Superoperator representations¶
This document summarizes our conventions for the different superoperator representations. We show how to apply the channels to states in these representations and how to convert channels between a subset of representations. By combining these conversion methods you can convert between any of the channel representations.
This document is not intended to be a tutorial or a comprehensive review. At the bottom of the document there is a list of references with more information. This document was influenced by [IGST] and we recommend reading [GRAPTN] to gain deeper understanding (see also [QN], [SVDMAT], [MATQO], [DUAL] listed at the bottom of this document). Additionally, these references explain, for example, how to determine if a channel is unital or completely positive in the different representations.
vec and unvec¶
Consider an \(m\times m\) matrix
where \(i\) is a row and \(j\) is a column index.
We define vec to be column stacking
were \(T\) denotes a transpose. Clearly an inverse operation,
unvec can be defined so that
Of course unvec() generally depends on the dimensions of \(A\),
which are not recoverable from vec(A). We often focus on square A,
but for generality, we require the dimensions for \(A\), defaulting
to the square root of the dimension of vec(A). Column stacking
corresponds to how matrices are stored in memory for column major
storage conventions.
Similarly we can define a row vectorization to be row stacking \({\rm vec_r}(A) = (a_{11}, a_{12}, \ldots, a_{1m}, a_{21},\ldots, a_{mm})^T\) . Note that \({\rm vec}(A) = {\rm vec_r}(A^T)\). In any case we will not use this row convention.
Matrix multiplication in vectorized form¶
For matrices \(A,B,C\)
which is sometimes called Roth’s lemma.
Eq. 2 is useful in representing quantum operations on mixed quantum states. For example consider
We can use Eq. 1 to write this as
so
where \(\mathcal U = U^*\otimes U\). The nice thing about this is the operator (the state) has become a vector and the superoperator (the left right action of \(U\)) has become an operator.
Some other useful results related to vectorization are
$ {}([A,X])= (IA - A^TI) {}(X)$
\({\rm vec}(ABC) = (I\otimes AB) {\rm vec}( C ) = (C^T B^T\otimes I) {\rm vec}(A)\)
\({\rm vec}(AB) = (I\otimes A) {\rm vec}(B) = (B^T\otimes I) {\rm vec}(A)\).
Matrix operations on Bipartite matrices: Reshuffling, SWAP, and tranposition¶
This section is based on the Wood et al. presentation in [GRAPTN].
As motivation for this section consider the Kraus representation theorem. It shows that a quantum channel can be represented as a partial trace over a unitary operation on a larger Hilbert space. Actually the unitary is on a bipartite Hilbert space.
When representing quantum channels one insight is used many times.
Consider two Hilbert spaces \(\mathbb H_A\) and \(\mathbb H_B\) with dimensions \(d_A\) and \(d_B\) respectively. An abstract quantum process matrix \(\mathcal Q\) lives in the combined (bipartite) space of \(\mathbb H_A \otimes \mathbb H_B\) so \(\mathcal Q\) is a \(d_A^2\times d_B^2\) matrix.
We can represent the process as a tensor with components
where \(|n,\nu\rangle = |n\rangle \otimes |\nu\rangle\), \(m,n\in \{0,\ldots, d_A-1\}\), \(\mu,\nu\in \{0,\ldots, d_B-1\}\) and all vectors are in the standard basis.
With respect to these indices some useful operations are [GRAPTN]:
Transpose \(T\): \(\mathcal Q_{m,\mu;n,\nu} \mapsto \mathcal Q_{n,\nu;m,\mu},\)
SWAP: \(\mathcal Q_{m,\mu;n,\nu} \mapsto \mathcal Q_{\mu,m;\nu,n},\)
Row-reshuffling \(R_r\): \(\mathcal Q_{m,\mu;n,\nu} \mapsto \mathcal Q_{m,n;\mu,\nu},\)
Col-reshuffling \(R\): \(\mathcal Q_{m,\mu;n,\nu} \mapsto \mathcal Q_{\nu,\mu;n,m}.\)
The importance of understanding reshuffling can be understood as understanding the relationship between
where \(G\) and \(\Gamma\) are matrices that act on \(\mathbb H_A\) and \(\mathbb H_B\) respectively, as explained in [VECQO].
