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QuTIpy : Quantum Theory of Information for Python
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States

Quantum states are the key mathematical objects in quantum theory. Quantum states are states of knowledge, representing uncertainty about the real physical state of the system.
Quantum mechanics is the backbone for quantum information processing, and many aspects of it cannot be explained by classical reasoning.
For example, there is no strong classical analogue for pure quantum states or entanglement, and this leads to stark differences between what is possible in the classical and quantum worlds.
However, at the same time, it is important to emphasize that all of classical information theory is subsumed by quantum information theory, so that whatever is possible with classical information processing is also possible with quantum information processing. As such, quantum information subsumes classical information while allowing for richer possibilities.

Quantum Systems

A quantum system
AA
 is associated with a Hilbert space
HAH_{A}
. The state of the system
AA
 is described by a density operator, which is a unit-trace, positive semi-definite linear operator acting on
HAH_{A}
.

Bipartite Quantum Systems

For distinct quantum systems
AA
 and
BB
 with associated Hilbert spaces
HAH_{A}
 and
HBH_{B}
 , the composite system
ABAB
 is associated with the Hilbert space
HA⊗HBH_{A} \otimes H_{B}
. This joint state is described by a bipartite quantum state
ρAB∈D(HA⊗HB)\rho_{AB} \in D(H_{A} \otimes H_{B})
. For brevity, the joint Hilbert space
HA⊗HBH_A \otimes H_B
 of the composite system
ABAB
 is denoted by
HABH_{AB}
 .
The measurement of a quantum system
AA
 is described by a Positive Operator Valued Measure (POVM)
{Mx}x∈X\{M_{x}\}_{x \in {X}}
, which is defined to be a collection of positive semi-definite operators indexed by a finite alphabet satisfying
∑x∈XMx=1HA\sum_{x \in X} {M_x} = 1_{H_A}
.
If the system is in the state
ρ\rho
, then the probability
Pr[x]Pr[x]
 of obtaining the outcome
xx
 is given by the Born rule as
Pr[x]=Tr[Mxρ]Pr[x] = Tr[M_x \rho ]
.

​Evolution​

The evolution of the state of a quantum system is described by a quantum channel, which is a linear, completely positive, and trace-preserving map acting on the state of the system.

Quantum States

A quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. The state of a quantum system is described by a density operator acting on the underlying Hilbert space of the quantum system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior.
A density operator is a unit-trace, positive semi-definite linear operator. We denote the set of density operators on a Hilbert space
HH
 as
D(H)D(H)
​
We typically use the Greek letters
ρ\rho
,
σ\sigma
,
τ\tau
, or
ω\omega
 to denote quantum states.

QuTIpy States

QuTIpy contains the definitions of these states, inside the states sub-module, and can be imported as such
from qutipy.states import (
Bell_state,
GHZ_state,
MaxEnt_state,
MaxMix_state,
RandomDensityMatrix,
RandomStateVector,
Werner_state,
Werner_twirl_state,
graph_state,
isotropic_state,
isotropic_twirl_state,
singlet_state
)

Maximally Entangled State

A pure state
ψAB=∣ψ⟩⟨ψ∣AB\displaystyle \psi_{AB} = |\psi\rangle\langle\psi|_{AB}
, for two systems
AA
 and
BB
 of the same dimension
dd
, is called Maximally Entangled if the Schmidt coefficients of
∣ψ⟩AB\displaystyle |\psi\rangle_{AB}
 are all equal to
1d\frac{1}{\sqrt{d}}
 , with
dd
 being the Schmidt rank of
∣ψ⟩AB\displaystyle |\psi\rangle_{AB}
.
In other words,
ψAB\psi_{AB}
 is called maximally entangled if
∣ψ⟩AB\displaystyle |\psi\rangle_{AB}
 has the Schmidt decomposition,
∣ψ⟩AB=1d∑k=1d∣ek⟩A⊗∣fk⟩B\displaystyle |\psi\rangle_{AB} = \frac{1}{\sqrt{d}}\sum_{k=1}^{d} |e_k\rangle_A \otimes |f_k\rangle_B
 for some orthonormal sets
{∣ek⟩A:1≤k≤d}\displaystyle \{ |e_k\rangle_A : 1 \le k \le d \}
 and
{∣fk⟩B:1≤k≤d}\displaystyle \{ |f_k\rangle_B : 1 \le k \le d \}
.
​
In simple terms, the Maximally Entangled can be written as
(1d)∗(∣0⟩∣0⟩+∣1⟩∣1⟩+...+∣d−1⟩∣d−1⟩)\displaystyle (\frac{1}{\sqrt{d}})*(|0\rangle|0\rangle+|1\rangle|1\rangle+...+|d-1\rangle|d-1\rangle)
 and can be created using the MaxEnt_state function.
# This will create a Macimally Entangled State for a 3 dimensional system.
# The resultant matrix will be of shape 9x9.
MaxEnt_state(3)

Bell State

​EPR Pairs​
A Bell state is defined as a maximally entangled quantum state of two qubits. It can be described as one of four entangled two qubit quantum states, known collectively as the four "Bell states".
​
∣ϕ+⟩≡∣ϕ0,0⟩=12(∣0,0⟩+∣1,1⟩)\displaystyle |\phi^{+}\rangle \equiv |\phi_{0, 0}\rangle = \frac{1}{\sqrt{2}} (|0, 0\rangle + |1, 1\rangle)
​
​
∣ϕ−⟩≡∣ϕ1,0⟩=12(∣0,0⟩−∣1,1⟩)\displaystyle |\phi^{-}\rangle \equiv |\phi_{1, 0}\rangle = \frac{1}{\sqrt{2}} (|0, 0\rangle - |1, 1\rangle)
​
​
∣ψ+⟩≡∣ϕ0,1⟩=12(∣0,1⟩+∣1,0⟩)\displaystyle |\psi^{+}\rangle \equiv |\phi_{0, 1}\rangle = \frac{1}{\sqrt{2}} (|0, 1\rangle + |1, 0\rangle)
​
​
∣ψ−⟩≡∣ϕ1,1⟩=12(∣0,1⟩−∣1,0⟩)\displaystyle |\psi^{-}\rangle \equiv |\phi_{1, 1}\rangle = \frac{1}{\sqrt{2}} (|0, 1\rangle - |1, 0\rangle)
​
A generalized version of the above Bell States is explained below,
Using the operators
XX
,
ZZ
, and
ZXZX
, we define the following set of four entangled two-qubit state vectors
∣ϕz,x⟩=(ZzXx⊗I)∣ϕ+⟩\displaystyle |\phi_{z,x}\rangle = (Z^zX^x \otimes I)|\phi^{+}\rangle
 for
z,x∈0,1z, x \in {0, 1}
.
To generates a
dd
-dimensional Bell State with
0<=z0 <= z
,
x<=d−1x <= d-1
, we can simply call the module Bell_state that was imported above.
# This will create a Bell State for a 2 dimensional system.
# The resultant matrix will be of shape 4x4.
Bell_state(d=2, z=1, x=1)

Singlet State

A singlet state is defined as
1(d2−d)×(I(d2)−F)\frac{1}{(d^2-d)} \times (I_{(d^2)}-F)
 where
FF
 is a Swap Operator.
Generating a singlet state is as easy as writing a single word,
# This will create a Singlet State for a 3 dimensional system.
# The resultant matrix will be of shape 9x9.
singlet_state(3)
​
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