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The relationship between entanglement of vector states to matrix operations


How to keep track of entanglements when emulating quantum computation?Entanglement distillation by local operations and post-selection using one entanglement pairHow is a single qubit fundamentally different from a classical coin spinning in the air?Matrix representation and CX gateRelation between quantum entanglement and quantum state complexityHow to analyze highly entangled quantum circuits?Composing the CNOT gate as a tensor product of two level matricesIs it correct to say that we need controlled gates because unitary matrices are reversible?Decomposition of any 2-level matrix into single qubit and CNOT gatesWhat is the meaning of the state $|1rangle-|1rangle$?













1












$begingroup$


I don't understand something which is I believe pretty fundamental. It's said that an operation represented by a matrix A is an entanglement if A can't be written as a tensor product of other matrices. On the other hand, I just learned lately that a vector state too can be determined by these criteria. That is, it's entangled if and only if it can't be written as a tensor product of other vector states.



Here comes the confusion. Suppose you take a CNOT for example. This obviously can't be written as a tensor product of other matrices as it's known to be entangled. What're the consequences of applying a CNOT on a vector? I don't understand it, if you take a vector like (1,0,0,0) or (0,0,10) you will get after CNOT (1,0,0,0) or (0,0,0,1) respectively. Which obviously can be written as the tensor product of other vectors. So what's going on here? Is CNOT supposed to entangle the vector state or not?



Why is it that the gate itself can't be decomposed but the vector which it acted upon can? Or if it isn't necessary then, what's the definition or requirements for entanglement which I am missing?










share|improve this question











$endgroup$

















    1












    $begingroup$


    I don't understand something which is I believe pretty fundamental. It's said that an operation represented by a matrix A is an entanglement if A can't be written as a tensor product of other matrices. On the other hand, I just learned lately that a vector state too can be determined by these criteria. That is, it's entangled if and only if it can't be written as a tensor product of other vector states.



    Here comes the confusion. Suppose you take a CNOT for example. This obviously can't be written as a tensor product of other matrices as it's known to be entangled. What're the consequences of applying a CNOT on a vector? I don't understand it, if you take a vector like (1,0,0,0) or (0,0,10) you will get after CNOT (1,0,0,0) or (0,0,0,1) respectively. Which obviously can be written as the tensor product of other vectors. So what's going on here? Is CNOT supposed to entangle the vector state or not?



    Why is it that the gate itself can't be decomposed but the vector which it acted upon can? Or if it isn't necessary then, what's the definition or requirements for entanglement which I am missing?










    share|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      I don't understand something which is I believe pretty fundamental. It's said that an operation represented by a matrix A is an entanglement if A can't be written as a tensor product of other matrices. On the other hand, I just learned lately that a vector state too can be determined by these criteria. That is, it's entangled if and only if it can't be written as a tensor product of other vector states.



      Here comes the confusion. Suppose you take a CNOT for example. This obviously can't be written as a tensor product of other matrices as it's known to be entangled. What're the consequences of applying a CNOT on a vector? I don't understand it, if you take a vector like (1,0,0,0) or (0,0,10) you will get after CNOT (1,0,0,0) or (0,0,0,1) respectively. Which obviously can be written as the tensor product of other vectors. So what's going on here? Is CNOT supposed to entangle the vector state or not?



      Why is it that the gate itself can't be decomposed but the vector which it acted upon can? Or if it isn't necessary then, what's the definition or requirements for entanglement which I am missing?










      share|improve this question











      $endgroup$




      I don't understand something which is I believe pretty fundamental. It's said that an operation represented by a matrix A is an entanglement if A can't be written as a tensor product of other matrices. On the other hand, I just learned lately that a vector state too can be determined by these criteria. That is, it's entangled if and only if it can't be written as a tensor product of other vector states.



      Here comes the confusion. Suppose you take a CNOT for example. This obviously can't be written as a tensor product of other matrices as it's known to be entangled. What're the consequences of applying a CNOT on a vector? I don't understand it, if you take a vector like (1,0,0,0) or (0,0,10) you will get after CNOT (1,0,0,0) or (0,0,0,1) respectively. Which obviously can be written as the tensor product of other vectors. So what's going on here? Is CNOT supposed to entangle the vector state or not?



      Why is it that the gate itself can't be decomposed but the vector which it acted upon can? Or if it isn't necessary then, what's the definition or requirements for entanglement which I am missing?







      quantum-gate quantum-state entanglement






      share|improve this question















      share|improve this question













      share|improve this question




      share|improve this question








      edited 1 hour ago









      Blue

      6,24531355




      6,24531355










      asked 2 hours ago









      bilanushbilanush

      976




      976






















          1 Answer
          1






          active

          oldest

          votes


















          2












          $begingroup$

          There are 3 different questions you need to ask yourself. First is the operator a tensor product of smaller matrices. Then is your input state a tensor product of smaller vectors. Finally is the result a tensor product of smaller vectors.



