Bra-ket notation to Matrix for entangled pairs

In summary, The notation |ab> for an entangled pair is the same as the tensor product |a> \otimes |b>. However, there is confusion when using corresponding matrices and applying a Hadamard operator twice to the input state |01>. The statement is ambiguous and it is not clear which qubit the Hadamard transformation should be applied to. When applying the transformation to a single qubit in a two-qubit system, one must use ##H_1 \otimes 1_2## in matrix form. The Hadamard matrix is symmetric and unitary, but it is unclear what happens when it is applied twice in a row.
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nomadreid
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I am confused about the the notation |ab> for an entangled pair. Isn't this the same as the tensor product |a> [itex]\otimes[/itex] |b>? If so, I run into another confusion when using the corresponding matrices. I read that I should apply a Hadamard operator H twice to the input state |01>. Does this mean (H|0>)[itex]\otimes[/itex] (H|1>)? I don't see how it could mean H(H(|01>), since |01> is represented by a four-by-one matrix, whereas H is represented by a two-by-two matrix. So what does this mean? Thanks
 
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No, unfortunately.
 
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nomadreid said:
I am confused about the the notation |ab> for an entangled pair. Isn't this the same as the tensor product |a> [itex]\otimes[/itex] |b>?
Yes.

nomadreid said:
If so, I run into another confusion when using the corresponding matrices. I read that I should apply a Hadamard operator H twice to the input state |01>. Does this mean (H|0>)[itex]\otimes[/itex] (H|1>)? I don't see how it could mean H(H(|01>), since |01> is represented by a four-by-one matrix, whereas H is represented by a two-by-two matrix. So what does this mean?
The statement is ambiguous. The use of the word "twice" mean for me that the same thing has to be done two times. If you were to apply a Hadamard transformation to both qubits, the language would be different, I think. However, what is not said is which qubit the Hadamard transformation must be applied to.

To apply the transformation to a single qubit in a two-qubit system, one must use ##H_1 \otimes 1_2## where the subscript refer to which qubit is operated on. In matrix form, this is a 4x4 matrix.

That said, the Hadamard matrix is symmetric and unitary. What happens when it is applied twice in row?
 
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1. What is "Bra-ket notation" and how is it used in quantum mechanics?

Bra-ket notation, also known as Dirac notation, is a mathematical notation used to describe states in quantum mechanics. It consists of a combination of a "bra" vector ⟨A| and a "ket" vector |B⟩ to create a scalar product ⟨A|B⟩. This notation is used to represent quantum states, operators, and measurements.

2. How can "Bra-ket notation" be translated into a matrix representation for entangled pairs?

In order to represent entangled quantum states using matrices, we can use the outer product of two ket vectors. For example, if we have two entangled qubits |A⟩ = a|0⟩ + b|1⟩ and |B⟩ = c|0⟩ + d|1⟩, their matrix representation would be |A⟩⟨B| = ac|00⟩ + ad|01⟩ + bc|10⟩ + bd|11⟩.

3. How does entanglement affect the matrix representation of quantum states?

Entanglement is a phenomenon in quantum mechanics where two or more particles become connected in such a way that the state of one particle cannot be described without considering the state of the other particle. This means that the matrix representation of entangled states cannot be written as a simple product of individual states, but must instead be represented using the outer product of the individual states.

4. What is the significance of using matrix representation for entangled quantum states?

Matrix representation is a useful tool in quantum mechanics because it allows us to perform calculations and make predictions about the behavior of entangled particles. By representing entangled states as matrices, we can apply mathematical operations to them and determine the outcomes of measurements made on the particles.

5. Can "Bra-ket notation" and matrix representation be used for multi-particle entangled states?

Yes, "Bra-ket notation" and matrix representation can be used for multi-particle entangled states. In this case, the outer product will involve more than two ket vectors, and the resulting matrix will have more rows and columns. This allows us to represent complex entangled states involving multiple particles and perform calculations on them using matrices.

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