On some classes of bipartite unitary operators

Introduction

In this work, we answer the question above for several classes $\mathcal{L}$ of relevance in quantum information theory: unitary conjugations $V\cdot V^{*}$, constant channels ($L(\rho)$ does not depend on the quantum state $\rho$), unital channels, ($L(I)=I$), mixed unitary channels (convex combinations of unitary conjugations), PPT channels (channels for which the Choi matrix has a positive semidefinite partial transpose), and entanglement breaking channels (channel which, acting on one half of any entangled state, yield separable states). In particular within our work, we give a partial positive answer to the conjecture formulated in . In , it was conjectured that convex combination of unitary conjugations can be obtained only by a very specific class of bipartite unitary operators (the set $\mathcal{U}_{mixed}$ in the present work). We show that this conjecture is true, under some additional assumptions.

It is very important to state at this time that we are not concerned with classes of channels, but with classes of bipartite unitary operators. Although these classes are defined in terms of channels, we are interested in characterizing the “interaction” unitaries $U$ with the property that, for all ancilla states $\beta$, the channel $L$ given by (1) has some fixed set of properties. A similar question was studied in , where the authors characterize the unitary operators $U$ having the property that the only matrices $\beta$ which give quantum channels in (1) are quantum states.

The paper is structured as follows. In Section 1 we define the classes of unitary operators we are interested in, and we present some general properties. Sections 2 - 6 deal each with one or more of these classes. In Section 7, we collect some more relations between the different classes, and present some equality cases. We close the work with a section containing some open problems. Finally, in Appendix A, we discuss the block singular value decomposition of operators.

Acknowledgements. We would like to thank Stéphane Attal for suggesting the question, and Denis Bernard and Michael Kech for inspiring discussions. We are also grateful to Siddharth Karumanchi for pointing out to us reference and sharing with us the unpublished notes . I.N.’s research has been supported by a von Humboldt fellowship and by the ANR projects OSQPI 2011 BS01 008 01 and RMTQIT ANR-12-IS01-0001-01. The three authors have been supported by the ANR project StoQ ANR-14-CE25-0003-01.

Some classes of unitary operators

Let us introduce some classes of quantum channels having the unitary invariance property:

We present next a list of such classes of channels, leaving the task of verifying the unitary invariance property as an exercise for the reader:

Positive partial transpose (PPT) channels

We move next to the main definition of this paper, the classes of bipartite unitary channels we are interested in. These classes are defined in a natural way as the set of unitary operators inducing, via the Stinesping formula (2), independent of the state $\beta$ of the environment system $B$, quantum channels belonging to one of the classes above. This exact notion, in the case of degradable channels and entanglement breaking channels, has been considered respectively in [14, Definition 15] and . More precisely, we define, for any $*\in\{aut,const,unital,mixed,PPT,EB\}$,

In relation to the class $\mathcal{U}_{aut}$, we also define (see Section 2)

One of the original motivations of this work was to obtain a characterization of the set $\mathcal{U}_{mixed}$. As stepping stones towards a description of this set, we introduce the following classes of bipartite unitary operators:

Let us mention at this point a simple but fundamental property of the sets of bipartite unitary matrices we have just introduced.

The local unitary group $\mathcal{U}_{n}\times\mathcal{U}_{k}$ acts by left and right multiplication on $\mathcal{U}_{*}$, for all $*\in\{aut,const,unital,mixed,PPT,EB,single,prob,prob-lin,block-diag-A,block-diag-B\}$:

As suggested by the property above, the subgroup $\mathcal{U}_{n}\otimes\mathcal{U}_{k}\subseteq\mathcal{U}_{nk}$ plays an important role in our study. It turns out that the flip operator

will also be of particular importance, in light of the following classical theorem.

