\rangle \rangle$ with $U,V\in \mathfrak{T}\left(\mathcal{H}\right)$.

We introduce tensors e and E in the form

$$\begin{equation}\mathbf{e}=\left[\begin{matrix}{c}\hfill \vert {e}_{1}\rangle \rangle \dots \vert {e}_{{d}^{2}}\rangle \rangle \hfill \end{matrix}\right],\quad \mathbf{E}=\left[\begin{matrix}\hfill \langle \langle {E}_{1}\vert \hfill \\ \hfill {\vdots}\hfill \\ \hfill \langle \langle {E}_{{d}^{2}}\vert \hfill \end{matrix}\right].\end{equation} \tag{ 65 }$$

Then one has

$$\begin{equation}\vert \rho \rangle \rangle =\mathbf{e}p,\quad \mathbf{E}\mathbf{e}=\mathbf{1},\quad \mathbf{e}\mathbf{E}=\sum _{i=1}^{{d}^{2}}\vert {e}_{i}\rangle \rangle \langle \langle {E}_{i}\vert .\end{equation} \tag{ 66 }$$

By using this notation, the matrix H can be written as follows:

$$\begin{equation}\mathbf{H}=-{\imath}\mathbf{E}\left(H\otimes {\mathbf{I}}_{d}-{\mathbf{I}}_{d}\otimes \bar{H}\right)\mathbf{e}.\end{equation} \tag{ 67 }$$

Let ${\left\{{\sigma }^{\left(i\right)}\right\}}_{i=1}^{{d}^{2}-1}$ be a linearly independent set of operators, satisfying following conditions: (i) operators are traceless $\enspace \mathrm{T}\mathrm{r}\left({\sigma }^{\left(i\right)}\right)=0$; (ii) operators are Hermitian ${\sigma }^{\left(i\right)}={\left({\sigma }^{\left(i\right)}\right)}^{{\ast}}$; (iii) $\enspace \mathrm{T}\mathrm{r}\left({\sigma }^{\left(i\right)}{\sigma }^{\left(j\right)}\right)=2{\delta }_{ij}$. In a two-dimensional Hilbert space (d = 2) these such a set can be presented with Pauli matrices. Then the Hamiltonian can be represented as follows:

$$\begin{equation}H={\nu }_{0}{\mathbf{I}}_{d}+\sum _{i=1}^{{d}^{2}-1}{\nu }_{i}{\sigma }^{\left(i\right)},\end{equation} \tag{ 68 }$$

where ${\nu }_{0}=\enspace \mathrm{T}\mathrm{r}\left(H\right)/d$ and ${\nu }_{i}=\enspace \mathrm{T}\mathrm{r}\left(H{\sigma }^{\left(i\right)}\right)/2$.

Let ${\left\{{\mathbf{H}}^{\left(i\right)}\right\}}_{i=1}^{{d}^{2}}$ be a set of matrices corresponding to ${\left\{{\sigma }^{\left(i\right)}\right\}}_{i=1}^{{d}^{2}-1}$:

$$\begin{equation}{\mathbf{H}}_{lk}^{\left(i\right)}={\imath}\enspace \mathrm{T}\mathrm{r}\left({\sigma }^{\left(i\right)}\left[{E}_{l},\enspace {e}_{k}\right]\right).\end{equation} \tag{ 69 }$$

We note the probability representation of a Hamiltonian equal to identity matrix gives zero matrix. Therefore, we have

$$\begin{equation}\mathbf{H}=\sum _{i=1}^{{d}^{2}-1}{\nu }_{i}{\mathbf{H}}^{\left(i\right)}.\end{equation} \tag{ 70 }$$

Next, we can see that

$$\begin{equation}\enspace \mathrm{T}\mathrm{r}\left({\mathbf{H}}^{\left(i\right)}{\mathbf{H}}^{\left(j\right)}\right)=-\enspace \mathrm{T}\mathrm{r}\left({\sigma }^{\left(i\right)}{\sigma }^{\left(j\right)}\otimes {\mathbf{I}}_{d}+{\mathbf{I}}_{d}\otimes \bar{{\sigma }^{\left(i\right)}}\bar{{\sigma }^{\left(j\right)}}\right)=-4d{\delta }_{ij}.\end{equation} \tag{ 71 }$$