A note on numerical implementations¶
Most linear algebra (or tensor) libraries have the ability to reshape
a matrix and swapaxes (or sometimes it is called permute_dims).
If you are trying to reshuffle indices usually the first job is to
write your matrix in tensor form. This requires reshaping a
\(d_A^2\times d_B^2\) matrix into a
\(d_A\times d_A\times d_B \times d_B\) tensor. Next you
permute_dims or swapaxes. Often \(d_A = d_B\) so we
reshape to a Matrix that has the same dimensions as the original
\(d_A^2\times d_A^2\) matrix.
The \(n\)-qubit Pauli basis¶
The \(n\)-qubit Pauli basis is denoted \(\mathcal P^{\otimes n} \)mathcal P = { I, X, Y, Z }` are the usual Pauli matrices. It is an operator basis for the \(d = 2^n\) dimensional Hilbert space and there are \(d^2 = 4^n\) operators in \(\mathcal P^{\otimes n} \)sqrt{d}` the basis is orthonormal with respect to the Hilbert-Schmidt inner product.
It is often convenient to index the \(d^2\) operators with a single label, e.g. \(P_1=I^{\otimes n},\, \ldots,\, P_{d^2}= Z^{\otimes n}\) (or \(P_0=I^{\otimes n}\) if you like zero indexing). In anycase, as these operators are Hermitian and unitary they obey \(P_i^2=I^{\otimes n}\).
To be explicit, for two qubits \(d=4\) and we have 16 operators e.g. \(\{II, IX, IY, IZ, XI, XX, XY, ..., ZZ\}\) were \(II\) should be interpreted as \(I\otimes I\) etc. The single index would be \(\{P_1, P_2, P_3, P_4, P_5, P_6, P_7, ..., P_{16}\}\).
Quantum channels in the Kraus decomposition (or operator-sum representation)¶
A completely positive map on the state \(\rho\) can be written using a set of Kraus operators \(\{ M_k \}\) as
where \(\rho'\) is the state at the output of the channel.
If \(\sum_k M_k^\dagger M_k= I \) where \(d\) is the Hilbert space dimension e.g. \(d=2^n\) for \(n\) qubits. Kraus operators are not necessarily unique, sometimes there is a unitary degree of freedom in the Kraus representation.
Kraus to \(\chi\) matrix (aka chi or process matrix)¶
We choose to represent the \(\chi\) matrix in the Pauli basis. So we expand each of the Kraus operators in the \(n\) qubit Pauli basis
\(M_k = \sum^{d^2}_{j=1}c_{kj}\,P_j\)
where \(\mathcal P_j \in \mathcal P ^{\otimes n}\).
Now the channel \(\mathcal E\) can be written as
\(\mathcal E (\rho) = \sum_{i,j=1}^{d^2} \chi_{i,j} P_i\rho P_j ,\)
where
is an element of the process matrix \(\chi\) of size \(d^2 \times d^2\). If the channel is CP the \(\chi\) matrix is a Hermitian and positive semidefinite.
The \(\chi\) matrix can be related to the (yet to be defined) Choi matrix via a change of basis. Typically the Choi matrix is defined in the computational basis, while the \(\chi\) matrix uses the Pauli basis. Moreover, they may have different normalization conventions.
In this light, after reviewing the Kraus to Choi conversion it is simple to see that the above is equivalent to first defining
then
Kraus to Pauli-Liouville matrix (Pauli transfer matrix)¶
We begin by defining the Pauli vector representation of the state \(\rho\)
where \(P_j \in \mathcal P^{\otimes n}\) and \(c_j = (1/d) \langle\langle P_j|\rho \rangle\rangle\).
The Pauli-Liouville or Pauli transfer matrix representation of the channel \(\mathcal E\) is denoted by \(R_{\mathcal E}\). The matrix elements are
Trace preservation implies \((R_{\mathcal E})_{0,j} = \delta_{0,j}\), i.e. the first row is one and all zeros. Unitality implies \((R_{\mathcal E})_{i,0} = \delta_{i,0}\), the first column is one and all zeros.
