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Strong\(^*\) convergence of quantum channels

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Abstract

Recently, the existence of a strongly converging sequence of quantum channels that cannot be represented a reduction of a sequence of unitary channels strongly converging to a unitary channel was shown. In this work, we obtain several characterizations of sequences of quantum channels that have the above representation. The corresponding convergence is called the strong\(^*\) convergence, since we have found that it is related to the convergence of selective Stinespring isometries in the strong\(^*\) operator topology. We present the arguments showing that the strong\(^*\) convergence is a proper, physically motivated type of convergence of infinite-dimensional quantum channels. Some properties of the strong\(^*\) convergence of quantum channels are considered. It is shown that for Bosonic Gaussian channels the strong\(^*\) convergence coincides with the strong convergence.

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Notes

  1. It seems reasonable to assume that all physical perturbations of a unitary channel \(\rho \mapsto U\rho U^*\) are properly described by continuous deformations of the unitary U in the strong operator topology (coinciding in this case with other basic operator topologies excepting the norm topology [12]).

  2. \(s\text {-}\!\lim \nolimits _{n\rightarrow \infty } X_n=X_0\) denotes the convergence of a sequence \(\{X_n\}\) to an operator \(X_0\) in the strong operator topology.

  3. \({\Phi }^*\) is the dual map to the channel \({\Phi }\) defined by relation (6).

  4. We write \(I_X\) and \(\mathrm {Id}_X\) instead of \(I_{{\mathcal {H}}_X}\) and \(\mathrm {Id}_{{\mathcal {H}}_X}\) (where \(X=A,B,..\)) to simplify notation.

  5. Similar statements for the strong convergence are proved explicitly in [9].

  6. I am grateful to R.F.Werner for pointing me this result and its proof in the case \({\Phi }_0=\mathrm {Id}_A\).

  7. It means that \(s\text {-}\!\lim \nolimits _{n\rightarrow \infty } W_n=P_0\) and \(s\text {-}\!\lim \nolimits _{n\rightarrow \infty } W^*_n=P_0\) [12].

  8. Gaussian unitary channel is a channel \(\rho \mapsto U_T\rho U^*_T\), where \(U_T\) is the canonical unitary corresponding to a symplectic transformation T [1, Ch.12].

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Acknowledgements

I am grateful to A.S. Holevo, A.V. Bulinski and G.G. Amosov for useful comments. Special thanks to the participants of the workshop “Quantum information, statistics, probability” September 2019, Steklov Mathematical Institute, Moscow (especially, to R.F.Werner), for the useful discussion and suggestions. I am also grateful to the unknown referee for the valuable suggestions, which led to improvements of the initial version of this paper.

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Correspondence to M. E. Shirokov.

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This work was funded by Russian Federation represented by the Ministry of Science and Higher Education (Grant No. 075-15-2020-788)

Appendix

Appendix

Below, we present results concerning possibility to dilate a strongly converging sequence of partial isometries to strongly converging sequence of unitary operators.

Proposition 3

Let \(\{V_n\}\) be a sequence of partial isometries on a separable Hilbert space \({\mathcal {H}}\) strongly converging to a partial isometry \(V_0\) such that \(V_n^*V_n=V_0^*V_0=P\) and \(\dim \mathrm {Ker}P=\dim \mathrm {Ker}Q_n\le +\infty ,\,\) where \(Q_n=V_nV^*_n\), for all \(n\ge 0\). The following properties are equivalent:

  1. (i)

    there exists a sequence \(\{U_n\}\) of unitaries on \({\mathcal {H}}\) strongly converging to a unitary operator \(U_0\) such that \(U_nP=V_n\) for all \(n\ge 0\);

  2. (ii)

    the sequence \(\{Q_n\}\) strongly converges to the operator \(Q_0\);

  3. (iii)

    the sequence \(\{V^*_n\}\) strongly converges to the operator \(V^*_0\).

Proof

The implications \(\mathrm (i)\Rightarrow (iii)\) and \(\mathrm (iii)\Rightarrow (ii)\) are obvious.