Thus, it is possible to define the projector ${\mathcal{P}}_{\text{unit}}\left(\cdot \right)$ on the space ${\mathrm{M}\mathrm{a}\mathrm{t}}_{{d}^{2}{\times}{d}^{2}}\left(\mathbb{R}\right)$ that correspond to Hamiltonians in the following explicit form:

$$\begin{equation}{\mathcal{P}}_{\text{unit}}\left(\mathbf{M}\right)=-\frac{1}{4d}\sum _{i=1}^{{d}^{2}-1}\enspace \mathrm{T}\mathrm{r}\left(\mathbf{M}\cdot {\mathbf{H}}^{\left(i\right)}\right){\mathbf{H}}^{\left(i\right)}\end{equation} \tag{ 72 }$$

(here M is some d² × d² matrix ). Finally, the matrix H corresponds to a Hamiltonian, if and only if it satisfies the identity

$$\begin{equation}{\mathcal{P}}_{\text{unit}}\left(\mathbf{H}\right)=\mathbf{H}.\end{equation} \tag{ 73 }$$

It also worth to note the ${\mathcal{P}}_{\text{unit}}$ turns to be a useful tool for studying experimental data in quantum process tomography experiments, since it allows extracting the unitary part of generator for a reconstructed process [35].

Here we generalize the results of the previous section on the case of the GKSL equation [47, 48]. Consider the Markovian master equation in the form $\dot {\rho }\left(t\right)=L\left(\rho \right)$ with

$$\begin{equation}L\left(\rho \right)=-{\imath}\left[H,\enspace \rho \left(t\right)\right]+\sum _{k}\left({A}_{k}\rho \left(t\right){A}_{k}^{{\dagger}}-\frac{1}{2}\left\{{A}_{k}^{{\dagger}}{A}_{k},\enspace \rho \left(t\right)\right\}\right),\end{equation} \tag{ 74 }$$

where $\left\{\cdot ,\enspace \cdot \right\}$ is an anticommutator, and A_k are some arbitrary operators describing dissipative evolution, also known as noise operators.

Let us introduce a CP map ${\Psi}:M{\mapsto}{\sum }_{k}{A}_{k}M{A}_{k}^{{\dagger}}$. One can think about Ψ(⋅) as a quantum channel without trace-preserving property. The second term of Equation (74) can be written in the following form:

$$\begin{equation}{\Psi}\left(\rho \right)-\frac{1}{2}\left\{{{\Psi}}^{{\ast}}\left({\mathbf{I}}_{d}\right),\enspace \rho \right\}\equiv D\left(\rho \right),\end{equation} \tag{ 75 }$$

where Ψ* is a map dual to Ψ.

Let $\mathbf{S}\in {\mathrm{M}\mathrm{a}\mathrm{t}}_{{d}^{2}{\times}{d}^{2}}\left(\mathbb{R}\right)$ be MIC-POVM-based representation of Ψ, i.e. ${\mathbf{S}}_{ij}=\enspace \mathrm{T}\mathrm{r}\left({E}_{i}{\Psi}\left({e}_{j}\right)\right)$. Then by using Equations (23), (24) and (49) one obtains

$$\begin{equation}\enspace \mathrm{T}\mathrm{r}\left({E}_{i}D\left({e}_{j}\right)\right)={\mathbf{S}}_{ij}-\sum _{k,l=1}^{{d}^{2}}{\mathbf{S}}_{kl}\frac{{\tilde {{\Lambda}}}_{i,l}^{\left(j\right)}+{\tilde {{\Lambda}}}_{l,i}^{\left(j\right)}}{2}\equiv {\mathbf{D}}_{ij}\end{equation} \tag{ 76 }$$

for i, j = 1, ..., d². Using the probability representation of the first term of 74 from the previous section, we obtain the GKSL equation in the MIC-POVM-based probability as follows:

$$\begin{equation}\dot {p}\left(t\right)=\mathbf{L}p\left(t\right),\quad \mathbf{L}=\mathbf{H}+\mathbf{D}.\end{equation} \tag{ 77 }$$

One can easily verify that $\mathbf{D}\in {\mathrm{M}\mathrm{a}\mathrm{t}}_{{d}^{2}{\times}{d}^{2}}\left(\mathbb{R}\right)$ and ${\sum }_{i=1}^{{d}^{2}}{\mathbf{D}}_{ij}=0$ for every j = 1, ..., d².

