On linear structure of non-commutative operator graphs

TL;DR

This paper investigates the linear structure of non-commutative operator graphs generated by group actions, providing explicit formulas and descriptions for graphs associated with the circle and Heisenberg-Weyl groups.

Contribution

It offers a new description of the linear structure of certain non-commutative operator graphs, including explicit formulas for their dimensions.

Findings

Graphs generated by the circle group have unitary generators permuting basis vectors.

Explicit formula for the dimension of graphs generated by the Heisenberg-Weyl group.

New characterization of the linear structure of these operator graphs.

Abstract

We continue the study of non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary actions of the circle group and the Heisenber-Weyl group as well. It is shown that the graphs generated by the circle group has the system of unitary generators fulfilling permutations of basis vectors. For the graph generated by the Heisenberg-Weyl group the explicit formula for a dimension is given. Thus, we found a new description of the linear structure for the operator graphs introduced in our previous works.

Equations

$$A\in \mathcal{V}\Rightarrow A^{\ast }\in \mathcal{V};I\in \mathcal{V}.$$

$$M(\varnothing )=0,M(G)=I,$$

$$M({\cup }_j{B}_j)=\sum _{j}M(B_j),B_k\cap B_l=\varnothing \text{for}k\ne l,B_j\in \mathfrak{B}.$$

$${\rho }_g(A)=U_{g}A{U}_g^{\ast }.$$

$${\rho }_g(M(B))=M(gB).$$

$$\mathcal{V}=\overline{span\{M(B),B\in \mathfrak{B}\}}.$$

$$|{\psi }_{sn}⟩=\frac{1}{\sqrt{d}}\sum _{k=1}^d{e}^{\frac{2\pi isk}{d}}|kk-nmodd⟩,1\le s\le d,$$

$$H_s=span\{|{\psi }_{sn}⟩,1\le n\le d\}.$$

$$\phi \to U_{\phi }=\sum _{s=1}^d{e}^{i\phi s}{P}_s,\phi \in \mathbb{T}.$$

$$Q_j=\sum _{k=1}^d|jj-kmodd⟩⟨jj-kmodd|,1\le j\le d.$$

$$W_n=\sum _{l=1}^d\sum _{r=1}^d|{\psi }_{l+nmoddr}⟩⟨{\psi }_{lr}|,0\le n\le d-1.$$

$$U_{\phi }{Q}_j{U}_{\phi }^{\ast }=\sum _{s=1}^d\sum _{l=1}^d{e}^{i\phi (s-l)}{P}_s{Q}_j{P}_l=$$

$$\sum _{s=1}^d\sum _{l=1}^d\sum _{r=1}^d{e}^{i\phi (s-l)}{P}_s|jj-rmodd⟩⟨jj-rmodd|P_l=$$

$$\sum _{s=1}^d\sum _{l=1}^d\sum _{r=1}^d\sum _{n=1}^d\sum _{h=1}^d{e}^{i\phi (s-l)}|{\psi }_{sn}⟩⟨{\psi }_{sn}⟩jj-rmodd⟨jj-rmodd⟩{\psi }_{lh}⟨{\psi }_{lh}|=$$

$$\frac{1}{d}\sum _{s=1}^d\sum _{l=1}^d\sum _{r=1}^d\sum _{n=1}^d\sum _{h=1}^d{e}^{i\phi (s-l)}{\delta }_{nr}{\delta }_{hr}{e}^{-\frac{2\pi ij(s-l)}{d}}|{\psi }_{sn}⟩⟨{\psi }_{lh}|=$$

$$\frac{1}{d}\sum _{s=1}^d\sum _{l=1}^d{e}^{i\phi (s-l)}{e}^{-\frac{2\pi ij(s-l)}{d}}\sum _{r=1}^d|{\psi }_{sr}⟩⟨{\psi }_{lr}|=\frac{1}{d}\sum _{n=0}^{d-1}{e}^{i\left(\phi -\frac{2\pi j}{d}\right)n}{W}_n,$$

$$\frac{1}{d}\sum _{n=0}^{d-1}{e}^{i\frac{2\pi kn}{d}}{W}_n=U_{{\phi }_{kj}}{Q}_j{U}_{{\phi }_{kj}}^{\ast }\in {\mathcal{V}}_j$$