To show that \(\mathrm (ii)\Rightarrow (i)\) consider the sequence \(\{W_n=V_nV_0^*\}\) strongly converging to the projector \(Q_0=V_0V_0^*\). Since \(V_n^*V_n=V_0^*V_0\) for all n, all the operators \(W_n\) are partial isometries. Note that \(W_nW_n^*=Q_n\) and \(W^*_nW_n=Q_0\) for all n. By the implication \(\mathrm (ii)\Rightarrow (i)\) in Lemma 2 below, there exists a sequence \(\{T_n\}\) of unitary operators strongly converging to the unit operator \(I_{{\mathcal {H}}}\) such that \(T_n Q_0=W_n\) for all n. Since \(\dim \mathrm {Ker}P=\dim \mathrm {Ker}Q_0\), there is a unitary operator \(U_0\) such that \(U_0P=V_0\). It is easy to see that the sequence of unitary operators \(U_n=T_nU_0\) strongly converging to the unitary operator \(U_0\) has the required properties. \(\square \)

Remark 3

A sequence \(\{V_n\}\) of partial isometries satisfying the assumptions of Proposition 3 for which the equivalent properties \(\mathrm{(i)}\)\(\mathrm{(iii)}\) do not hold can be found in the proof of Corollary 14 in [10].

Lemma 2

Let \(\{S_n=\{\varphi ^n_i\}_{i\in I}\}_{n\ge 0}\) be a sequence of orthonormal systems of vectors in a separable Hilbert space \({\mathcal {H}}\) such that \(\dim S_n^{\bot }=\,\dim S_0^{\bot }\le +\infty \) for all n, where I is a finite or countable set of indexes and \(S_n^{\bot }\) is the orthogonal complement of \(S_n\). Let \(P_n=\sum _{i\in I}|\varphi ^n_i\rangle \langle \varphi ^n_i|\,\) be the projector on the subspace \({\mathcal {H}}_n\) generated by \(S_n\) and \(W_n=\sum _{i\in I}|\varphi ^n_i\rangle \langle \varphi ^0_i|\,\) a partial isometry. Assume that \(\lim _{n\rightarrow \infty } \varphi ^n_i=\varphi ^0_i\) for each \(i\in I\). The following properties are equivalent:

  1. (i)

    for each \(n\ge 0\,\) there is an orthonormal basis \(S^\mathrm {e}_n=\{\varphi ^n_i\}_{i\in I}\cup \{\psi ^n_j\}_{j\in J}\) in \({\mathcal {H}}\) obtained by extension of the system \(S_n\) such that \(\lim _{n\rightarrow \infty } \psi ^n_j=\psi ^0_j\) for each \(j\in J\);

  2. (ii)

    the sequence \(\{P_n\}\) strongly converges to the operator \(P_0\);

  3. (iii)

    the sequence \(\{W^*_n\}\) strongly converges to the operator \(P_0\);

Proof. \(\mathrm (i)\Rightarrow (iii)\). It follows from \(\mathrm (i)\) that

$$\begin{aligned} U_n=\sum _{i\in I}|\varphi ^n_i\rangle \langle \varphi ^0_i|+\sum _{j\in J}|\psi ^n_j\rangle \langle \psi ^0_j| \end{aligned}$$

is an unitary operator strongly converging to the unit operator \(I_{{\mathcal {H}}}\) as \(n\rightarrow \infty \). Then, the unitary operator \(U^*_n\) strongly converges to the unit operator as well, i.e.,

$$\begin{aligned} \sum _{i\in I}|\varphi ^0_i\rangle \langle \varphi ^n_i|\theta \rangle \oplus \sum _{j\in J}|\psi ^0_j\rangle \langle \psi ^n_j|\theta \rangle \;\rightarrow \; \sum _{i\in I}|\varphi ^0_i\rangle \langle \varphi ^0_i|\theta \rangle \oplus \sum _{j\in J}|\psi ^0_j\rangle \langle \psi ^0_j|\theta \rangle \end{aligned}$$

as \(n\rightarrow \infty \) for any vector \(\theta \) in \({\mathcal {H}}\). Hence, \(W^*_n\) strongly converges to \(P_0\).

