Spin polarization-scaling quantum maps and channels

TL;DR

This paper introduces a new class of quantum maps for spin-$j$ particles that model the decrease of spin polarization, analyzing their positivity, entanglement-breaking properties, and differences across spin values.

Contribution

It defines spin polarization-scaling maps for arbitrary spin-$j$ particles and characterizes their positivity and entanglement properties, highlighting differences from spin-$rac{1}{2}$ cases.

Findings

Conditions for positivity and complete positivity are derived.

Maps can be entanglement breaking under certain parameters.

Specific analysis for spin-1 particles is provided.

Abstract

We introduce a spin polarization-scaling map for spin-$j$ particles, whose physical meaning is the decrease of spin polarization along three mutually orthogonal axes. We find conditions on three scaling parameters under which the map is positive, completely positive, entanglement breaking, 2-tensor-stable positive, and 2-locally entanglement annihilating. The results are specified for maps on spin-$1$ particles. The difference from the case of spin-$\frac{1}{2}$ particles is emphasized.

Equations

$$\Phi [X]=\frac{1}{2j+1}\mathrm{tr}[X]I+\frac{3}{j(j+1)(2j+1)}\sum _{i=1}^3{\lambda }_i\mathrm{tr}[X{J}_i]J_i,$$

$$p_i\mapsto {\lambda }_i{p}_i,i=1,2,3.$$

$$\frac{1}{2j+1}\left(\mathrm{tr}[X]-\frac{3}{j+1}\sqrt{\sum _{i=1}^3{\left({\lambda }_i\mathrm{tr}[X{J}_i]\right)}^2}\right).$$

$$X=x_{0}I+\sum _{i=1}^3{x}_i{J}_i,x_0⩾0,x_i\in \mathbb{R},x_0⩾j\sqrt{\sum _{i=1}^3{x}_i^2}.$$

$$I\otimes I+\frac{3}{j(j+1)}\left({\lambda }_1{J}_1\otimes J_1-{\lambda }_2{J}_2\otimes J_2+{\lambda }_3{J}_3\otimes J_3\right)⩾0.$$

$$(2j+1{)}^2(\Phi \otimes \mathrm{Id})[|{\psi }_{+}⟩⟨{\psi }_{+}|]$$

$$=\sum _{m,m^{\prime }=-j}^j\Phi [|jm⟩⟨j{m}^{\prime }|]\otimes |jm⟩⟨j{m}^{\prime }|$$

$$=\sum _{m=-j}^j\left(I+\frac{3m}{j(j+1)}{\lambda }_3{J}_3\right)\otimes |jm⟩⟨jm|$$

$$+\sum _{m=-j}^j\frac{3}{2j(j+1)}\sqrt{(j-m)(j+m+1)}$$

$$\times ({\lambda }_1{J}_1-i{\lambda }_2{J}_2)\otimes |m⟩⟨m+1|$$

$$+\sum _{m=-j}^j\frac{3}{2j(j+1)}\sqrt{(j+m)(j-m+1)}$$

$$\times ({\lambda }_1{J}_1+i{\lambda }_2{J}_2)\otimes |m⟩⟨m-1|$$

$$=I\otimes I+\frac{3}{j(j+1)}\left({\lambda }_1{J}_1\otimes J_1-{\lambda }_2{J}_2\otimes J_2+{\lambda }_3{J}_3\otimes J_3\right).$$

$$J_{1}=\frac{1}{2}\left(\begin{array}[]{cc}0&1\\ 1&0\\ \end{array}\right),\quad J_{2}=\frac{1}{2}\left(\begin{array}[]{cc}0&-i\\ i&0\\ \end{array}\right),\quad J_{3}=\frac{1}{2}\left(\begin{array}[]{cc}1&0\\ 0&-1\\ \end{array}\right)$$

$$\displaystyle J_{1}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\ 1&0&1\\ 0&1&0\\ \end{array}\right),\quad\ J_{2}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-i&0\\ i&0&-i\\ 0&i&0\\ \end{array}\right),$$

$$\displaystyle J_{3}=\left(\begin{array}[]{ccc}1&0&0\\ 0&0&0\\ 0&0&-1\\ \end{array}\right)$$

$$I\otimes I+\frac{3}{j(j+1)}\left({\lambda }_1{J}_1\otimes J_1\pm {\lambda }_2{J}_2\otimes J_2+{\lambda }_3{J}_3\otimes J_3\right)⩾0.$$