We also discuss a necessary and sufficient properties of L to be corresponded to some Liouvillian L(⋅). It is known (see reference [49]) that L(⋅) is a generator of CPTP maps if and only if

$$\begin{equation}{\bar{P}}_{\sigma }\left(\mathrm{I}\mathrm{d}\otimes L\right)\left(\sigma \right){\bar{P}}_{\sigma }{\geqslant}0,\end{equation} \tag{ 78 }$$

where $\sigma \in \mathfrak{S}\left(\mathcal{H}\otimes \mathcal{H}\right)$ is a maximally entangled state [see e.g. Equation (32)] and ${\bar{P}}_{\sigma }={I}_{{d}^{2}}-\sigma$. We then consider d⁴-dimensional vectors s and p_s with the following components:

$$\begin{equation}\begin{aligned}\hfill {s}_{\left(ij\right)}& =\enspace \mathrm{T}\mathrm{r}\left({E}_{i}\otimes {E}_{j}\sigma \right),\hfill \\ \hfill {\bar{p}}_{s,\left(ij\right)}& =\enspace \mathrm{T}\mathrm{r}\left({E}_{i}\right)\enspace \mathrm{T}\mathrm{r}\left({E}_{j}\right)-{s}_{\left(ij\right)}.\hfill \end{aligned}\end{equation} \tag{ 79 }$$

The probability representation of the expression in left-hand side of Equation (78) takes the form

$$\begin{equation}{\bar{p}}_{s}{\ast}\left({\mathbf{I}}_{{d}^{2}}\otimes \mathbf{D}\right)s{\ast}{\bar{p}}_{s}\equiv {p}_{\mathbf{D}}.\end{equation} \tag{ 80 }$$

Thus, one can employ the algorithm from section 2.2 to check that p_D corresponds to non-normalized state with respect to the MIC POVM E ⊗ E, and thus condition 78 is fulfilled.

Up to this point, we described evolution equation for states from the viewpoint of the Schrödinger picture. Here we show how to adapt the MIC-POVM-based probability representation to the Heisenberg picture, where measurement operators evolve. As a master equation, we consider GKSL equation from the previous section.

Consider a POVM $M={\left\{{M}_{i}\right\}}_{i=1}^{m}$. Remember, that in the MIC-POVM-based representation it is defined by m × d² pseudostocastic matrix M. The probability vector of measurement outcomes at time t ⩾ 0 is given by

$$\begin{equation}\mathbf{M}p\left(t\right)=\mathbf{M}{e}^{\mathbf{L}t}{p}_{0}.\end{equation} \tag{ 81 }$$

Now we can introduce a Heisenberg representation of M: M^Heis(t) := Me^Lt, that is solution of the equation

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}t}{\mathbf{M}}^{\text{Heis}}\left(t\right)={\mathbf{M}}^{\text{Heis}}\left(t\right)\mathbf{L},\quad {\mathbf{M}}^{\text{Heis}}\left(0\right)=\mathbf{M}\left(0\right).\end{equation} \tag{ 82 }$$

The probabilities of outcomes are given by q(t) = M^Heis(t)p₀ We note that Equation (83) can be also adapted to an evolution of a particular effect (row of M) instead of full matrix M. In the case of Hermitian observable one can write the following equation for an row-vector allowing to compute mean value of $\langle O\left(t\right)\rangle ={\mathbf{O}}_{\text{mean}}^{\text{Heis}}\left(t\right){p}_{0}$:

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}t}{\mathbf{O}}_{\text{mean}}^{\text{Heis}}\left(t\right)={\mathbf{O}}_{\text{mean}}^{\text{Heis}}\left(t\right)\mathbf{L},\quad {\mathbf{O}}_{\text{Heis}}^{\text{mean}}\left(0\right)={\mathbf{O}}_{\text{mean}}.\end{equation} \tag{ 83 }$$

Let us consider a process of quantum computations from the viewpoint of MIC-POVM based probability representation. Within this section we fix a single MIC-POVM E constructed with tensor product operation from SIC-POVM effects 90:

$$\begin{equation}E={\left\{{E}_{{i}_{1}}^{\text{sym}}\otimes \dots \otimes {E}_{{i}_{n}}^{\text{sym}}\right\}}_{{i}_{1},\dots ,{i}_{n}\in \left\{1,2,3,4\right\}}.\end{equation} \tag{ 102 }$$

The initial 4ⁿ-dimensional probability vector corresponding to |ψ^init⟩ then takes the form

$$\begin{equation}{p}^{\text{init}}={p}^{\left(0\right)}\otimes \dots \otimes {p}^{\left(0\right)},\end{equation} \tag{ 103 }$$

where

$$\begin{equation}{p}^{\left(0\right)}={\left[\begin{matrix}\hfill \frac{3+\sqrt{3}}{12}\frac{3+\sqrt{3}}{12}\frac{3-\sqrt{3}}{12}\frac{3-\sqrt{3}}{12}\hfill \end{matrix}\right]}^{\top }\end{equation} \tag{ 104 }$$

is probability vector corresponding to the state |0⟩. We note that in the considered probability representation each qubit corresponds to a classical two-bit string, and the whole n-qubit state can be considered as a probability distribution over all possible values of 2n-bit string.

