Vector Notation of Quantum States for 2 Qubit System

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SUMMARY

The vector notation for quantum states in a 2 qubit system is derived using the tensor product of two vector spaces, each of dimension 2. The states are represented as follows: |00⟩ = (1; 0; 0; 0), |01⟩ = (0; 1; 0; 0), |10⟩ = (0; 0; 1; 0), and |11⟩ = (0; 0; 0; 1). Understanding this notation is essential for working with quantum computing concepts and linear algebra. Resources such as Wikipedia and various lecture notes provide further clarity on the tensor product method.

PREREQUISITES
  • Understanding of Dirac notation for quantum states
  • Basic knowledge of linear algebra concepts
  • Familiarity with tensor products in vector spaces
  • Introduction to quantum computing principles
NEXT STEPS
  • Study the tensor product of vector spaces in detail
  • Learn about quantum state representation in higher dimensions
  • Explore quantum computing resources such as lecture notes from CMU and Berkeley
  • Practice calculating tensor products with different quantum states
USEFUL FOR

Students and professionals in quantum computing, physicists, and anyone interested in understanding the mathematical representation of quantum states in multi-qubit systems.

MrMuscle
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How can I write the vector notation of quantum states when I have a 2 qubit system?
I am confused about the vector notation of quantum states when I have a 2 qubit system.

For 1 qubit, I just write l1> = (0 ;1 ) for representing 1,

and l0> = (1;0) for representing 0.

Dirac notation is straightforward
However when it comes to representing two qubits in linear algebra I am confused :(

What is the vector notations for the following ones? Can you suggest a method to calculate them I don't want to memorize.

l00> = ?
l11>= ?
l10>= ?
l01>= ?

Thanks in advance!
 
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MrMuscle said:
Summary: How can I write the vector notation of quantum states when I have a 2 qubit system?

I am confused about the vector notation of quantum states when I have a 2 qubit system.

For 1 qubit, I just write l1> = (0 ;1 ) for representing 1,

and l0> = (1;0) for representing 0.

Dirac notation is straightforward
However when it comes to representing two qubits in linear algebra I am confused :(

What is the vector notations for the following ones? Can you suggest a method to calculate them I don't want to memorize.

l00> = ?
l11>= ?
l10>= ?
l01>= ?

Thanks in advance!
What you are doing is making a tensor product of two vector spaces, each of dimension 2. Read the first part of this
https://en.wikipedia.org/wiki/Tensor_productor may be this is clearer
https://www.math3ma.com/blog/the-tensor-product-demystified
 
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