          Stick with the two qubit example and CNOT that you had.



          First question: CNOT is not able to be written as $A otimes B$ with $A$ and $B$ operating on 1 qubit each.
          Second question: The examples you gave were $(1,0,0,0)=|0 0 rangle$ which can be broken up as $|0 rangle otimes |0 rangle$. The other example can as well. It is $|1 rangle otimes |0 rangle$.
          Third question: The result states were $(1,0,0,0)$ which was not entangled again. The other was $(0,0,0,1)$ which is $|1 rangle otimes |1 rangle$ also not entangled.



          The implication goes differently than you seem to think.



          If the operator was not entangled and written as $A otimes B$ and the input state was not entangled and written as $v otimes w$ with the same dimensions as the operator, then you would get $Av otimes Bw$ as the output. So not entangled operator AND not entangled input vector with the same dimensionality imply not entangled on the output.



          So the example shows if you only have operator entangled but input not, the output can still be not entangled.



          As another example, take the identity operator. It can be written as $I otimes I$ so it is not entangled. Apply it to an entangled input, you get the same state out. That gives the case of unentangled operator and entangled input giving entangled output.






          share|improve this answer











          $endgroup$













          • $begingroup$
            So why is it matter to call CNOT an entanglement if it doesn't act it on the state? since this is what we mostly care about. I mean, is it possible to say that entanglement state is the only one which can turn an unentangled state to an entangled one? Can you show me a state which is unentangled but the CNOT turns it into one? Because otherwise who cares about entangled operation
            $endgroup$
            – bilanush
            18 mins ago











          Your Answer





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          $begingroup$

          There are 3 different questions you need to ask yourself. First is the operator a tensor product of smaller matrices. Then is your input state a tensor product of smaller vectors. Finally is the result a tensor product of smaller vectors.



          Stick with the two qubit example and CNOT that you had.



          First question: CNOT is not able to be written as $A otimes B$ with $A$ and $B$ operating on 1 qubit each.
          Second question: The examples you gave were $(1,0,0,0)=|0 0 rangle$ which can be broken up as $|0 rangle otimes |0 rangle$. The other example can as well. It is $|1 rangle otimes |0 rangle$.
          Third question: The result states were $(1,0,0,0)$ which was not entangled again. The other was $(0,0,0,1)$ which is $|1 rangle otimes |1 rangle$ also not entangled.



          The implication goes differently than you seem to think.



          If the operator was not entangled and written as $A otimes B$ and the input state was not entangled and written as $v otimes w$ with the same dimensions as the operator, then you would get $Av otimes Bw$ as the output. So not entangled operator AND not entangled input vector with the same dimensionality imply not entangled on the output.



          So the example shows if you only have operator entangled but input not, the output can still be not entangled.



          As another example, take the identity operator. It can be written as $I otimes I$ so it is not entangled. Apply it to an entangled input, you get the same state out. That gives the case of unentangled operator and entangled input giving entangled output.






          share|improve this answer











          $endgroup$













          • $begingroup$
            So why is it matter to call CNOT an entanglement if it doesn't act it on the state? since this is what we mostly care about. I mean, is it possible to say that entanglement state is the only one which can turn an unentangled state to an entangled one? Can you show me a state which is unentangled but the CNOT turns it into one? Because otherwise who cares about entangled operation
            $endgroup$
            – bilanush
            18 mins ago
















          2












          $begingroup$

          There are 3 different questions you need to ask yourself. First is the operator a tensor product of smaller matrices. Then is your input state a tensor product of smaller vectors. Finally is the result a tensor product of smaller vectors.



          Stick with the two qubit example and CNOT that you had.



          First question: CNOT is not able to be written as $A otimes B$ with $A$ and $B$ operating on 1 qubit each.
          Second question: The examples you gave were $(1,0,0,0)=|0 0 rangle$ which can be broken up as $|0 rangle otimes |0 rangle$. The other example can as well. It is $|1 rangle otimes |0 rangle$.
          Third question: The result states were $(1,0,0,0)$ which was not entangled again. The other was $(0,0,0,1)$ which is $|1 rangle otimes |1 rangle$ also not entangled.



          The implication goes differently than you seem to think.



          If the operator was not entangled and written as $A otimes B$ and the input state was not entangled and written as $v otimes w$ with the same dimensions as the operator, then you would get $Av otimes Bw$ as the output. So not entangled operator AND not entangled input vector with the same dimensionality imply not entangled on the output.



          So the example shows if you only have operator entangled but input not, the output can still be not entangled.



