Understand the Outer Product of two qubits

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SUMMARY

The discussion focuses on the outer product of two qubits, specifically the expression |1>_a<1|, where |1>_a represents the ket for qubit 'a' and <1| is the bra for another qubit. It is established that the outer product of a bra and a ket forms an operator, exemplified by the notation ##|\alpha \rangle \langle \beta |##, which maps a ket ##|\gamma \rangle## to the resulting ket ##|\alpha \rangle \langle \beta | \gamma \rangle##. Furthermore, it is confirmed that this operator can be expressed in terms of Pauli matrices, applicable to any operator in quantum mechanics.

PREREQUISITES
  • Understanding of quantum mechanics concepts, specifically qubits
  • Familiarity with bra-ket notation in quantum mechanics
  • Knowledge of operators in quantum mechanics
  • Basic understanding of Pauli matrices and their applications
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  • Research the mathematical formulation of the outer product in quantum mechanics
  • Learn about the properties and applications of Pauli matrices
  • Explore the concept of tensor products in quantum systems
  • Study the role of operators in quantum state transformations
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Homework Statement
Understanding |1>_a<1|
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|1>_a<1|
Hi, I'm trying to understand an outer product |1>_a<1| where |1>_a is the ket for one qubit (a) and <1| is the bra for another qubit. Does this make sense and is it possible to express it in terms of tensor products or pauli matrices?
 
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I'm not sure about qubits specifically but in general the outer product of bra and a ket is an operator. E.g

##|\alpha \rangle \langle \beta |##

Is an operator that maps a ket ##|\gamma \rangle## to the ket ##|\alpha \rangle \langle \beta | \gamma \rangle##.

In that sense, you can express the operator in terms of Pauli matrices, as you can for any operator.
 

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