$$0⩽(2j+1{)}^2(\Phi \otimes \Phi )[|{\psi }_{+}⟩⟨{\psi }_{+}|]$$

$$=(2j+1{)}^2(\mathrm{Id}\otimes \Phi )\circ (\Phi \otimes \mathrm{Id})[|{\psi }_{+}⟩⟨{\psi }_{+}|]$$

$$\displaystyle=({\rm Id}\otimes\Phi)\bigg{[}I\otimes I+\frac{3}{j(j+1)}$$

$$\displaystyle\qquad\qquad\quad\times\left(\lambda_{1}J_{1}\otimes J_{1}-\lambda_{2}J_{2}\otimes J_{2}+\lambda_{3}J_{3}\otimes J_{3}\right)\bigg{]}$$

$$=I\otimes I+\frac{3}{j(j+1)}\left({\lambda }_1^2{J}_1\otimes J_1-{\lambda }_2^2{J}_2\otimes J_2+{\lambda }_3^2{J}_3\otimes J_3\right)$$

$$=(2j+1{)}^2({\Phi }^2\otimes \mathrm{Id})[|{\psi }_{+}⟩⟨{\psi }_{+}|],$$

Full text

Spin polarization, qubit, qutrit, positive map, quantum channel, entanglement breaking, 2-tensor-stable properties

I Introduction

Quantum states of a spin-$j$ particle are described by $(2j+1)\times(2j+1)$ density matrices $\varrho\in\mathcal{S}(\mathcal{H})$ satisfying the properties $\varrho^{{\dagger}}=\varrho$, ${\rm tr}[\varrho]=1$, and $\langle\varphi|\varrho|\varphi\rangle\geqslant 0$ for all $|\varphi\rangle\in\mathcal{H}$, ${\rm dim}\mathcal{H}=2j+1$. Taking into account the normalization condition, the density matrix $\varrho$ is defined by $(2j+1)^{2}-1$ real parameters, which are usually treated as components of the generalized Bloch vector petz-2009 ; checinska-2009 ; karimipour-2011 ; byrd-2011 ; goyal-2016 . However, many physical phenomena can be explained and visualized via a spin polarization vector ${\bf p}\in\mathbb{R}$ with components $p_{i}={\rm tr}[\varrho J_{i}]$, where $J_{1},J_{2},J_{3}$ are usual $(2j+1)$-dimensional representations of angular momentum operators (see, e.g., varshalovich ). Angular momentum operators are Hermitian and satisfy the commutation relation $[J_{k},J_{l}]=ie_{klm}J_{m}$, where $e_{klm}$ is the conventional Levi-Civita symbol and the summation over $m$ being assumed. Note that the spin-polarization vector ${\bf p}$ does not contain the full information about the quantum state if $j\geqslant 1$. Despite this fact, it is of great use in quantum physics and chemistry as its components represent average spin projections onto three orthogonal axes and are experimentally measurable. Linear transformations of the spin polarization vector include rotations and scaling. Rotations are attributed to the unitary evolution, so we do not consider them in the present paper. Physically motivated scaling of the spin polarization vector is described by a map $\Phi:\mathcal{B}(\mathcal{H})\mapsto\mathcal{B}(\mathcal{H})$ of the following form:

[TABLE]

where $I\in\mathcal{B}(\mathcal{H})$ is the identity operator and $\lambda_{i}\in\mathbb{R}$. The factors take into account that ${\rm tr}[I]=2j+1$ and ${\rm tr}[J_{k}J_{l}]=\frac{1}{3}j(j+1)(2j+1)\delta_{kl}$, $\delta_{kl}$ is the Kronecker delta. The map (1) is trace-preserving and unital, i.e. ${\rm tr}\left[\Phi[X]\right]={\rm tr}[X]$ and $\Phi[I]=I$. Note that the map (1) differs in general from other classes of unital maps nathanson-2007 ; landau-1993 . Physical meaning of Eq. (1) is the transformation of the spin polarization

[TABLE]