          As another example, take the identity operator. It can be written as $I otimes I$ so it is not entangled. Apply it to an entangled input, you get the same state out. That gives the case of unentangled operator and entangled input giving entangled output.






          share|improve this answer











          $endgroup$













          • $begingroup$
            So why is it matter to call CNOT an entanglement if it doesn't act it on the state? since this is what we mostly care about. I mean, is it possible to say that entanglement state is the only one which can turn an unentangled state to an entangled one? Can you show me a state which is unentangled but the CNOT turns it into one? Because otherwise who cares about entangled operation
            $endgroup$
            – bilanush
            18 mins ago














          2












          2








          2





          $begingroup$

          There are 3 different questions you need to ask yourself. First is the operator a tensor product of smaller matrices. Then is your input state a tensor product of smaller vectors. Finally is the result a tensor product of smaller vectors.



          Stick with the two qubit example and CNOT that you had.



          First question: CNOT is not able to be written as $A otimes B$ with $A$ and $B$ operating on 1 qubit each.
          Second question: The examples you gave were $(1,0,0,0)=|0 0 rangle$ which can be broken up as $|0 rangle otimes |0 rangle$. The other example can as well. It is $|1 rangle otimes |0 rangle$.
          Third question: The result states were $(1,0,0,0)$ which was not entangled again. The other was $(0,0,0,1)$ which is $|1 rangle otimes |1 rangle$ also not entangled.



          The implication goes differently than you seem to think.



          If the operator was not entangled and written as $A otimes B$ and the input state was not entangled and written as $v otimes w$ with the same dimensions as the operator, then you would get $Av otimes Bw$ as the output. So not entangled operator AND not entangled input vector with the same dimensionality imply not entangled on the output.



          So the example shows if you only have operator entangled but input not, the output can still be not entangled.



          As another example, take the identity operator. It can be written as $I otimes I$ so it is not entangled. Apply it to an entangled input, you get the same state out. That gives the case of unentangled operator and entangled input giving entangled output.






          share|improve this answer











          $endgroup$



          There are 3 different questions you need to ask yourself. First is the operator a tensor product of smaller matrices. Then is your input state a tensor product of smaller vectors. Finally is the result a tensor product of smaller vectors.



          Stick with the two qubit example and CNOT that you had.



          First question: CNOT is not able to be written as $A otimes B$ with $A$ and $B$ operating on 1 qubit each.
          Second question: The examples you gave were $(1,0,0,0)=|0 0 rangle$ which can be broken up as $|0 rangle otimes |0 rangle$. The other example can as well. It is $|1 rangle otimes |0 rangle$.
          Third question: The result states were $(1,0,0,0)$ which was not entangled again. The other was $(0,0,0,1)$ which is $|1 rangle otimes |1 rangle$ also not entangled.



          The implication goes differently than you seem to think.



          If the operator was not entangled and written as $A otimes B$ and the input state was not entangled and written as $v otimes w$ with the same dimensions as the operator, then you would get $Av otimes Bw$ as the output. So not entangled operator AND not entangled input vector with the same dimensionality imply not entangled on the output.



          So the example shows if you only have operator entangled but input not, the output can still be not entangled.



          As another example, take the identity operator. It can be written as $I otimes I$ so it is not entangled. Apply it to an entangled input, you get the same state out. That gives the case of unentangled operator and entangled input giving entangled output.







          share|improve this answer














          share|improve this answer



          share|improve this answer








          edited 1 hour ago









          Blue

          6,24531355




          6,24531355










          answered 2 hours ago









          AHusainAHusain

          1,8391311




          1,8391311












          • $begingroup$
            So why is it matter to call CNOT an entanglement if it doesn't act it on the state? since this is what we mostly care about. I mean, is it possible to say that entanglement state is the only one which can turn an unentangled state to an entangled one? Can you show me a state which is unentangled but the CNOT turns it into one? Because otherwise who cares about entangled operation
            $endgroup$
            – bilanush
            18 mins ago


















          • $begingroup$
            So why is it matter to call CNOT an entanglement if it doesn't act it on the state? since this is what we mostly care about. I mean, is it possible to say that entanglement state is the only one which can turn an unentangled state to an entangled one? Can you show me a state which is unentangled but the CNOT turns it into one? Because otherwise who cares about entangled operation
            $endgroup$
            – bilanush
            18 mins ago
















          $begingroup$
          So why is it matter to call CNOT an entanglement if it doesn't act it on the state? since this is what we mostly care about. I mean, is it possible to say that entanglement state is the only one which can turn an unentangled state to an entangled one? Can you show me a state which is unentangled but the CNOT turns it into one? Because otherwise who cares about entangled operation
          $endgroup$
          – bilanush
          18 mins ago




          $begingroup$
          So why is it matter to call CNOT an entanglement if it doesn't act it on the state? since this is what we mostly care about. I mean, is it possible to say that entanglement state is the only one which can turn an unentangled state to an entangled one? Can you show me a state which is unentangled but the CNOT turns it into one? Because otherwise who cares about entangled operation
          $endgroup$
          – bilanush
          18 mins ago


















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