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.

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}
.

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
HAHBH_{A} \otimes H_{B}
. This joint state is described by a bipartite quantum state
ρABD(HAHB)\rho_{AB} \in D(H_{A} \otimes H_{B})
. For brevity, the joint Hilbert space
HAHBH_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}xX\{M_{x}\}_{x \in {X}}
, which is defined to be a collection of positive semi-definite operators indexed by a finite alphabet satisfying
xXMx=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 ]
.

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.

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 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
)

A pure state
ψAB=ψψAB\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
ψAB\displaystyle |\psi\rangle_{AB}
are all equal to
1d\frac{1}{\sqrt{d}}
, with
dd
being the Schmidt rank of
ψAB\displaystyle |\psi\rangle_{AB}
.
In other words,
ψAB\psi_{AB}
is called maximally entangled if
ψAB\displaystyle |\psi\rangle_{AB}
has the Schmidt decomposition,
ψAB=1dk=1dekAfkB\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
{ekA:1kd}\displaystyle \{ |e_k\rangle_A : 1 \le k \le d \}
and
{fkB:1kd}\displaystyle \{ |f_k\rangle_B : 1 \le k \le d \}
.
In simple terms, the Maximally Entangled can be written as
(1d)(00+11+...+d1d1)\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)

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)
ϕϕ1,0=12(0,01,1)\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)
ψϕ1,1=12(0,11,0)\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=(ZzXxI)ϕ+\displaystyle |\phi_{z,x}\rangle = (Z^zX^x \otimes I)|\phi^{+}\rangle
for
z,x0,1z, x \in {0, 1}
.
To generates a
dd
-dimensional Bell State with
0<=z0 <= z
,
x<=d1x <= 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)

A singlet state is defined as
1(d2d)×(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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Quantum Systems
Quantum States
Maximally Entangled State
Bell State
Singlet State