In case of spin-$\frac{1}{2}$ particles, formula (1) transforms into a well-known Pauli qubit map $\Phi[X]=\frac{1}{2}\left({\rm tr}[X]I+\sum_{i=1}^{3}\lambda_{i}{\rm tr}[X\sigma_{i}]\sigma_{i}\right)$, where $(\sigma_{1},\sigma_{2},\sigma_{3})$ is the conventional set of Pauli matrices (see, e.g., nielsen-chuang ; heinosaari-ziman ). The qubit ($j=\frac{1}{2}$) map $\Phi$ is known to be positive if and only if $|\lambda_{i}|\leqslant 1$, completely positive if and only if $1\pm\lambda_{3}\geqslant|\lambda_{1}\pm\lambda_{2}|$, entanglement breaking if and only if $|\lambda_{1}|+|\lambda_{2}|+|\lambda_{3}|\leqslant 1$, 2-local-entanglement-annihilating if and only if $\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}\leqslant 1$, 2-tensor-stable positive if and only if $1\pm\lambda_{3}^{2}\geqslant|\lambda_{1}^{2}\pm\lambda_{2}^{2}|$ filippov-rybar-ziman-2012 ; filippov-2014 ; filippov-magadov . Similar characterization for higher spins ($j\geqslant 1$) is still missing, so the goal of the present paper is to analyze analogous properties of such maps and illustrate them for qutrits ($j=1$).

The paper is organized as follows. In Sec. II, we analyze positivity of the map (1). In Sec. III, the criterion of complete positivity of such a map is presented. In Sec. IV, the entanglement breaking property is partially characterized. In Sec. V, we review the positivity and entanglement annihilation behaviour of the map $\Phi\otimes\Phi$. In Sec. VI, brief conclusions are presented.

II Positivity

We will refer to an operator $R$ as positive-semidefinite and write $R\geqslant 0$ if $\langle\varphi|R|\varphi\rangle\geqslant 0$ for all $|\varphi\rangle\in\mathcal{H}$. A map $\Phi$ is called positive if $\Phi[X]\geqslant 0$ for all $X\geqslant 0$ stormer-1963 .

Let us now analyze positivity of the spin polarization-scaling map (1).

Since each $J_{i}$ is a spin projection operator with eigenvalues $j,j-1,\ldots,-j$, eigenvalues of the operator $a_{1}J_{1}+a_{2}J_{2}+a_{3}J_{3}$ are $|{\bf a}|\{j,j-1,\ldots,-j\}$, where $|{\bf a}|=\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$. Therefore, the minimal eigenvalue of $\Phi[X]$ reads

[TABLE]

Suppose $X\geqslant 0$. As $|{\rm tr}[XJ_{i}]|\leqslant j{\rm tr}[X]$, the minimal value of (3) is non-negative if $1-\frac{3j}{j+1}\sqrt{\sum_{i=1}^{3}\lambda_{i}^{2}}\geqslant 0$. Thus, we have found sufficient condition for positivity of the map $\Phi$.

Proposition 1**.**

Spin-polarization-scaling map $\Phi$ is positive if $\sum_{i=1}^{3}\lambda_{i}^{2}\leqslant\left(\frac{j+1}{3j}\right)^{2}$.

The necessary condition for positivity of the map (1) follows from the particular form of the positive-semidefinite operator

[TABLE]

In fact, if $X$ is given by formula (4), then $\Phi[X]=x_{0}I+\sum_{i=1}^{3}\lambda_{i}x_{i}J_{i}$ and $\Phi[X]\geqslant 0$ if and only if $x_{0}\geqslant j\sqrt{\sum_{i=1}^{3}\lambda_{i}^{2}x_{i}^{2}}$. Suppose $x_{0}=jx_{1}$ and $x_{2}=x_{3}=0$, then $\Phi[X]\geqslant 0$ if $|\lambda_{1}|\leqslant 1$. Similarly, necessary conditions $|\lambda_{2}|\leqslant 1$ and $|\lambda_{3}|\leqslant 1$ appear for choices $x_{0}=jx_{2}$, $x_{1}=x_{3}=0$ and $x_{0}=jx_{3}$, $x_{1}=x_{2}=0$, respectively.

Proposition 2**.**

Suppose the spin polarization-scaling map $\Phi$ is positive, then $|\lambda_{i}|\leqslant 1$, $i=1,2,3$.

III Complete positivity

A linear map $\Phi$ is called completely positive if $\Phi\otimes{\rm Id}_{k}$ is positive for all $k=1,2,\ldots$. Here ${\rm Id}_{k}$ is the identity transformation of $k$-dimensional operators $\mathcal{B}(\mathcal{H}_{k})$.

Proposition 3**.**

The spin polarization-scaling map $\Phi$ is completely positive if and only if

[TABLE]

Proof.