In this representation the channel is applied to the state by multiplication
Kraus to Superoperator (Liouville)¶
We already saw an example of this in the section on vec-ing. There we
re-packaged conjugation by unitary evolution into the action of a matrix
on a vec’d density operator. Unitary evolution is simply the case of a
single Kraus operator, so we generalize this by taking a sum over all
Kraus operators.
Consider the set of Kraus operators \(\{ M_k \}\). The corresponding quantum operation is
Using the vec operator (see Eq. 1) this implies a superoperator
which acts as \(\mathcal E |\rho\rangle \rangle\) using Equation 2.
Note In quantum information a superoperator is an abstract concept. The object above is a concrete representation of the abstract concept in a particular basis. In the NMR community this particular construction is called the Liouville representation. The Pauli-Liouville representation is attained from Liouville representation by a change of basis, so the similarity in naming makes sense.
Kraus to Choi¶
Define $ | = _{i=0}^{d-1}|i,i $
One can show that
\(|A\rangle \rangle = {\rm vec}(A) = \sqrt{d} (I\otimes A) |\eta\rangle\).
The Choi state is
An often quoted equivalent expression is
\(\begin{align} \mathcal C &= I\otimes \mathcal E (|\eta \rangle \langle \eta|) \\\\ &=\sum_{ij} |i\rangle \langle j| \otimes \mathcal E (|i \rangle \langle j | ). \end{align}\)
\(\chi\) matrix to Pauli-Liouville matrix¶
Superoperator to Pauli-Liouville matrix¶
The standard basis on \(n\) qubits is called the computational basis. It is essentially all the strings \(|c_1\rangle=|0..0\rangle\) through to \(|c_{\rm max}\rangle = |1...1\rangle\). To convert between a superoperator and the Pauli-Liouville matrix representation we need to do a change of basis from the computational basis to the Pauli basis. This is achieved by the unitary
The we have
Superoperator to Choi¶
The conversion from the superoperator to a Choi matrix \(\mathcal C\) is simply a (column) reshuffling operation
It turns out that $ E = R(C)$ which means that \(\mathcal E= R(R(\mathcal E))\).
Pauli-Liouville matrix to Superoperator¶
To convert between the Pauli-Liouville matrix and the superoperator representation we need to to a change of basis from the Pauli basis to the computational basis. This is achieved by the unitary
which is simply \(U_{c2p}^\dagger\).
The we have
Pauli-Liouville to Choi¶
We obtain the normalized Choi matrix using the expression
Choi to Kraus¶
This is simply the reverse of the Kraus to Choi procedure.
Given the Choi matrix \(\mathcal C\) we find its eigenvalues \(\{\lambda_i\}\) and vectors \(\{|M_i\rangle\rangle \}\). Then the Kraus operators are
For numerical implementation one usually puts a threshold on the eigenvalues, say \(\lambda> 10^{-10}\), to prevent numerical instabilities.
Choi to Pauli-Liouville¶
First we normalize the Choi representation
Then the matrix elements of the Pauli-Liouville matrix representation of the channel can be obtained from the Choi state using
Choi to Superoperator¶
The conversion from a Choi matrix \(\mathcal C\) to a superoperator is simply a (column) reshuffling operation
It turns out that $ C = R(E)$ which means that \(\mathcal C= R(R(\mathcal C))\).
Examples: One qubit channels¶
Some observations:
- The Choi matrix of a unitary process always has rank 1.
- The superoperator / Liouville representation of a unitary process is always full rank.
- The eigenvalues of a Choi matrix give you an upper bound to the probability a particular (canonical) Kraus operator will occur (generally that probability depends on the state). This is helpful when sampling Kraus operators (you can test for which occurred according to the order of these eigenvalues).
- The \(\chi\) matrix (in the Pauli basis) is very convenient for computing the result of Pauli twirling or Clifford twirling the corresponding process.
Unitary Channels or Gates¶
As an example we look at two single qubit channels \(R_z(\theta) = \exp(-i \theta Z/2)\) and \(H\). The Hadamard is is a nice channel to examine as it transforms \(X\) and \(Z\) to each other
which can be easily seen in some of the channel representations.
