Quantum States and Amplitudes

In[]:=
<<Wolfram`QuantumFramework`

Key Concepts

◼
  • Quantum Amplitudes
    • ◼
  • State Vector
    • ◼
  • Density Matrix

    • State Vectors and Amplitudes

    You have already seen bra-ket notation for quantum states like
    |0〉
    or
    |1〉
    . What about other kinds of quantum states? For example, the result of applying a Hadamard gate to the register state:
    In[]:=
    state1=QuantumOperator["H"]@QuantumState["0"]
    Out[]=
    QuantumState
    Pure state
    Qudits: 1
    Type: Vector
    Dimension: 2
    
    In[]:=
    state1["BlochPlot"]
    Out[]=
    You can see from the Bloch plot that this state is plainly neither
    |0〉
    nor
    |1〉
    .
    In[]:=
    state1["Formula"]
    Out[]=
    1
    2
    |0〉+
    1
    2
    |1〉
    This state is instead a particular combination of
    |0〉
    and
    |1〉
    . Since it is a linear combination, this suggests that quantum states can be represented by vectors and operators can be represented by matrices.
    In[]:=
    state1["StateVector"]//MatrixForm//TraditionalForm
    Out[]//TraditionalForm=
    1
    2
    1
    2
    In[]:=
    QuantumOperator["H"]["Matrix"]//MatrixForm//TraditionalForm
    Out[]//TraditionalForm=
    1
    2
    1
    2
    1
    2
    -
    1
    2
    Notice the relationship between the vector coefficients and the probabilities for each classical outcome.
    In[]:=
    state1["Probabilities"]
    Out[]=
    0
    1
    2
    ,1
    1
    2
    
    The computational basis is called a basis because the states
    |0〉
    and
    |1〉
    can be used as basis elements for the complex vector space
    2
    
    :
    In[]:=
    QuantumState["0"]["StateVector"]//MatrixForm//TraditionalForm
    Out[]//TraditionalForm=
    
    1
    0
    
    In[]:=
    QuantumState["1"]["StateVector"]//MatrixForm//TraditionalForm
    Out[]//TraditionalForm=
    
    0
    1
    
    The coefficients of each basis element are known as the quantum amplitudes. A quantum state can be defined by giving its amplitudes:
    In[]:=
    state2=QuantumState[{a,b}]
    Out[]=
    QuantumState
    Pure state
    Qudits: 1
    Type: Vector
    Dimension: 2
    
    This can be written in Dirac notation:
    In[]:=
    state2["Formula"]//TraditionalForm
    Out[]//TraditionalForm=
    a|0〉+b|1〉
    The norm squared of the normalized amplitudes give the probability for measuring each of the basis elements as the outcome of a measurement interaction:
    In[]:=
    state2["Probabilities"]
    Out[]=
    0
    2
    Abs[a]
    2
    Abs[a]
    +
    2
    Abs[b]
    ,1
    2
    Abs[b]
    2
    Abs[a]
    +
    2
    Abs[b]
    
    Typically, qubit amplitudes are already normalized such that
    2
    a
    +
    2
    b
    =1
    .
    In[]:=
    state2["Probabilities"]//Map[#/.Abs[a]^2+Abs[b]^2->1&]
    Out[]=
    |0〉
    2
    Abs[a]
    ,|1〉
    2
    Abs[b]
    
    This quantum state can also be represented as a state vector:
    In[]:=
    state2["StateVector"]//MatrixForm//TraditionalForm
    Out[]//TraditionalForm=
    
    a
    b
    
    Some quantum states require a matrix representation known as the density matrix:
    In[]:=
    state2["DensityMatrix"]//MatrixForm//TraditionalForm
    Out[]//TraditionalForm=
    a
    *
    a
    a
    *
    b
    b
    *
    a
    b
    *
    b
    Keep in mind that the amplitudes can be complex numbers. The following plot lets you explore the relationship between the probability of measuring each state in the computational basis and the complex phase difference between each amplitude:
    Out[]=
    ​
    P(​|0〉​)
    Relative Phase
    Wolfram`QuantumFramework`QuantumState[{0.707107,0.707107}]BlochPlot,PlotLabel
    Wolfram`QuantumFramework`QuantumState
    p
    ,
    θ
    
    1-p
    [Formula]

    Example

    Consider the quantum state
    1
    2
    |0〉+
    3
    2
    |1〉
    .
    What is the state vector, the density matrix, the Bloch plot, and the measurement distribution in the computational basis?
    Solution
    Define the state with the given amplitudes:
    In[]:=
    state3=QuantumState12,
    3
    2
    Out[]=
    QuantumState
    Pure state
    Qudits: 1
    Type: Vector
    Dimension: 2
    
    Probabilities are the (normalized) norm squared of these amplitudes:
    In[]:=
    state3["Probabilities"]
    Out[]=
    0
    1
    4
    ,1
    3
    4
    
    Compute the state vector, density matrix, and Bloch plot:
    In[]:=
    state3["StateVector"]//MatrixForm//TraditionalForm
    Out[]//TraditionalForm=
    1
    2
    3
    2
    In[]:=
    state3["DensityMatrix"]//MatrixForm//TraditionalForm
    Out[]//TraditionalForm=
    1
    4
    3
    4
    3
    4
    3
    4
    The Bloch vector can be given in either spherical or Cartesian coordinates:

    States vs Measurements

    While a measurement operation is always necessary to obtain practical results from quantum circuits, it is often helpful to analyze what happens to quantum states as each gate operation is applied to the qubits.
    Consider the following quantum circuit:
    What happens to the initial state after each gate is applied?
    Recall that amplitudes do not have to be real numbers. They can take on complex values. What about a similar circuit but different initial state?
    Even though the measurement distributions for each of the two initial states were identical, they are different quantum states. Thus, this circuit has very different effects on them.