General Functions

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General Functions
In Hilbert Space, nobody can hear you scream...
Hilbert Space
​Hilbert Space​
The primary mathematical object in quantum theory is the Hilbert space. We consider only finite-dimensional Hilbert spaces, denoted by H\mathcal{H}H
. Although we will be considering finite-dimensional spaces exclusively, we note here that many of the statements and claims extend directly to the case of separable, infinite-dimensional Hilbert spaces, especially for operationally-defined tasks and information quantities.
A ddd
-dimensional Hilbert space (1≤d<∞)(1 \le d < \infty)(1≤d<∞)
 is defined to be a complex vector space equipped with an inner product. We use the notation ∣ψ⟩{\displaystyle |\psi\rangle}∣ψ⟩
to denote a vector in H\mathcal{H}H
. More generally, a Hilbert space is a "complete inner product" space.
Completeness is an issue that pops up only in infinite-dimensional spaces, so all finite-dimensional inner-product spaces are Hilbert spaces.
Bra-Ket Notation
​Driac's Notation​
A ket is of the form ∣v⟩{\displaystyle |v\rangle } ∣v⟩
 . Mathematically it denotes a vector, v{\displaystyle {\boldsymbol {v}}}v
, in an abstract (complex) vector space V{\displaystyle V}V
, and physically it represents a state of some quantum system. An example of a Ket can be ∣r⟩=[xyz]{\displaystyle |r\rangle } = \begin{bmatrix} x \\ y\\ z\end{bmatrix} ∣r⟩=⎣⎡​xyz​⎦⎤​
 represents a vector r⃗=[xyz]{\displaystyle \vec{r} } = \begin{bmatrix} x \\ y\\ z\end{bmatrix}r=⎣⎡​xyz​⎦⎤​
.
A bra is of the form ⟨f∣{\displaystyle \langle f|}⟨f∣
. Mathematically it denotes a linear form f:V→C{\displaystyle f:V\to \mathbb {C} }f:V→C
, i.e. a linear map that maps each vector in V{\displaystyle V}V
 to a number in the complex plane C{\displaystyle \mathbb {C} }C
. Letting the linear functional ⟨f∣{\displaystyle \langle f|}⟨f∣
 act on a vector ∣v⟩{\displaystyle |v\rangle }∣v⟩
 is written as ⟨f∣v⟩∈C{\displaystyle \langle f|v\rangle \in \mathbb {C} }⟨f∣v⟩∈C
. The bra is similar to the ket, but the values are in a row, and each element is the complex conjugate of the ket's elements.
​
In the simple case where we consider the vector space  Cn{\displaystyle \mathbb {C} ^{n}}Cn
, a ket can be identified with a column vector, and a bra as a row vector.
​
Meaning : 
​⟨A∣=[A1A2A3…]{\displaystyle \langle A| }=\begin{bmatrix}A_1&A_2&A_3&\dots\end{bmatrix}⟨A∣=[A1​​A2​​A3​​…​]
     &     ∣B⟩=[B1B2B3⋮]{\displaystyle |B\rangle }=\begin{bmatrix}B_1\\B_2\\B_3\\\vdots\end{bmatrix}∣B⟩=⎣⎡​B1​B2​B3​⋮​⎦⎤​
​
Example:
​∣0⟩=[10]{\displaystyle |0\rangle }=\begin{bmatrix}1\\0\end{bmatrix}∣0⟩=[10​]
,  for two dimensional Hilbert Space ,
​
Defining a basis state ∣0⟩{\displaystyle |0\rangle }∣0⟩
, we can use the ket module like this:
from qutipy.general_functions import ket
​
# Defining a ket 0 in a 2Dimensional Hilbert space,
# The first argument takes a dimension of the Hilbert space,
# while the secind argument takes the ket value.
v = ket(2,0)
Here we have defined the ket v for ∣v⟩=[10]{\displaystyle |v\rangle } = \begin{bmatrix} 1 \\ 0 \end{bmatrix}∣v⟩=[10​]
. In numpy, defining the same would need one to define the matrix manually, just as shown in the Overview section.
Partial Transpose
​Peres–Horodecki criterion 
The Partial Transpose plays an important role in quantum information theory due to its connection with entanglement. In fact, it leads to a sufficient condition for a bipartite state to be entangled.
Given quantum systems AAA
 and BBB
, the partial transpose on BBB
 is denoted by TB≡idA⊗TBT_B\equiv id_A \otimes T_BTB​≡idA​⊗TB​
, and it is defined as, 
​TB(XAB):=∑j,j′=0dB−1(1A⊗∣i⟩⟨i′∣B)XAB(1A⊗∣i⟩⟨i′∣B)T_B(X_{AB})  :=  \sum\limits^{d_B-1}_{j, j'=0}   (\mathbf{1}_A  \otimes  \ket{i}\bra{i'}_B)  X_{AB}  (\mathbf{1}_A  \otimes  \ket{i}\bra{i'}_B) TB​(XAB​):=j,j′=0∑dB​−1​(1A​⊗∣i⟩⟨i′∣B​)XAB​(1A​⊗∣i⟩⟨i′∣B​)
​
​
partial_transpose(...) is a function that computes the partial transpose of a matrix. The transposition may be taken on any subset of the subsystems on which the matrix acts. 
Defining a state X with [ ... ]
import numpy as np
​
X = np.array(
    [
        [ 1,  2,  3,  4],
        [ 5,  6,  7,  8],
        [ 9, 10, 11, 12],
        [13, 14, 15, 16]
    ]
)
Now we can apply the partial_transpose function over our state X : 
from qutipy.general_functions import partial_transpose
​
pt = partial_transpose(X, [1], X.shape)
​
Pauli
States
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Hilbert Space
Bra-Ket Notation
Partial Transpose