Differentiate between Local and Global UnitariesIf all quantum gates must be unitary, what about...
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Differentiate between Local and Global Unitaries
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$begingroup$
Just like we have the PPT, NPT criteria for checking if states can be written in Tensor form or not, is there any criteria, given the matrix of a unitary acting on 2 qubits, to check if it is local or global (can be factored or not)?
mathematics matrix-representation unitarity
asked 5 hours ago
Mahathi VempatiMahathi Vempati
4638
$endgroup$
add a comment |
$begingroup$
Just like we have the PPT, NPT criteria for checking if states can be written in Tensor form or not, is there any criteria, given the matrix of a unitary acting on 2 qubits, to check if it is local or global (can be factored or not)?
mathematics matrix-representation unitarity
asked 5 hours ago
Mahathi VempatiMahathi Vempati
4638
$endgroup$
add a comment |
$begingroup$
Just like we have the PPT, NPT criteria for checking if states can be written in Tensor form or not, is there any criteria, given the matrix of a unitary acting on 2 qubits, to check if it is local or global (can be factored or not)?
mathematics matrix-representation unitarity
asked 5 hours ago
Mahathi VempatiMahathi Vempati
4638
$endgroup$
Just like we have the PPT, NPT criteria for checking if states can be written in Tensor form or not, is there any criteria, given the matrix of a unitary acting on 2 qubits, to check if it is local or global (can be factored or not)?
mathematics matrix-representation unitarity
mathematics matrix-representation unitarity
asked 5 hours ago
Mahathi VempatiMahathi Vempati
4638
asked 5 hours ago
Mahathi VempatiMahathi Vempati
4638
asked 5 hours ago
Mahathi VempatiMahathi Vempati
4638
asked 5 hours ago
Mahathi VempatiMahathi Vempati
4638
asked 5 hours ago
Mahathi VempatiMahathi Vempati
4638
4638
add a comment |
add a comment |
1 Answer
1
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$begingroup$
This is actually a much easier problem. In the case of states, you're trying to use the PPT criterion, or others, to distinguish if $rho$ can be written in the form
$$
rho=sum_ip_isigma^A_iotimessigma^B_i,
$$
where $sum_ip_i=1$ and the $sigma^A_i$ and $sigma^B_i$ are valid states on single sites. The difficulty actually comes from the freedom that the summation over $i$ provides.
In the case of unitaries (or more general operations), you're only trying to ascertain if $U$ can be written in the form
$$
U=U^Aotimes U^B
$$
or not. This is something that you can do very mechanically. For example, if we make matrices
$$
sum_{k,l}U_{ik,jl}|kranglelangle l|,
$$
then each of these ought to be of the form $U^A_{ij}U^B$, in other words, the same up to a constant. If they're not, it's not of tensor product form.
answered 3 hours ago
DaftWullieDaftWullie
14.2k1540
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
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oldest
votes
$begingroup$
This is actually a much easier problem. In the case of states, you're trying to use the PPT criterion, or others, to distinguish if $rho$ can be written in the form
$$
rho=sum_ip_isigma^A_iotimessigma^B_i,
$$
where $sum_ip_i=1$ and the $sigma^A_i$ and $sigma^B_i$ are valid states on single sites. The difficulty actually comes from the freedom that the summation over $i$ provides.
In the case of unitaries (or more general operations), you're only trying to ascertain if $U$ can be written in the form
$$
U=U^Aotimes U^B
$$
or not. This is something that you can do very mechanically. For example, if we make matrices
$$
sum_{k,l}U_{ik,jl}|kranglelangle l|,
$$
then each of these ought to be of the form $U^A_{ij}U^B$, in other words, the same up to a constant. If they're not, it's not of tensor product form.
answered 3 hours ago
DaftWullieDaftWullie
14.2k1540
$endgroup$
add a comment |
$begingroup$
This is actually a much easier problem. In the case of states, you're trying to use the PPT criterion, or others, to distinguish if $rho$ can be written in the form
$$
rho=sum_ip_isigma^A_iotimessigma^B_i,
$$
where $sum_ip_i=1$ and the $sigma^A_i$ and $sigma^B_i$ are valid states on single sites. The difficulty actually comes from the freedom that the summation over $i$ provides.
In the case of unitaries (or more general operations), you're only trying to ascertain if $U$ can be written in the form
$$
U=U^Aotimes U^B
$$
or not. This is something that you can do very mechanically. For example, if we make matrices
$$
sum_{k,l}U_{ik,jl}|kranglelangle l|,
$$
then each of these ought to be of the form $U^A_{ij}U^B$, in other words, the same up to a constant. If they're not, it's not of tensor product form.
answered 3 hours ago
DaftWullieDaftWullie
14.2k1540
$endgroup$
add a comment |
$begingroup$
This is actually a much easier problem. In the case of states, you're trying to use the PPT criterion, or others, to distinguish if $rho$ can be written in the form
$$
rho=sum_ip_isigma^A_iotimessigma^B_i,
$$
where $sum_ip_i=1$ and the $sigma^A_i$ and $sigma^B_i$ are valid states on single sites. The difficulty actually comes from the freedom that the summation over $i$ provides.
In the case of unitaries (or more general operations), you're only trying to ascertain if $U$ can be written in the form
$$
U=U^Aotimes U^B
$$
or not. This is something that you can do very mechanically. For example, if we make matrices
$$
sum_{k,l}U_{ik,jl}|kranglelangle l|,
$$
then each of these ought to be of the form $U^A_{ij}U^B$, in other words, the same up to a constant. If they're not, it's not of tensor product form.
answered 3 hours ago
DaftWullieDaftWullie
14.2k1540
$endgroup$
This is actually a much easier problem. In the case of states, you're trying to use the PPT criterion, or others, to distinguish if $rho$ can be written in the form
$$
rho=sum_ip_isigma^A_iotimessigma^B_i,
$$
where $sum_ip_i=1$ and the $sigma^A_i$ and $sigma^B_i$ are valid states on single sites. The difficulty actually comes from the freedom that the summation over $i$ provides.
In the case of unitaries (or more general operations), you're only trying to ascertain if $U$ can be written in the form
$$
U=U^Aotimes U^B
$$
or not. This is something that you can do very mechanically. For example, if we make matrices
$$
sum_{k,l}U_{ik,jl}|kranglelangle l|,
$$
then each of these ought to be of the form $U^A_{ij}U^B$, in other words, the same up to a constant. If they're not, it's not of tensor product form.
answered 3 hours ago
DaftWullieDaftWullie
14.2k1540
answered 3 hours ago
DaftWullieDaftWullie
14.2k1540
answered 3 hours ago
DaftWullieDaftWullie
14.2k1540
answered 3 hours ago
DaftWullieDaftWullie
14.2k1540
14.2k1540
add a comment |
add a comment |
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