([8, Theorem 3.1]) Let $G$ be a compact group such that $\mathcal{U}_{n}\otimes\mathcal{U}_{k}\subseteq G\subseteq\mathcal{U}_{nk}$. Then $G$ is one of the following

$G=\mathcal{U}_{n}\otimes\mathcal{U}_{k}$;

If $n=k$, $G=\langle\mathcal{U}_{n}\otimes\mathcal{U}_{n},F_{n}\rangle$.

By direct computation, one can show that the following chain of inclusions holds :

Note that one can define “$B$”-versions of the above sets, in an obvious way, by swapping the tensor factors $A$ and $B$ ( above inclusions are still true for $\mathcal{U}^{B}_{*}$).

One of the main focuses of the current work will be to understand which are the inclusions above which are strict and which are actually equalities.

Let $U,V\in\mathcal{U}_{nk}$ be two bipartite unitary operators. The following two assertions are equivalent:

There exists a unitary operator $W\in\mathcal{U}_{k}$ such that $U=(I\otimes W)V$.

We only prove “1 $\implies$ 2”, since the converse follows from direct calculation. We start form the hypothesis,

and we conclude the proof by noticing that $W$ has to be unitary, since $UV^{*}$ is. ∎

Bipartite unitary operators producing unitary conjugations and constant channels

In this section we provide characterizations of the sets $\mathcal{U}_{aut}$ (3), $\mathcal{U}_{single}$ (9), and $\mathcal{U}_{const}$ (4), showing that only tensor products of unitary operators (resp. flipped tensor products) belong to these classes.

We start with the set of automorphisms, that is the set of unitary conjugation channels.

The proof is easy, and consists of two steps: we show first that the unitary operators $V_{\beta}$ can be chosen to not depend on $\beta$, and then, using Lemma 1.3, we show that the unitary $U$ has the required form.

Let us first introduce some notation. To any matrix

At the level of Choi matrices, the above equation reads

On the left side of the equation above the operator is a rank one operator. This way, in order that the sum in the right side is a rank one operator, the vector has to be proportional. Since all the involved vectors are of norm $\sqrt{n}$, it follows that they should all be the same, up to a phase. The same holds for the unitary operators, which concludes the first step of the proof.

We show next that the class $\mathcal{U}_{single}$, i.e. the set of unitary operators $U\in\mathcal{U}_{nk}$ with the property that the map $\beta\mapsto L_{U,\beta}$ is constant, is actually identical to $\mathcal{U}_{aut}$.

For the proof of Theorem 2.3, we need the following lemma.

The non-trivial implication follows from the following equation

By using linearity, the hypothesis translates to the following equality

By reshaping the operator $U$, the previous equality can be re-written as

we get $U=V\otimes W$. The fact that $V\in\mathcal{U}_{n}$ follows from the unitarity of $U$, and this concludes the proof of the first implication. The fact that tensor product unitary operators belong to $\mathcal{U}_{single}$ can be verified by direct computation. ∎

For the case of $\mathcal{U}_{const}$, it is easy to see that it is related to the previous case via a flip operation, although there is a slight technical complication. This question has also been considered in .

If $k\neq rn$ for $r=1,2,\ldots$, then $\mathcal{U}_{const}$ is empty.

If $k=rn$ for some positive $r$, then we have

where $F_{n}\in\mathcal{U}_{n^{2}}$ denotes the flip operator (14); see Figure 2 for the diagrammatic representation of such an operator.

Let us start with the easy implication, considering an operator $U$ as in (16). By direct computation, one can see that

proving that, for arbitrary $\beta$, the channel $L_{U,\beta}$ is constant.

The proof of the difficult implication starts in the same way as the one of Theorem 2.3. The hypothesis that the channels $L_{U,\beta}$ are constant translates to the following condition:

If $n=k$ in the previous result, equation (16) can be written as $U=F_{n}(V\otimes W)$ for a pair of unitary operators $V,W\in\mathcal{U}_{n}$, so in this case we have

Bipartite unitary operators producing unital channels

In this section we study the set $\mathcal{U}^{A}_{unital}$ of bipartite unitary operations which yield unital channels for every choice of the state on the auxiliary space.