Kraus
As the channel is unitary there is only one Kraus operator used in the operator sum representation. However we express them in the Pauli basis to make some of the below manipulations easier
:math:`chi` matrix (process)
Pauli-Liouville matrix
Superoperator
Choi
Pauli Channels¶
Pauli channels are nice because they are diagonal in two representations and they have the depolarizing channel as a special case.
In the operator sum representation a single qubit Pauli channel is defined as
where \(p_x,p_y,p_z\ge 0\) and \(p_x+p_y+p_z\le 1\).
If we define \(p' = p_x+p_y+p_z\) then
The Pauli channel specializes to the depolarizing channel when
for \(0\le p \le 1\).
Kraus
The Kraus operators used in the operator sum representation are
:math:`chi` matrix (process)
Pauli-Liouville matrix
Superoperator
So
Choi
Amplitude Damping or the \(T_1\) channel¶
Amplitude damping is an energy dissipation (or relaxation) process. If a qubit it in its excited state \(|1\rangle\) it may emit energy, a photon, and transition to the ground state \(|0\rangle\). In device physics an experiment that measures the decay over some time \(t\), with functional form \(\exp(-\Gamma t)\), is known as a \(T_1\) experiment (where \(T_1 = 1/\Gamma\)).
From the perspective of quantum channels the amplitude damping channel is interesting as is an example of a non-unital channel i.e. one that does not have the identity matrix as a fixed point \(\mathcal E_{AD} (I) \neq I\).
Kraus
The Kraus operators are
where \(\sigma_- = (\sigma_+)^\dagger= \frac 1 2 (X +i Y) =|0\rangle \langle 1| \) process we make the decay rate time dependant \(\gamma(t) = \exp(-\Gamma t)\).
:math:`chi` matrix (process)
Pauli-Liouville matrix
Superoperator
Choi
Examples: Two qubit channels¶
This section will not be as comprehensive we only consider two channels and two representations the operator sum representation (Kraus) and the superoperator representation.
A unitary channel on one qubit
\[\mathcal U_{IZ}(\rho) = U_{IZ} \rho U_{IZ}^\dagger\]with Kraus operator \(U_{IZ} = I\otimes Z = IZ\).
A dephasing channel on one qubit
\[\mathcal E_{IZ}(\rho) = (1-p)II \rho II + p IZ \rho IZ,\]with Kraus operators \(M_0=\sqrt{1-p}II\) and \(M_1= \sqrt{p}IZ\).
References¶
| [IGST] | Introduction to Quantum Gate Set Tomography. Greenbaum. arXiv:1509.02921, (2015). https://arxiv.org/abs/1509.02921 |
| [QN] | Quantum Nescimus. Improving the characterization of quantum systems from limited information. Harper. PhD thesis University of Sydney, 2018. https://ses.library.usyd.edu.au/handle/2123/17896 |
| [GRAPTN] | (1, 2, 3) Tensor networks and graphical calculus for open quantum systems. Wood et al. Quant. Inf. Comp. 15, 0579-0811 (2015). https://arxiv.org/abs/1111.6950 |
| [SVDMAT] | Singular value decomposition and matrix reorderings in quantum information theory. Miszczak. Int. J. Mod. Phys. C 22, No. 9, 897 (2011). https://dx.doi.org/10.1142/S0129183111016683 https://arxiv.org/abs/1011.1585 |
| [VECQO] | Vectorization of quantum operations and its use. Gilchrist et al., arXiv:0911.2539, (2009). https://arxiv.org/abs/0911.2539 |
| [MATQO] | On the Matrix Representation of Quantum Operations. Nambu et al. arXiv: 0504091 (2005). https://arxiv.org/abs/quant-ph/0504091 |
| [DUAL] | On duality between quantum maps and quantum states. Zyczkowski et al. Open Syst. Inf. Dyn. 11, 3 (2004). https://dx.doi.org/10.1023/B:OPSY.0000024753.05661.c2 https://arxiv.org/abs/quant-ph/0401119 |