We consider a representation of single-qubit and two-qubit gates. A pseudostochastic matrix of a single-qubit gate U₁ in the SIC-POVM representation reads:

$$\begin{equation}\mathbf{S}\left({U}_{1}\right)=3\mathbf{s}\left({U}_{1}\right)-2{\mathbf{J}}_{1},\end{equation} \tag{ 105 }$$

where

$$\begin{equation}\mathbf{s}{\left({U}_{1}\right)}_{ij}=2\mathrm{T}\mathrm{r}\left({E}_{i}^{\text{sym}}{U}_{1}{E}_{j}^{\text{sym}}{U}_{1}^{{\dagger}}\right),\quad i,j=1,2,3,4,\end{equation} \tag{ 106 }$$

and J₁ is a 4 × 4 matrix with all entities equal to 1/4. It is easy to see that s(U₁) is a bistochastic matrix (note that ρ_j can be considered as a fair quantum state). The matrix J₁ is also bistochastic and outputs maximally chaotic state for any input probability vector. Pseudostochastic matrices for some common single-qubit gates are provided in appendix A. We also demonstrate an application of the Hadamard gate to the state |0⟩ in figure 4.

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In the case of n ⩾ 2 qubits, the pseudostochastic matrix corresponding to an action of U₁ on a particular qubit can be obtained by tensor product with identity matrix (matrices). As an illustrative example consider the case where U₁ acts on the second qubit among n = 4 qubits. Since an absence of operation corresponds to identity stochastic matrix the resulting pseudostochastic matrix reads

$$\begin{equation}{\mathbf{I}}_{4}\otimes \mathbf{S}\left({U}_{1}\right)\otimes {\mathbf{I}}_{4}\otimes {\mathbf{I}}_{4}=3{\mathbf{I}}_{4}\otimes \mathbf{s}\left({U}_{1}\right)\otimes {\mathbf{I}}_{16}-2{\mathbf{I}}_{4}\otimes {\mathbf{J}}_{1}\otimes {\mathbf{I}}_{16}.\end{equation} \tag{ 107 }$$

We note that the right-hand side of Equation (107) is a difference of two scaled stochastic matrices.

We see that the probability vector resulting from the action of the considered pseudostochastic matrix appears to be a linear combination of two probability vectors obtained from the initial one by acting with different bistochastic matrices. The sum of coefficients of this linear combination is equal to one thus forming affine combinations. Due to the negative elements inside, it is very different from a convex hull typical for a classical randomization process.

The situation with a two-qubit entangling gate U₂ (e.g. CX or CZ) is a bit more complex. The corresponding pseudostochastic matrix reads

$$\begin{equation}\mathbf{S}\left({U}_{2}\right)=9{\mathbf{s}}_{\mathrm{I}}\left({U}_{2}\right)-12{\mathbf{s}}_{\text{II}}\left({U}_{2}\right)+4{\mathbf{J}}_{2},\end{equation} \tag{ 108 }$$

where

$$\begin{equation}\begin{aligned}\hfill {\mathbf{s}}_{\mathrm{I}}{\left({U}_{2}\right)}_{\left(ij\right),\left(kl\right)}& =4\mathrm{T}\mathrm{r}\left({E}_{i}^{\text{sym}}\otimes {E}_{j}^{\text{sym}}{U}_{2}{E}_{k}^{\text{sym}}\otimes {E}_{l}^{\text{sym}}{U}_{2}^{{\dagger}}\right),\hfill \\ \hfill {\mathbf{s}}_{\text{II}}{\left({U}_{2}\right)}_{\left(ij\right),\left(kl\right)}& =\mathrm{T}\mathrm{r}\left({E}_{i}^{\text{sym}}\otimes {E}_{j}^{\text{sym}}{U}_{2}{\tilde {\rho }}_{kl}{U}_{2}^{{\dagger}}\right),\hfill \\ \hfill {\tilde {\rho }}_{kl}& ={E}_{k}^{\text{sym}}\otimes {\rho }^{\text{mix}}+{\rho }^{\text{mix}}\otimes {E}_{l}^{\text{sym}},\quad i,j,k,l=1,2,3,4,\hfill \end{aligned}\end{equation} \tag{ 109 }$$