Using linearity, one can extend the definition (5) to the whole space of $k\times k$ complex matrices:

In particular, $\mathcal{U}^{B}_{unital}=\mathcal{U}^{A}_{unital}$.

Choosing $B=e_{i}e_{j}^{*}$, we get $W_{ij}=\delta_{ij}I_{n}$ and hence $W=I_{nk}$. In other words, $V=U^{\Gamma}\in\mathcal{U}_{nk}$, which finishes the proof of the first inclusion.

The second inclusion follows by working backwards the previous arguments: since $V=U^{\Gamma}\in\mathcal{U}_{nk}$, equations (20) and (19) hold. ∎

Since both sets $\mathcal{U}_{nk}$ and $\mathcal{U}_{nk}^{\Gamma}$ are algebraic varieties (i.e. they can be describes as the zero-set of a system of polynomial equations), we obtain the following corollary.

The set $\mathcal{U}_{unital}$ is a real algebraic variety.

We are going to investigate next $\mathcal{U}_{unital}$, as an algebraic variety. We compute first the dimension of the “enveloping tangent space” of $\mathcal{U}_{unital}$ at a point which is a block-diagonal unitary

The notion of enveloping tangent spaces was introduced in (also called defect), and it is simply defined by (see also )

The dimension of the enveloping tangent space of $\mathcal{U}_{unital}$ at a point which is a block-diagonal unitary of the form

where $\Lambda_{ij}$ is the set $\{(\lambda_{x},d_{x})\}$ where $\lambda_{x}$ is an eigenvalue of the unitary operator $U_{i}U_{j}^{*}$ having multiplicity $d_{x}$.

the first condition $UA^{*}+AU^{*}=0$ is equivalent to the following system of equations

while the condition $U(A^{\Gamma})^{*}+A^{\Gamma}U^{*}=0$ is equivalent to

First, let us note that the diagonal blocks $A_{ii}$ appear only in two identical equations

The general solution to the equation above is $A_{ii}=B_{ii}U_{i}$, where $B_{ii}$ is an arbitrary anti-hermitian matrix ($B_{ii}+B_{ii}^{*}=0$). Hence, the total dimension of the diagonal blocks of $A$ is $kn^{2}$. Note that this corresponds to the case $i=j$ in formula (21): there, $\Lambda_{ii}=\{(1,n)\}$.

Let us now study off-diagonal blocks of $A$. Again, the equations are decoupled: for $i<j$, one has to solve

From the first equation, one finds $A_{ji}=-U_{j}A_{ij}^{*}U_{i}$. Plugging this into the second equation, we have to solve now

where $R=U_{i}U_{j}^{*}$ and $S=U_{j}^{*}U_{i}$. This is the well-known Sylvester equation. From the analysis in [10, Chapter VIII], the dimension of the solution space of this homogenous equation depends of the Jordan block structure of the matrices $R$ and $S$. Since in our case both $R$ and $S$ are unitary (hence diagonalizable), the Jordan blocks have unit dimension. Moreover, $R$ and $S$ have the same spectrum $\Lambda_{ij}$. It follows from [10, Chapter VIII, eq. (19)] that the complex dimension of the solutions of the system (22)-(23) is precisely

The proof above can be adapted mutatis mutandis to the case of $(B)$-classical unitary operators, as follows.

The same result holds for $B$-block-diagonal unitary operators of the from

The dimension of the enveloping tangent space of $\mathcal{U}_{unital}$ at a product unitary operator $U=V\otimes W$ is $n^{2}k^{2}$, which is also the dimension of $\mathcal{U}_{nk}$.

For $k=2$, the dimension of the enveloping tangent space of $\mathcal{U}_{unital}$ at a point $U=I\oplus V=I\otimes e_{1}f_{1}^{*}+V\otimes e_{2}f_{2}^{*}$ is

where $d_{\lambda}$ are the multiplicities of the eigenvalues $\lambda$ of $V$.