A linear map $\Phi:\mathcal{B}(\mathcal{H}_{d})\mapsto\mathcal{B}(\mathcal{H}_{d})$ is known to be completely positive if and only if its Choi matrix $\Omega_{\Phi}=(\Phi\otimes{\rm Id}_{d})[|\psi_{+}\rangle\langle\psi_{+}|]$ is positive-semidefinite choi-1975 (see also pillis-1967 ; jamiolkowski-1972 ; jiang-2013 ; majewski-2013 ), where $|\psi_{+}\rangle=\frac{1}{\sqrt{d}}\sum_{i=1}^{d}|i\rangle\otimes|i\rangle$ is the maximally entangled state and $\{|i\rangle\}_{i=1}^{d}$ is some orthonormal basis in $\mathcal{H}_{d}$.

For our construction of the Choi operator let us choose eigenvectors of the operator $J_{3}$ as the basis, namely, $J_{3}|jm\rangle=m|jm\rangle$ and ${\langle jm^{\prime}|}{jm\rangle}=\delta_{m^{\prime}m}$, $d=2j+1$. Introduce auxiliary operators $J_{\pm}=J_{1}\pm iJ_{2}$, then $J_{\pm}|jm\rangle=\sqrt{(j\mp 1)(j\pm m+1)}|jm\pm 1\rangle$. Some algebra yields

[TABLE]

Thus, $\Phi$ is completely positive if and only if the operator (III) is positive-semidefinite. ∎

Example 1**.**

If $j=\frac{1}{2}$, then angular momentum operators are given by matrices

[TABLE]

in the basis $\{|\frac{1}{2}\frac{1}{2}\rangle,|\frac{1}{2}-\frac{1}{2}\rangle\}$. The condition (5) reduces to $1\pm\lambda_{3}\geqslant|\lambda_{1}\pm\lambda_{2}|$. Geometrically, these inequalities correspond a tetrahedron with vertices $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, and $(-1,-1,1)$ in the parameter space $(\lambda_{1},\lambda_{2},\lambda_{3})$ ruskai-2002 .

Example 2**.**

If $j=1$, then angular momentum operators are given by matrices

[TABLE]

in the basis $\{|11\rangle,|10\rangle,|1-1\rangle\}$. The condition (5) reduces to $4-9(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2})+27\lambda_{1}\lambda_{2}\lambda_{3}\geqslant 0$, $|\lambda_{i}|\leqslant\frac{2}{3}$, $i=1,2,3$. Geometrical figure corresponding to such inequalities is depicted in Fig. 1(a).

IV Entanglement breaking

A positive-semidefinite operator $R\in\mathcal{B}(\mathcal{H}_{1})\otimes\mathcal{B}(\mathcal{H}_{2})$ is called separable (non-entangled) if there exist positive-semidefinite operators $R_{1}^{(k)}$ and $R_{2}^{(k)}$ such that $R=\sum_{k}R_{1}^{(k)}\otimes R_{2}^{(k)}$ werner-1989 ; horodecki-2009 . A linear map $\Phi:\mathcal{H}_{1}\mapsto\mathcal{H}_{1}$ is called entanglement breaking if $(\Phi\otimes{\rm Id})[\varrho]$ is separable for all $\varrho\in\mathcal{S}(\mathcal{H}_{1}\otimes\mathcal{H}_{2})$ and identity transformation ${\rm Id}:\mathcal{H}_{2}\mapsto\mathcal{H}_{2}$ holevo-1998 ; king-2002 ; shor-2002 ; ruskai-2003 ; horodecki-2003 . The well-known result is that $\Phi$ is entanglement breaking if and only if the Choi matrix $\Omega_{\Phi}$ is separable.

The necessary condition for separability of $\Omega_{\Phi}$ is that $\Omega_{\Phi}^{\Gamma}\geqslant 0$, where $X^{\Gamma}=\sum_{j,j^{\prime}}I\otimes|j^{\prime}\rangle\langle j|XI\otimes|j^{\prime}\rangle\langle j|$ is the partially transposed operator, $X\in\mathcal{B}(\mathcal{H}_{1})\otimes\mathcal{B}(\mathcal{H}_{2})$ peres-1996 ; horodecki-1996 . Applying such a condition to the Choi matrix (III) and taking into account that in conventional basis $\{jm\}_{m=-j}^{j}$ the matrices $J_{x}^{\top}=J_{x}$, $J_{y}^{\top}=-J_{y}$, $J_{z}^{\top}=J_{z}$, we obtain the following result.

Proposition 4**.**

Suppose the spin polarization-scaling map $\Phi$ is completely positive and entanglement breaking, then

[TABLE]

Note that the requirement (19) is sufficient for the channel $\Phi$ to be entanglement binding horodecki-2000 .

