In this chapter of the Qiskit Tutorial, you will learn about the Bra-Ket Notation. Bra-Ket notation is used ubiquitously in the field of Quantum Mechanics, Quantum Information, and Quantum Computing.
Ket
A Ket is a column vector. Any column vector a with elements a1 a2 a3 ... an can be written as |a>
Note– |a> is commonly also written as ket(a).
Bra
A Bra is a row vector. Any row vector b with elements b1 b2 b3 ... bn can be written as <b|
Note– <b| is commonly also written as bra(b).
Bras as Hermitian Conjugates of Kets
The Bra is a Hermitian conjugate of its corresponding Ket. For example, Bra- <b| is the hermitian conjugate of it’s corresponding Ket- |b>. The Hermitian conjugate of a Ket is the transpose of the complex conjugate of its elements. For a Ket |b> with elements b1 b2 b3 ... bn , the corresponding Bra <b|will have elements b1* b2* b3* ... bn* . This is further illustrated below.
Note– The complex conjugate of a complex number can be obtained by reversing the sign of the imaginary part of the complex number. The complex conjugate of a complex number a + ib is a - ib and that of a - ib is a + ib. All real numbers(i.e, those without an imaginary part) are their own complex conjugates.
Note– The Hermitian conjugate of a Ket- |b> is also written as |b>†
Example
For a Ket – |r> with elements x1 + i y1 and x2 - i y2
Bra-Ket
Bra-Ket is the multiplication of a Bra(a row vector) and a Ket(a column vector). The product of a Bra and Ket is a complex number. The Bra-Ket notation of a row vector <b| multiplied by a column vector |a> is <b|a>.
Example
In this example, we will multiply a Bra- <b|with elements b1 and b2 with a Ket- |a> with elements a1 and a2. The multiplication follows the rules of matrix multiplication. The product of a Bra and Ket is a complex number. The product of a Bra and a Ket is also called inner product.
Ket-Bra
Ket-Bra is the multiplication of a Ket(a column vector) and a Bra(a row vector). The product of a Ket and Bra is a matrix. The Ket-Bra notation of a column vector |a> multiplied by a row vector <b| is |a><b|.
Example
In this example, we will multiply a Ket- |a> with elements a1 and a2 with a Bra- <b|with elements b1 and b2 . The multiplication follows the rules of matrix multiplication. The product of a Ket and Bra is a matrix. The product of a Ket and a Bra is also called outer product.