Consider a block-diagonal unitary operator

where the operators $U_{i}$ are in generic position:

The dimension of the enveloping tangent space of $\mathcal{U}_{unital}$ at $U$ is then

Note that the expression above is symmetric in $n$ and $k$.

We conjecture that the expression (24) is the dimension of $\mathcal{U}_{unital}$, as an algebraic variety, see Conjecture 8.1.

Bipartite unitary operators producing PPT channels

We consider in this section $PPT$ channels and bipartite unitary operators which produce such channels via the Stinespring formula, independent of the state of the environment.

Recall that the maximally entangled state is the matrix (here, we drop the normalization constant)

A quantum channel $L$ is said to be PPT if and only if its Choi matrix

is PPT, i.e. $C_{L}^{\Gamma}\geq 0$. Hence, the set $\mathcal{U}_{PPT}$ admits the following characterization:

Since the structure of positive maps between matrix algebras is rather poorly understood, we focus for the moment on a subset of $\mathcal{U}_{PPT}$, namely

We have the following description of the set $\mathcal{U}_{CPPT}$, in which, remarkably, the partial transpose of $U$ plays a special role.

We use again the fact that complete positivity is characterized by the fact that the Choi matrix is positive semidefinite. In Figure 4, we have depicted in the left image the matrix $C_{L_{U,\beta}}^{\Gamma}$, while in the center panel we have the Choi matrix of the map $\beta\mapsto C_{L_{U,\beta}}^{\Gamma}$. The right-most panel contains the diagram of the same Choi matrix, where we have replaced $U$ by its partial transpose $U^{\Gamma}$, in order to obtain a nicer expression. The equality of the last two panels contains the proof of the claim. ∎

In order to further simplify the description given above, by conjugating the above expression by the pseudo-inverse of the matrix $U^{\Gamma}$, we are focusing next on the study of the set

We have gathered the following properties of the set $\mathcal{P}_{CPPT}$; we leave the proofs of these simple facts to the reader.

$I_{nk}\notin\mathcal{P}_{CPPT}$. In other words, no product unitary lies inside $\mathcal{U}_{CPPT}$, nor inside $\mathcal{U}_{PPT}$. As a consequence, we have $\mathcal{U}_{unital}\cap\mathcal{U}_{PPT}=\emptyset$.

Since $\mathcal{U}_{const}\subseteq\mathcal{U}_{PPT}$, if $k=nr$, for any unitary operator $V\in\mathcal{U}_{nr}$, $P_{V}\in\mathcal{P}_{CPPT}$, where $P_{V}$ is depicted in Figure 5.

At the level of examples, the only observation here is that $\mathcal{U}_{const}\subseteq\mathcal{U}_{PPT}$. We refer the reader to Section 8 for some related open problems.

Bipartite unitary operators producing mixed unitary channels

In this section we investigate the set $\mathcal{U}_{mixed}$. We provide necessary conditions for a bipartite unitary operator $U$ to belong to $\mathcal{U}_{mixed}$, and we show that in the case of qubit channels ($n=2$), the sets $\mathcal{U}_{mixed}$ and $\mathcal{U}_{unital}$ are equal.

For any choice of $f$ and $\{e_{i}\}$, the operators

are Kraus operators for the channel $L_{U,ff^{*}}$. Since the channel is mixed unitary, it follows from [22, Section IV] that the linear span of the $E_{i}$ should contain a unitary operator. ∎

Note that in the statement above, the set

does not depend on the particular choice of the basis $\{e_{i}\}$, but only on the vector $f$.

As a direct consequence of the above result, we obtain the following simple criterion for deciding if a given unitary matrix $U$ is an element of $\mathcal{U}_{mixed}$.