where ρ^mix = I₂/2 is a maximally mixed state, J₂ is a matrix with all entities equal to 1/16. One can check that all matrices s_I(U₂), s_II(U₂), and J₂ are bistochastic. We observe again a linear combination of stochastic matrices with coefficient giving a total of unity and having negative elements. As an example we show the construction of a two-node cluster state in figure 5. We also provide pseudostochastic matrices corresponding to different two-qubit gates in appendix A. The pseudostochastic matrix of two-qubit operation acting on a particular pair of n ⩾ 3 qubits can be obtained by employing tensor product with identity matrix (matrices) in a similar way as in the case of the single-qubit gate.

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The pseudostochastic matrix of the single-qubit projective measurement in the computation basis is given by the following expression:

$$\begin{equation}{\mathbf{M}}_{\text{pr}}=3{\mathbf{m}}_{\text{pr}}-2{\mathbf{J}}_{\text{pr}},\end{equation} \tag{ 110 }$$

where

$$\begin{equation}{\mathbf{m}}_{\text{pr}}=\frac{1}{2}\left[\begin{matrix}{cccc}\hfill 1+\frac{1}{\sqrt{3}}\hfill & \hfill 1+\frac{1}{\sqrt{3}}\hfill & \hfill 1-\frac{1}{\sqrt{3}}\hfill & \hfill 1-\frac{1}{\sqrt{3}}\hfill \\ \hfill 1-\frac{1}{\sqrt{3}}\hfill & \hfill 1-\frac{1}{\sqrt{3}}\hfill & \hfill 1+\frac{1}{\sqrt{3}}\hfill & \hfill 1+\frac{1}{\sqrt{3}}\hfill \end{matrix}\right]\end{equation} \tag{ 111 }$$

and J_pr is 2 × 4 stochastic matrix with all elements equal to 1/2. One can see that the form of Equation (110) is similar to Equation (105). However, the important difference is that within the projective measurement we have a reduction of probability space dimensionality. An example of the projective measurement of a state |1⟩ in the probability representation is shown in figure 6. In the case of n > 1 qubits the pseudostochastic matrix of the single-qubit measurement can be obtained in a similar fashion as in the case single-qubit gate [see example in Equation (107)] a pseudostochasitc matrix of the projective measurement of several qubits can be obtained as a sequence of measurements on individual qubits.

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We see that the running of n-qubit quantum circuit can be considered as a kind of random walk of 2n-bit string. In each step corresponding to an implementation of single-qubit or two-qubit quantum gates a single or a pair of 2 bit chunks are affected (each chunk consists of (2k)th and (2k + 1)th position in the string with k = 0, 1, ..., n − 1). The final step of readout measurement corresponds to a compressive random mapping of each 2 bit chunk, corresponding to measured qubit, to 1 bit value. The quantum nature of this randomized process manifests itself by the fact that all steps are described with pseudostochastic matrices given by Equations (105), (108) and (110). On the one hand the negative conditional probabilities of pseudostochastic matrices prevent us from straightforward emulating of quantum processes within quantum computation with classical randomized algorithms, and on the hand they actually underlie the advantage of quantum computers over classic ones.

In order to provide another illustrative example of the probabilistic representation of quantum computing processes, we consider a two-qubit Grover's algorithm [54] with a classical oracle function

$$\begin{equation}\chi \left(10\right)=1,\quad \chi \left(00\right)=\chi \left(01\right)=\chi \left(11\right)=0.\end{equation} \tag{ 112 }$$

The corresponding circuit which allows finding the 'secret string' 10 by a single query to the two-qubit quantum oracle

$$\begin{equation}{U}_{\chi }\vert xy\rangle ={\left(-1\right)}^{\chi \left(xy\right)}\vert xy\rangle ,\quad x,y\in \left\{0,1\right\}\end{equation} \tag{ 113 }$$

is shown in figure 7(a). The evolution of the probability representation of corresponding four-bit string is shown in figure 7(b). One can observe how the information about the secret string first gets into the state after applying the oracle and then is extracted with the diffusion gate and projective measurement.

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