$$S|j⟩=|j+1modd⟩,M|j⟩=e^{\frac{2\pi i}{d}j}|j⟩$$

$$h_k^0=\frac{1}{\sqrt{n}}\sum _{j=0}^{d-1}{e}^{\frac{2\pi ikj}{d}}|jj⟩,$$

$$h_k^j=(I\otimes S^j)h_k^0.$$

$$H_j=span\{h_k^j,0\le k\le d-1\}.$$

$$\pi (S)h_k^j=h_{k+1modd}^j,$$

$$\pi (M)h_k^j=e^{\frac{2\pi i}{d}k}{h}_k^j.$$

$$y_{ml}=\sum _{k=0}^{d-1}|h_m^k⟩⟨h_l^k|$$

$$h_0=\sum _{m=0}^{d-1}{y}_{mm}\equiv I,$$

$$h_p=\sum _{m=0}^{d-1}{y}_{m+pmoddm}+y_{mm+pmodd},1\le p\le d-1.$$

$$\{\begin{array}{ll}& \dim \mathcal{V}=\frac{d-1}{2}+1\text{ for odd }d\\ & \dim \mathcal{V}=\frac{d}{2}+1\text{ for even }d\end{array}$$

$$h_p=h_{p^{{}^{\prime }}}$$

$$\sum _{m=0}^{d-1}{y}_{m+pmoddm}+\sum _{m=0}^{d-1}{y}_{mm+pmodd}=\sum _{m^{{}^{\prime }}=0}^{d-1}{y}_{m^{{}^{\prime }}+p^{{}^{\prime }}modd{m}^{{}^{\prime }}}+\sum _{m^{{}^{\prime }}=0}^{d-1}{y}_{m^{{}^{\prime }}{m}^{{}^{\prime }}+p^{{}^{\prime }}modd}$$

$$p^{\prime }=d-p$$

Full text

Keywords non-commutative operator graphs, covariant resolution of identity, quantum anticliques

1 Introduction

In this article we give a clarification of our current results on non-commutative operator graphs generated by covariant resolutions of identity [1, 2]. A linear subspace $\mathcal{V}$ in the algebra of all bounded linear operators on the Hilbert space $H$ is called a non-commutative operator graph if

[TABLE]

These objects allow to determine a possibility of zero-error correction for a system of unitary operators representing errors in quantum information transmission (for explicit description of this theory see [4, 3]). Basically, we interested only in graphs satisfying the Knill-Laflamme condition [5]. It is said that the graph $\mathcal{V}$ satisfies the Knill-Laflamme condition, if there is an orthogonal projection $P_{K}$ on the subspace $K\subset H$ such that $P_{K}\mathcal{V}P_{K}=\mathbb{C}P_{K}$, the projection $P_{K}$ is called an anticlique and the subspace $K$ is known as a quantum error correcting code.

Some properties and illustrating examples of non-commutative operators graphs could be given in terms of resolutions of identity covariant with respect to an action of locally-compact groups. Studying of non-commutative operator graphs generated by covariant resolution of identity was initiated in [6]. Let $G$ be a locally compact group and $\mathfrak{B}$ is a sigma-algebra generated by compact subsets of G, then the set of positive operators $\{M(B),B\in\mathfrak{B}\}$ is said to be a resolution of identity if

[TABLE]

[TABLE]

Let $(U_{g})_{g\in G}$ be a projective unitary representation of $G$. Define the action of this group on operator $A$ as

[TABLE]

Then $M(B)$ is said to be covariant with respect to this action if

[TABLE]

Suppose that a graph $\mathcal{V}$ is generated by $M(B)$ in the following sense

[TABLE]

Note that if $H$ is finite dimensional there is no need to take a closure in the last equation.