Let $U\in\mathcal{U}_{nk}$ be a bipartite unitary operator with the following property:

With the help of the criterion above, we present next an example of an element $U\in\mathcal{U}^{B}_{block-diag}\setminus\mathcal{U}_{mixed}$, which shows, in particular, that the inclusion $\mathcal{U}_{mixed}\subset\mathcal{U}_{unital}$ is strict; this example is motivated by [16, Section 4.3] and [22, Example 1]. Let $U\in\mathcal{U}_{4\cdot 2}$ be

Obviously, $U\in\mathcal{U}^{B}_{cl}$. In the spirit of the criterion above, compute

Asking for the diagonal matrix above to be unitary leads to a contradiction, and thus, by Corollary 5.3, we conclude $U\notin\mathcal{U}_{mixed}$.

Let us now consider the qubit case $n=2$, which is special because the quantum Birkhoff result holds for qubits .

As a corollary, since every unital channel $L_{U,\beta}$ must be mixed unitary, we obtain the following result.

For $n=2$ and any $k\geq 2$, we have that

In particular. $U\in\mathcal{U}_{mixed}$ iff $U\in\mathcal{U}\cap\mathcal{U}^{\Gamma}$.

Block-diagonal bipartite unitary operations

In this section we study the set of block-diagonal operators, $\mathcal{U}_{block-diag}^{A,B}$. Before proving any results on this class, let us provide another way of writing equation (12), which has the benefit of being unique in a certain sense. As a corollary, we deduce that the only unitary transformations which are blcok-diagonal with respect to both sub-systems $A$ and $B$ are given by partial isometries.

A bipartite unitary transformation $U\in\mathcal{U}$ is an element of $\mathcal{U}_{block-diag}^{A}$ if and only if it can be written as

and $U_{i}\nsim U_{j}$ for all $i\neq j$.

Moreover, the decomposition (26) is unique, up to $\sim$ and permutation of the terms in the sum.

Consider two decompositions of a same operator in $\mathcal{U}_{block-diag}^{A}$ of the form of (26)

$U_{p}\nsim U_{q}$ for all $p\neq q$ and $V_{l}\nsim V_{m}$ for all $l\neq m$.

For all $i$ in $\{1,\dots,r\}$ and $j$ in $\{1,\dots,s\}$, applying $I\otimes R_{i}^{*}$ on the left and $I\otimes Q_{j}^{*}Q_{j}$ on the right, Equation (28) becomes

This implies that at least one of the terms in the sum is non-trivial. Moreover, since $V_{l}\nsim V_{m}$ for all $l\neq m$, the operator $U_{i}$ can be in relation with only one of the $V_{j}$’s. Therefore, we obtain $r=s$ and for all $i$, there exist a unique $j$ such that $U_{i}=\lambda_{ij}\,V_{j}$ and $R_{i}^{*}R_{i}Q_{j}^{*}Q_{j}=1/\lambda_{ij}\,R_{i}^{*}Q_{j}$. After following the same strategy with $I\otimes Q_{j}Q_{j}^{*}$ on the left and $I\otimes R_{i}^{*}$ on the right, we now can deduce that $R_{i}=1/\lambda_{ij}Q_{j}$. The result follows ∎

Another point of view on block-diagonal unitaries is the fact captured in the next proposition.

A bipartite unitary operation is block-diagonal if and only if it admits a block-singular value decomposition with respect to $B$:

where $\mathcal{M}_{block-diag}^{A}$ is the set (see Appendix A)

In particular, the sets $\mathcal{U}_{block-diag}^{A}$, $\mathcal{U}_{block-diag}^{B}$, and $\mathcal{U}_{block-diag}^{A}\cap\mathcal{U}_{block-diag}^{B}$ are algebraic varieties.

