This post reviews the theory and format of quantum bra-kets and ket-bras, which is also known as Dirac’s notation. Ket-bras are outer products in comparison to bra-kets which are inner products.
Definition
Wikipedia defines a bra-ket as follows:
‘In quantum mechanics, bra-ket notation, or Dirac notation, is used ubiquitously to denote quantum states…. A ket is in the form |v> and denotes a vector, v, in an abstract (complex) vector space V, and physically it represents a state of some quantum system.
A bra is in the form <f| and denotes a linear form, i.e. a linear map that maps each vector in V to a number in the complex plane. Letting the linear function f act on a vector v is written as <f||v> ε C.’
Qubits can be measured as being in only two states. The two states used allow the addressing of the first two dimensions of the electron’s movement. The use of i permits rotation out of these two dimensions into a third, allowing the electron’s movement to be addressed in three dimensions.
Quantum computing software allows the direction of the orbit of the electron in three dimensions. The Quantum computer hardware implements this in the qubit. Consequently, the location of the electron at any time relates to the instructions from the quantum program software. The hardware implements the program instructions. However, the measurement of a qubit only tells us if the electron is in the Northern or Southern hemisphere.
Bra-Ket <b|a>
Let a, b ε R2
Given the qubit states |a> = and |b> = , the bra-ket <b|a> is the dot or inner product of<b| and |a>.
It is necessary to reverse |b> in order to format the bra-ket <b|a> from |a> and |b>. To reverse |b> to get <b|, the first step is to transpose <b| to <b|t by switching the columns to a row. The superscript t represents transpose.
- |b> = .
- <b|t = (b1 b2)
As<b| is the complex conjugate of <b|t, the second step is to find <b| from <b|t. Note that the symbol * represents the complex conjugate, defined as follows:
- If b = c + d, b* = c + –d
So, <b| = |b>t*
- <b| = (b1 b2)* = (b1* b2*)
Note that b1 and b2 consist of ^/v and i^/iv Opposite Values and that b1 = c1 + d1, where c1 is a ^/v Opposite Value and d1 is an i^/iv Opposite Value. However, only the i Opposite Values have their signs flipped after complex conjugation.
The bra-ket can now be formed.
- bra-ket: <b|a> is the inner product of <b| and |a> and = (b1* b2*).(a1 a2) = a1b1* + a2b2*
Bra-Ket <a|b>
It is necessary to reverse the state |a> in order to format the bra-ket <a|b> from |a> and |b>.
- <a|t = (a1 a2)
- <a|t* = (a1 a2)* = (a1* b2*)
- bra-ket <a|b> = (a1* a2*).(b1 b2) = a1*b1 + a2*b2
- and <a|b>* = a1b1* + a2b2*.
- Note that the bra-ket for <a|b> is the same as the bra-ket for <b|a>
Ket-Bra |a><b|
Along with bra-kets in Dirac’s notation, there are ket-bras. A ket-bra is a product or outer product, unlike the bra-ket which is an inner product.
- ket-bra: |a><b| = ()*()
- =
- The result is a 2*2 matrix
Next: Basis and Born’s Rule
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