Quantum Computing Asked on August 20, 2021

In the textbook “Quantum Computation and Quantum Information” by Nielsen and Chuang, it is stated that there exists a set of unitaries $U_i$ and a probability distribution $p_i$ for **any** matrix A,

$$sum_i p_i U_i A U_i^dagger =tr(A) I/d,$$

where $d$ is the dimension of the Hilbert space. (This is on page 517; Exercise 11.19; equation (11.85)) The left-hand side is a Kraus representation given A.

But is this possible for a general non-diagonalizable (i.e. non-normal) matrix A? For a normal matrix A, I found it is indeed the case.

(* General result*) The main thing to keep in mind is that this is a result about

For such choice of matrices we have, for *any* matrix $rho$,
$$sum_i U_i rho U_i^dagger
= sum_{ijk ell m} lvert jrangle!langle krvert,, (U_i)_{jell}(U_i^*)_{km} rho_{ell m}
= sum_{jkell m} lvert jrangle!langle krvert,, delta_{jk} delta_{ell m}rho_{ell m}
\= sum_{jell} lvert jrangle!langle jrvert ,, rho_{ellell} = operatorname{tr}(rho) I. $$

Notice how the identity does not depend on what $rho$ is. It can be an arbitrary operator. You can test it yourself with a nondiagonalisable matrix such as $rho=begin{pmatrix}0&1\0&0end{pmatrix}$. It is a statement about the mapping $rhomapsto sum_i U_i rho U_i^dagger$, not about $rho$.

Notice also that I did not use any assumption on the $U_i$. They need not be unitaries (indeed, they *cannot* be unitaries in my choice of normalisation). To get the same factor on the RHS, you need only modify the normalisation of the matrices to have $operatorname{tr}(U_i U_j^dagger)=delta_{ij}/d$, and the rest follows.

(* Representations of the completely depolarising channel*)
Consider the linear map $Phi(X)=operatorname{tr}(X) I/d$. You can verify that it is a CPTP map and thus admits a Kraus decomposition.

Its natural representation reads $Phi_{i|j}^{k|ell}=K(Phi)_{ij,kell}=delta_{kell}delta_{ij}/d=lvert mrangle!langle mrvert$ with $|mrangle$ a maximally entangled state. The Kraus decomposition is then obtained as the spectral decomposition of the operator mapping $jell$ to $ik$. Stated more precisely, we need the spectral decomposition of the Choi operator $$J(Phi)equiv (Phiotimes I)lvert mrangle!langle mrvert=frac1 d Iotimes Iequiv I/d.$$

The eigendecomposition of this operator is trivial: its eigenvalues are all equal to $1/d$, thus any orthonormal set of vectors will be a suitable set of eigenvectors. Write these as $newcommand{bs}[1]{boldsymbol{#1}} {bs v_a}_a$, so that $J(Phi)bs v_a=frac1 d bs v_a$ for all $a=1,...,d^2$. In terms of the natural representation, these satisfy $$sum_{jell} K(Phi)_{ij,kell}(bs v_a)_{jell} = frac1 d(bs v_a)_{ik} Longleftrightarrow K(Phi) = frac1 d sum_a bs v_a otimes bs v_a^dagger.$$ $$K(Phi)_{ij,kell}=frac1 dsum_a (bs v_a)_{ik}(bs v_a^*)_{jell}.$$ Defining the operators $A_a$ as $(A_a)_{ij}equiv (bs v_a)_{ij}$ we thus get the Kraus decomposition $Phi(X) = sum_a A_a X A_a^dagger. $ Note that the orthogonality of the vectors $bs v_a$, $langle bs v_a,bs v_brangle=delta_{ab}$, translates into the orthogonality of the matrices $A_a$ in the $L_2$ norm: $operatorname{tr}(A_a A_b^dagger)=delta_{ab}$.