For $U\in\mathcal{M}_{block-diag}^{A}$, write

The above matrix is the identity if and only if each of its diagonal blocks $X_{i}X_{i}^{*}$ is the identity, and the claim follows. ∎

Let us now investigate the relation between the two classes $\mathcal{U}_{block-diag}^{A}$ and $\mathcal{U}_{block-diag}^{B}$. We start by presenting an algorithm allowing to check if a unitary matrix $U$ in $\mathcal{U}_{nk}$ belongs to $\mathcal{U}_{block-diag}^{A}$. This key result relies on Theorem A.1.

if and only if the families $\{X_{\alpha}X_{\beta}^{*}\}_{\alpha,\beta=1}^{n^{2}}$, resp. $\{X_{\alpha}^{*}X_{\beta}\}_{\alpha,\beta=1}^{n^{2}}$, consist of commuting, normal operators. Then the unitarity of the $(U_{i})_{i=1,\dots,p}$’s directly follows from the unitarity of $U$. ∎

Any element $U\in\mathcal{U}_{block-diag}^{B}$ can be written as

In order to apply Proposition 6.4 we consider the orthonormal basis $\{E_{ij}=f_{i}e_{j}^{*},i,j=1,2\}$. In particular

It is obvious that these sets consist of commuting, normal operators, finishing the proof. ∎

Note however that the inclusion in the above result is strict (in the case $n\geq 3$, $k=2$). For $n=3$, and arbitrary $k\geq 2$, we construct next an example of a unitary operator being in $\mathcal{U}^{B}_{block-diag}$ but not in $\mathcal{U}^{A}_{block-diag}$.

We immediately note that the operators $X_{11}^{*}X_{22}=V$ and $X_{11}^{*}X_{33}=W$ do not commute. Since this commutativity is necessary to be in $\mathcal{U}^{A}_{block-diag}$ (Proposition 6.4), we conclude that $U$ doesn’t belong to $\mathcal{U}^{A}_{block-diag}$.

Another class of interesting block-diagonal (with respect to the second system, $B$) operators are circulant unitary matrices.

We show next that the matrices $X_{A}$ are all circulant, fact which, by Theorem A.1, suffices to conclude, since all the matrices appearing in the theorem will be simultaneously diagonalizable in the Fourier basis.

The crucial observation is that the above quantity only depends on the difference $j-i$: indeed, if $[(j-i)=(j^{\prime}-i^{\prime})]_{n}$, then there exists some $r$ such that $j^{\prime}-i^{\prime}=j-i+nr$, and thus

showing that the matrix $X_{A}$ is circular, and finishing the proof.

The statement about circular unitary operators follows from the general case using Proposition 6.3. ∎

We turn next to the study of the unitary operators which are block-diagonal with respect to both systems $A$ and $B$.

A unitary operator $U$ is block diagonal with respect to both tensor factors $A$ and $B$ (i.e. $U\in\mathcal{U}^{A}_{block-diag}\cap\mathcal{U}^{B}_{block-diag}$) iff

Let $U$ be an element in the intersection $\mathcal{U}_{block-diag}^{A}\cap\mathcal{U}_{block-diag}^{B}$. Then, $U$ admits both decompositions

Applying $Q_{i}^{*}\otimes R_{j}^{*}$ on the left, we obtain

Then there exists $\mu_{ij}\neq 0$ such that

Now since the $Q_{i}$’s and the $R_{j}$’s satisfy (33) we end up with

Since the operators $U_{j}$ and $V_{i}$ are unitary, we conclude that $|\mu_{ij}|=1$ and that gives the result. ∎

Finally, we compute next the (real) dimension of $\mathcal{U}_{block-diag}^{A}$ and $\mathcal{U}_{block-diag}^{A}\cap\mathcal{U}_{block-diag}^{B}$.