Example 3**.**

If $j=\frac{1}{2}$, then Eq. (19) is equivalent to $|\lambda_{1}|+|\lambda_{2}|+|\lambda_{3}|\leqslant 1$.

Example 4**.**

If $j=1$, then Eq. (19) is equivalent to $4-9(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2})\pm 27\lambda_{1}\lambda_{2}\lambda_{3}\geqslant 0$.

V 2-tensor-stable properties

Some properties of linear maps do not change under tensoring the map with itself, for instance, $\Phi\otimes\Phi$ is completely positive if and only if $\Phi$ is completely positive. Similarly, $\Phi\otimes\Phi$ is entanglement breaking if and only if $\Phi$ is entanglement breaking (see, e.g., filippov-melnikov-ziman-2013 ). However, other properties of a map can change drastically under tensor power. For example, the map $\Phi\otimes\Phi$ can be non-positive even if $\Phi$ is positive filippov-magadov .

A linear map $\Phi$ is called 2-tensor-stable positive if $\Phi\otimes\Phi$ is positive muller-hermes-2016 .

Example 5**.**

It is shown in Ref. filippov-magadov that the spin polarization-scaling map $\Phi$ given by Eq. (1) for qubits ($j=\frac{1}{2}$) is 2-tensor-stable positive if and only if $\Phi^{2}$ is completely positive, i.e. $1\pm\lambda_{3}^{2}\geqslant|\lambda_{1}^{2}\pm\lambda_{2}^{2}|$.

For higher spins ($j\geqslant 1$) the result of Example 5 can be extended as follows.

Proposition 5**.**

If the spin polarization-scaling map $\Phi$ is 2-tensor-stable positive, then $\Phi^{2}$ is completely positive.

Proof.

Consider a positive-semidefinite operator $|\psi_{+}\rangle\langle\psi_{+}|$, where $|\psi_{+}\rangle=(2j+1)^{-1/2}\sum_{m=-j}^{j}|jm\rangle\otimes|jm\rangle$. The action of the positive map $\Phi\otimes\Phi$ on such an operator reads

[TABLE]

i.e. the Choi matrix $\Omega_{\Phi^{2}}$ is positive-semidefinite and $\Phi^{2}$ is completely positive. ∎

In contrast to the case $j=\frac{1}{2}$, for higher spins ($j\geqslant 1$) Proposition 5 provides the necessary condition only. For instance, it is not hard to see that for $j=1$ there exists a spin polarization-scaling map $\Phi$ such that $\Phi^{2}$ is completely positive but $(\Phi\otimes\Phi)[|\varphi\rangle\langle\varphi|]\ngeqslant 0$ for the Schmidt-rank-2 state $|\varphi\rangle=\frac{1}{\sqrt{2}}\left(|11\rangle+|1-1\rangle\right)$.

In the case $j=1$, the map $\Phi^{2}$ is completely positive if and only if $4-9(\lambda_{1}^{4}+\lambda_{2}^{4}+\lambda_{3}^{4})+27\lambda_{1}^{2}\lambda_{2}^{2}\lambda_{3}^{2}\geqslant 0$, which is depicted in Fig. 1(b).

A linear map $\Phi:\mathcal{B}(\mathcal{H})\mapsto\mathcal{B}(\mathcal{H})$ is called 2-locally entanglement annihilating moravcikova-ziman-2010 ; filippov-rybar-ziman-2012 ; filippov-ziman-2013 ; filippov-ziman-2014 if $(\Phi\otimes\Phi)[|\psi\rangle\langle\psi|]$ is separable for all $|\psi\rangle\in\mathcal{H}\otimes\mathcal{H}$. The same line of reasoning as for 2-tensor-stable positive maps leads to the following result.

Proposition 6**.**

If the spin polarization-scaling map $\Phi$ is 2-locally entanglement annihilating, then $\Phi^{2}$ is entanglement breaking.

VI Conclusions

We have considered the physically motivated sets of operator maps for spin systems. The physical meaning of such maps is the degradation of spin polarization with scaling parameters $\lambda_{1},\lambda_{2},\lambda_{3}$ along the axes $x,y,z$, respectively. We have found conditions (necessary, or sufficient, or both) under which the spin polarization-scaling map is positive, completely positive, entanglement breaking, 2-tensor-stable positive, 2-locally entanglement annihilating. These results can be of use in the analysis of data, where only spin polarization degrees of freedom are available. The crucial difference between the cases of spin-$\frac{1}{2}$ and spin-$1$ particles is illustrated in a series of examples.

Acknowledgements.

The study is supported by Russian Science Foundation under project No. 16-11-00084.