(* Result from Kraus representation*) This proves that, for any set of matrices $A_a$ such that $operatorname{tr}(A_a A_b^dagger)=delta_{ab}$, we have for

(* Finding Kraus decompositions made up of unitaries*) In the above, $A_a$ are not unitaries. However, the freedom in the choice of vectors $bs v_a$, or equivalently the freedom in the choice of $A_a$, can be used to find a decomposition in terms of Kraus operators that are (proportional to) unitaries.
A basis of unitaries can be constructed

Correct answer by glS on August 20, 2021

Since it hasn't been mentioned so far, and I think it's an interesting aspect: A weighted ensemble $(p_i,U_i)$ of unitaries in $U(d)$ such that
$$
sum_i p_i U_i X U_i^dagger = operatorname{tr}(X) mathbb{I}/d,
$$
is called a *weighted unitary 1-design*. If the weights can be chosen uniformly, i.e. $p_i equiv 1/N$ where $N$ is the size of the ensemble, this reduces to the definition of a "normal" unitary 1-design.

There are many examples for unitary 1-designs:

- Unitary 1-designs are exactly tight operator frames with frame constant $N/d$
- In particular, any orthogonal operator basis of unitaries is a unitary 1-design, e.g. the Weyl operators
- In fact any irreducible unitary representation of a group is a unitary 1-design, e.g. the Heisenberg-Weyl (=generalised Pauli) and Clifford group.
- For any large enough Haar-random ensemble of unitaries there are weights such that the above equation holds with high probability.

Answered by Markus Heinrich on August 20, 2021

If it holds for hermitian matrices, it holds for all matrices due to linearity: Over $mathbb C$, the hermitian matrices span the full matrix space.

Answered by Norbert Schuch on August 20, 2021

This problem can be approached without regards to Kraus representations (even if the motivation is to prove the convexity of entropy) or whether A is a normal matrix or not. Rather, this is a feature of the choice of ${ U_{j} }$. In particular, there exists a choice such that their action is to ``coarse-grain'' all the information in a state.

Here's a single qubit example to illustrate my point: consider the set $p_{j} = frac{1}{4}, U_{j} = sigma_{j}$ for $j in { 1,2,3,4 }$, where, $sigma_{j}$ are the Pauli matrices (with $sigma_{0} = mathbb{I}$). Then, its action on a single qubit is, $$ sumlimits_{j} p_{j} U_{j} rho U^{dagger}_{j} = frac{1}{4} left( mathbb{I} rho mathbb{I} + sigma_{x} rho sigma_{x} + sigma_{y} rho sigma_{y} + sigma_{z} rho sigma_{z} right) = cdots = operatorname{Tr}left( rho right) frac{mathbb{I}}{2},$$ where the $cdots$ can be evaluated using the anticommutativity of the Pauli matrices (Hint: use the relation $sigma_{j} sigma_{k} sigma_{j} = - sigma_{k}$ for $j neq k$).

Now, since any matrix $A$ can be written as $A = H + iK$ for hermitian matrices $H,K$; and any hermitian matrix $H$ can be written as $H = H_{1} - H_{2}$ for positive semidefinite matrices, you can write $A = H_{1} - H_{2} + i(K_{1} - K_{2})$. Rewriting each of the matrices as $H_{1} = operatorname{Tr}left( H_{1} right) (frac{1}{operatorname{Tr}left( H_{1} right)} H_{1})$, we have that $frac{1}{operatorname{Tr}left( H_{1} right)} H_{1}$ is a density matrix and so the above result applies. Continuing this, you'll find, using the linearity of trace, that for the $2 times 2$ case, the above unitaries give you $mathrm{Tr}(A) frac{mathbb{I}}{d}$.

The generalization to $n times n$ matrices is left as an exercise to the OP (where you need to find a set of unitaries analogous to the Pauli matrices).

Edit: One way to obtain the result in $d$ dimensions is to use the $d^2$ Heisenberg-Weyl operators (or the finite dimensional representation of the Heisenberg-Weyl algebra). If $X(i)Z(j)$ is the $(i,j)$th operator then, we have, $frac{1}{d^{2}} sum_{i, j=0}^{d-1} X(i) Z(j) rho Z^{dagger}(j) X^{dagger}(i)=frac{mathbb{I}}{d}$. See, for example, Page 176 of this book.

Answered by keisuke.akira on August 20, 2021

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