The real dimension of the algebraic variety $\mathcal{U}_{block-diag}^{A}$ is

The real dimension of the algebraic variety $\mathcal{U}_{block-diag}^{A}\cap\mathcal{U}_{block-diag}^{B}$

Let us first perform a heuristic parameter counting for a generic element

Let us now find the dimension of the intersection. Similarly, let us count parameters for a generic element of the form

The choice of the four orthonormal bases corresponds to a total of $2n^{2}+2k^{2}$ real parameters, the choice of the coefficients $nk=2nk-nk$. Since, in $\lambda_{ij}e_{i}f_{i}^{*}\otimes g_{j}h_{j}^{*}$, all the phases can be absorbed in the coefficient $\lambda_{ij}$, we have over counted $2n+2k$ real parameters. Again, the case $\min(n,k)=1$ is degenerated, since any unitary operator is of the desired form. ∎

Further relations between unitary classes

As discussed in the introduction, the following chain of inclusions holds:

We discuss in this section the situations when some of the above inclusions are equalities. See Section 8 for some related open questions.

The sets $\mathcal{U}_{block-diag}^{A}$ and $\mathcal{U}_{prob-lin}$ are equal.

Without loss of generality, we assume that the unitary operators $U_{i}$ are different up to a phase, i.e. $U_{i}\not\sim U_{j}$, for all $i\neq j$.

We then conclude that the matrices $M_{i}$ can be written as $M_{i}=R_{i}R_{i}^{*}=R_{i}^{*}R_{i}=R_{i}^{2}$ for some hermitian and positive semi-definite $R_{i}$.

By linearity, the previous equality gives

Then, by definition of the partial trace, we obtain the following equivalences

Note that the previous equation is actually a sum of non-negative terms equals to . Therefore, we conclude that for $i\neq j$

If $n=2$, then $\mathcal{U}_{block-diag}^{A}=\mathcal{U}_{unital}$. In particular, the chain of inclusions (34) collapses:

Let us now prove that $\{XY^{*}\,|\,X,Y=A,B,C,D\}$ and $\{X^{*}Y\,|\,X,Y=A,B,C,D\}$ consist of normal commuting matrices. It can be noticed that it is sufficient to check that, for all $X,Y,Z=A,B,C,D$,

Using the symmetry $A,D$ in (38) together with the symmetry $B,C$, the 64 different cases boil down to 11 non-trivial cases : $AA^{*}B,AA^{*}D,AB^{*}B,AB^{*}C,AB^{*}D,AD^{*}B,AD^{*}D,BA^{*}C,BA^{*}D,BB^{*}C$ and $BC^{*}D$. Each of them can be easily checked as for instance

A similar result has been obtained in [14, Theorem 9], under more stringent assumptions. More precisely, it is shown in that, when $n=2$, $\mathcal{U}_{block-diag}^{A}=\mathcal{U}_{prob}$, assuming that the unitary operators appearing in the mixed-unitary decomposition of channels are linearly independent.

Swapping the roles of $n$ and $k$, we obtain the following result.

If $k=2$, then $\mathcal{U}_{block-diag}^{B}=\mathcal{U}_{unital}\,.$

Conclusions and open questions

We end this work with a list of questions that we have left unanswered (or even untouched). We hope to get back to some of these problems in some future work.

We start with the problem of computing the dimension of the algebraic variety $\mathcal{U}_{unital}$; recall that previously, we have looked at the enveloping tangent space of this variety, at some particular points.

Show that $\dim\mathcal{U}_{unital}=kn^{2}+nk^{2}-nk$.

It has been showed in Theorem 7.1 that any operator in the set $\mathcal{U}_{prob-lin}$ (which is a subset of $\mathcal{U}_{mixed}$) is block diagonal, with respect to the system $A$. Moreover, in the qubit case $n=2$, we have $\mathcal{U}_{block-diag}^{A}=\mathcal{U}_{mixed}$, see Proposition 7.2. We conjecture that this equality always hold, and that the technical restrictions appearing in the definition of $\mathcal{U}_{prob-lin}$ are actually superfluous.

Regarding bipartite unitary operators producing PPT channels, we have left the following problem open.

This brings us to the problem of characterizing the set $\mathcal{U}_{EB}$ and comparing it to $\mathcal{U}_{PPT}$ (at the level of quantum states, this would be the fact that the PPT criterion for separability is necessary in all dimensions, and sufficient for $nk\leq 6$).

