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Differentiate between Local and Global Unitaries


If all quantum gates must be unitary, what about measurement?Why are quantum gates unitary and not special unitary?What do they mean by “qubit can't be copied”?Solving a circuit implementing a two-level unitary operationMatrix representation and CX gateHow to properly write the action of a quantum gate implementing an operator $U$ on the superposition of its eigenvectors?Extending a square matrix to a unitary matrixDecomposition of a unitary matrixGrover operator as a rotation matrixIs it correct to say that we need controlled gates because unitary matrices are reversible?













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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







share|improve this question









$endgroup$

















    3












    $begingroup$


    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







    share|improve this question









    $endgroup$















      3












      3








      3





      $begingroup$


      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







      share|improve this question









      $endgroup$




      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






      share|improve this question













      share|improve this question











      share|improve this question




      share|improve this question










      asked 4 hours ago









      Mahathi VempatiMahathi Vempati

      4638




      4638






















          1 Answer
          1






          active

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          2












          $begingroup$

          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.






          share|improve this answer









          $endgroup$













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            1






            active

            oldest

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            active

            oldest

            votes






            active

            oldest

            votes









            2












            $begingroup$

            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.






            share|improve this answer









            $endgroup$


















              2












              $begingroup$

              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.






              share|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                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.






                share|improve this answer









                $endgroup$



                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.







                share|improve this answer












                share|improve this answer



                share|improve this answer










                answered 2 hours ago









                DaftWullieDaftWullie

                14.2k1540




                14.2k1540






























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