\(\mathrm (iii)\Rightarrow (ii)\). Since \(W_n\) strongly converges to \(P_0\) by the assumption, it follows from \(\mathrm (iii)\) that \(P_n=W_nW_n^*\) strongly converges to \(P_0\).

\(\mathrm (ii)\Rightarrow (i)\). Let \(S^\mathrm {e}_0=\{\varphi ^0_i\}_{i\in I}\cup \{\psi ^0_j\}_{j\in J}\) be an orthonormal basis (o.n.b. in what follows) in \({\mathcal {H}}\) obtained by extension of the system \(S_0\). Sequentially applying Lemma 3, one can construct, for any natural n and \(m\le \dim S_0^{\bot }\), an orthonomal system \(\{ \alpha ^n_1,\ldots ,\alpha ^n_m \}\) in \(S_n^{\bot }\) in such a way that \(\lim _{n\rightarrow \infty }\alpha ^n_j=\psi ^0_j\) for all \(j=\overline{1,m}\). This gives the required sequence of o.n.b. \(S^\mathrm {e}_n=S_n\cup \{\psi ^n_j\}_{j\in J}\) in the case \(\dim S_0^{\bot }<+\infty \). If \(\dim S_0^{\bot }=+\infty \) this sequence can be constructed as follows:

$$\begin{aligned} \begin{array}{l} \psi ^1_1=\alpha ^1_1\, \text {and}\,\{\psi ^1_j\}_{j>1}\,\text { is}\,\text {any}\,\text {o.n.b.}\,\text {in }[\{ \alpha ^1_1\}\cup S_1]^{\bot },\\ \psi ^2_1=\alpha ^2_1, \psi ^2_2=\alpha ^2_2\,\text {and}\,\{\psi ^2_j\}_{j>2}\text { is any o.n.b. in }[\{ \alpha ^2_1,\alpha ^2_2 \}\cup S_2]^{\bot },\\ ..............................\\ \psi ^n_1=\alpha ^n_1,\ldots , \psi ^n_n=\alpha ^n_n\text {\, and\, }\{\psi ^n_j\}_{j>n}\,\text {is}\,\text {any}\,\text {o.n.b.}\,\text {in}\,[\{ \alpha ^n_1,\ldots ,\alpha ^n_n \}\cup S_n]^{\bot },\\ .............................. \end{array} \end{aligned}$$

Remark 4

A sequence \(\{S_n\}\) of orthonormal systems satisfying the assumptions of Lemma 2 for which properties \(\mathrm{(i)}\)\(\mathrm{(iii)}\) of this lemma do not hold can be easily constructed: let \(\{\tau _i\}\) be a countable orthonormal system of vectors, \(\varphi ^n_i=\tau _i\) for all \(i\ne n\) and \(\varphi ^n_n=\psi \), where \(\psi \) is any unit vector in \(\{\tau _i\}^{\bot }\).

Lemma 3

Let the assumptions of Lemma 2 hold and \(\psi _0\) be any unit vector in \(S_0^{\bot }\). If the sequence \(\{P_n\}\) strongly converges to the operator \(P_0\), then there is a sequence \(\{\psi _n\}\) of unit vectors converging to the unit vector \(\psi _0\) such that \(\psi _n\in S_n^{\bot }\) for all n.

Proof

Let \({\bar{P}}_n=I_{{\mathcal {H}}}-P_n\) and \(|\psi _n\rangle ={\bar{P}}_n|\psi _0\rangle /\Vert {\bar{P}}_n|\psi _0\rangle \Vert \) if \(\Vert {\bar{P}}_n|\psi _0\rangle \Vert \ne 0\) and \(|\psi _n\rangle \) be any unit vector in \(S_n^{\bot }\) otherwise. Since the sequence \(\{{\bar{P}}_n\}\) strongly converges to the operator \({\bar{P}}_0\) and \({\bar{P}}_0|\psi _0\rangle =|\psi _0\rangle \) the sequence \(\{|\psi _n\rangle \}_n\) has the required properties. \(\square \)

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Shirokov, M.E. Strong\(^*\) convergence of quantum channels. Quantum Inf Process 20, 145 (2021). https://doi.org/10.1007/s11128-021-03087-z

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