Provide a description of the set $\mathcal{U}_{EB}$. For which values of $n,k$, is it true that $\mathcal{U}_{PPT}=\mathcal{U}_{EB}$?

At the level of examples, beside the obvious inclusion $\mathcal{U}_{const}\subseteq\mathcal{U}_{EB}$, we also haveWe thank Siddharth Karumanchi for pointing this out to us., when $n=k$,

Indeed, for a unitary operator $U=\left(\sum_{i=1}^{n}U_{i}\otimes e_{i}f_{i}^{*}\right)\cdot F_{n}$, the corresponding quantum channel reads

Finally, we consider the following classe of bipartite unitary matrices yielding channels of interest in quantum information theory.

The study of these classes has been initiated in , where mainly the qubit case $n=2$ has been discussed. The structure of these operators in the general case remains open.

Characterize the sets $\mathcal{U}_{CQ}$, $\mathcal{U}_{QC}$, and $\mathcal{U}_{CC}$.

Appendix A Necessary and sufficient conditions for the existence of a block-SVD

In this section, we establish necessary and sufficient conditions for the existence of a block singular value decomposition of a bipartite operator with respect to the second sub-system $B$. Moreover, we present an algorithm for obtaining such a decomposition when it does exist. These results are inspired from , see also [12, Theorem (2.5.5) and Section 7.3, Problem 25].

We denote the set of matrices $X$ satisfying the above condition(s) by $\mathcal{M}_{block-diag}^{A}$

The set $\mathcal{M}_{block-diag}^{B}$ is defined in a similar way, by swapping the roles of the two tensor factors.

The implication $(2)\implies(3)$ is obvious. Let us first show $(1)\implies(2)$. For a matrix $X$ as in (32), we have

Let us now show $(3)\implies(1)$. The fact that the normal matrices $X_{\alpha}X_{\beta}^{*}$ commute implies they have the same set of eigenprojectors $P_{i}$:

In the same vein, we have, for another set of orthogonal eigenprojectors $Q_{i}$:

Letting $\alpha=\beta$ in (39) and (40) and using the fact that the matrices $X_{\alpha}X_{\alpha}^{*}$ and $X_{\alpha}^{*}X_{\alpha}$ have the same (positive) eigenvalues (counting multiplicities), we have that $p=p^{\prime}$ and there exists permutations $\sigma_{\alpha}\in\mathcal{S}_{p}$ and complex numbers $\lambda_{i}^{(\alpha)}$ such that

for some partial isometries $R_{i}^{(\alpha)}$ having initial projection $Q_{\sigma_{\alpha}(i)}$ and final projection $P_{i}$. Plugging the last expression into (39) and (40), we find that the permutations $\sigma_{\alpha}$ must be equal; we shall assume, by re-ordering the eigenprojectors $Q_{i}$, that these permutations are all equal to the identity. Using similar arguments, the partial isometries $R_{i}^{(\alpha)}$ cannot depend on $\alpha$, and we write $R_{i}^{(\alpha)}=R_{i}$. We have thus

The set $\mathcal{M}_{block-diag}^{A}$ is a real algebraic variety.

A matrix $X$ belongs to $\mathcal{M}_{block-diag}^{A}$ if and only if the two families of $n^{4}$ matrices $\{X_{\alpha}X_{\beta}^{*}\}$ and $\{X_{\alpha}^{*}X_{\beta}\}$ commute; these commutations conditions can be restated as (degree 4) polynomial conditions in the real and imaginary parts of the elements of $X$. ∎

The real dimension of the algebraic variety $\mathcal{M}_{block-diag}^{A}$ is $2k(n^{2}+k-1)$.

For the terminology and the results used in this proof, we refer the reader to [11, Chapter 11]. Let us introduce the flag manifold (see [11, Example 8.34] or [5, Section 4.9])

References