2 Linear structure of $\mathcal{V}$

At first let us define the graph introduced in [1]. Consider the finite dimensional Hilbert space $H={\mathbb{C}}^{d}\otimes{\mathbb{C}}^{d}$ and the orthonormal basis $\ket{k},\ 1\leq k\leq d$ in $\mathbb{C}^{d}$. Then, vectors

[TABLE]

are known as the generalized Bell states. Let $P_{s},\ 1\leq s\leq d$ be the projections on the subspaces

[TABLE]

Define a unitary representation of the circle group $\mathbb{T}=[0,2\pi]$ with the operation $+/mod(2\pi)$ by the formula

[TABLE]

Also consider a set of projections in $\mathbb{C}^{d}$ of the form

[TABLE]

The following corollary defines examples of the graphs mentioned above.

Corollary 1[1]. Given $j,\ 1\leq j\leq d,$ the projection (2) generates the graph $\mathcal{V}_{j}=span\{U_{\varphi}Q_{j}U_{\varphi}^{*},\varphi\in{\mathbb{T}}\}$ for which the projections $\{P_{s},\ 1\leq s\leq d\}$ are anticliques.

Let us consider a set of unitary operators defined as follows

[TABLE]

The unitaries from (3) act as permutations on the set of vectors (1).

Below we clarify a structure of the graphs determined by Corollary 1.

Theorem 1. All graphs $\mathcal{V}_{j}\equiv\mathcal{V}$ coincide for $1\leq j\leq d$. Moreover, one can choose (3) as generators of $\mathcal{V}$.

Proof.

For operators $U_{\varphi}Q_{j}U_{\varphi}^{*}$ we have such expansion in basis $\ket{\psi_{kn}}$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where unitaries $W_{n}$ are determined by (3). To complete the proof we need to show that $W_{n}\in{\mathcal{V}}_{j}$ for all $j,\ 1\leq j\leq d$. Substituting $\varphi=\varphi_{kj}=\frac{2\pi{(k+j)}}{d}$ into (4) we obtain

[TABLE]

for all $0\leq k\leq d-1$. The result follows. $\Box$

3 Examples for Heisenberg-Weyl Group

Now we want to give an analogous clarification of linear structure for the example of the non-commutative operator graph generated by the same technique implied for the Heisenberg-Weyl group. Unlike the circle group this group is non-commutative.

At first we need to introduce the construction from [2]. Let $\mathfrak{H},\ dim\mathfrak{H}=d$ be a finite-dimensional Hilbert space with the fixed orthogonal basis $\ket{j},\ 0\leq j\leq d-1$. Unitary operators $S$ and $M$ on $\mathfrak{H}$ given by the formulae

[TABLE]

define the representation of Heisenberg-Weyl group $G_{d}$ of rank $d$.

Consider $H=\mathfrak{H}\otimes\mathfrak{H}$ with the orthonormal basis $\ket{kj},0\leq k,j\leq d-1$. Also consider the orthogonal basis in $H$ consisting of entangled vectors

[TABLE]

[TABLE]

Define subspaces in $H$ spanned by systems of the following entangled vectors

[TABLE]

Let $\pi$ be the map transmitting $S$ and $U$ to unitary operators in $H$ as follows

[TABLE]

[TABLE]

It is shown in [2] that the map $\pi$ can be extended to a unitary representation of the group $G_{d}$.

Also we need special sums of matrix units

[TABLE]

and their linear combinations

[TABLE]

[TABLE]

Resulting linear structure of the operator graph generated by representation $\pi$ was given in [2] as follows $\mathcal{V}=span\{h_{p},0\leq p\leq d-1\}$. Here we give the explicit value for a dimension of this graph.

Theorem 2. A dimension of the graph is given by the formula

[TABLE]

Proof.

Equation

[TABLE]

or equivalently

[TABLE]

holds true if

[TABLE]

because a first summand in the left hand side equals to a second summand in the right hand side and vice-versa. Thus, $\dim\mathcal{V}$ expresses by (5). $\Box$

The next corollary follows from the proof of the theorem.

Corollary 2. The graph $\mathcal{V}$ has the system of generators

[TABLE]

4 Conclusion

We gave a clarification of the examples of the operator graphs given in [1]. It is shown that all these examples coincide and it is possible to choose unitary generators of a given graph, so we have the same situation as in [2]. For examples given in [2] we gave the explicit formula for dimensions of the graphs and specified their linear structure.

5 Acknowledgments

The work is supported by Russian Science Foundation under the grant no. 19-11-00086.