1
$\begingroup$

I have a statevector of 10 qubits. I want to check if the qubits 0,1 and 2,3 are EPR pairs/Bell pairs. The other qubits (4,5,6,7,8,9) have been measured.

There are four possible 2-qubit states that are Bell pairs. I created circuits for all possible tensor products of these 4 states with the following code (total 16 states):

num_qubits = 10

#measurement_string contains the measurement values of qubits 4,5,6,7,8,9
#example:measurement_string  = 001010 means q4 was measured = 0,q5 = 0, q6 = 1, q7 = 0,q8 = 1, q9 = 0

def epr_product_1(measurement_string):
  #first entangled pair
  qc = QuantumCircuit(num_qubits)
  qc.h(0)
  qc.cx(0,1)

  #second entangled pair
  qc.h(1)
  qc.cx(2,3)

  #qubits 4,5,6,7,8,9 were measured as either 0 or 1, create the circuit accordingly
  for index,s in enumerate(measurement_string):
    if s == '1':
        qc.x(index + 4)

  return Statevector.from_instruction(qc)


def epr_product_2(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)
    
    qc.x(0)
    qc.h(0)

    qc.cx(0,1)

    #second entangled pair
    qc.x(2)
    qc.h(2)

    qc.cx(2,3)
    
        
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)


def epr_product_1_2(measurement_string):
    #firdt entangled pair
    qc = QuantumCircuit(num_qubits)
    
    qc.h(0)

    qc.cx(0,1)

    #second entangled pair
    qc.x(2)
    qc.h(2)

    qc.cx(2,3)
    
        
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    
    return Statevector.from_instruction(qc)

def epr_product_2_1(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)

    #second entangled pair
    qc.x(0)
    qc.h(0)

    qc.cx(0,1)
    
    
    qc.h(2)

    qc.cx(2,3)
    
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)


def epr_product_3(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)
    
    qc.x(1)
    qc.h(0)

    qc.cx(0,1)

    #second entangled pair
    qc.x(3)
    qc.h(2)

    qc.cx(2,3)
                
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)

def epr_product_4(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)
    
    
    qc.x(1)
    qc.h(0)

    qc.z(0)
    qc.z(1)

    #second entangled pair
    qc.x(3)
    qc.h(2)

    qc.z(2)
    qc.z(3)
                
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)

def epr_product_3_4(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)
    
    qc.x(1)
    qc.h(0)

    qc.cx(0,1)

    #second entangled pair
    qc.x(3)
    qc.h(2)

    qc.z(2)
    qc.z(3)
                
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)


def epr_product_4_3(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)

    qc.x(1)
    qc.h(0)
    qc.z(0)
    qc.z(1)
    
    qc.x(3)
    qc.h(2)
    qc.cx(2,3)
        
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)


def epr_product_1_3(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)
    
    qc.h(0)

    qc.cx(0,1)
    
    #second entangled pair
    qc.x(3)
    qc.h(2)

    qc.cx(2,3)
    
                
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)


def epr_product_3_1(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)

    #second entangled pair
    qc.x(1)
    qc.h(0)
    
    # Apply a CNOT
    qc.cx(0,1)
    
    
    qc.h(2)

    qc.cx(2, 3)    
        
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)


def epr_product_2_3(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)
    
    qc.x(0)
    qc.h(0)

    qc.cx(0,1)

    #second entangled pair
    qc.x(3)
    qc.h(2)

    qc.cx(2,3)
            
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)


def epr_product_3_2(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)


    qc.x(1)
    qc.h(0)

    qc.cx(0,1)
    
    
    qc.x(2)
    qc.h(2)

    qc.cx(2,3)
    
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)


def epr_product_1_4(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)
    
    qc.h(0)

    qc.cx(0,1)

    #second entangled pair
    qc.x(3)
    qc.h(2)

    qc.z(2)
    qc.z(3)
                
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)


def epr_product_4_1(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)

    qc.x(1)
    qc.h(0)
    qc.z(0)
    qc.z(1)
    
    
    qc.h(2)

    qc.cx(2,3)
                
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)


def epr_product_2_4(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)
    
    qc.x(0)
    qc.h(0)

    qc.cx(0,1)

    #second entangled pair
    qc.x(3)
    qc.h(2)

    qc.z(2)
    qc.z(3)
                
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)


def epr_product_4_2(measurement_string):
    #first entangled pair
    qc = QuantumCircuit(num_qubits)

    qc.x(1)
    qc.h(0)

    qc.z(0)
    qc.z(1)
    
    
    qc.x(2)
    qc.h(2)

    qc.cx(2,3)
                
    for index,s in enumerate(measurement_string):
        if s == '1':
            qc.x(index + 4)
            
    return Statevector.from_instruction(qc)

Then I compared the statevectors obtained from the above circuit with the statevector that I have(called 'my_state_vec' in code) using the equiv function:

def check_epr_pairs():

    measurement = '010101'

    epr_products = []
    epr_products.append(epr_product_1(measurement))
    epr_products.append(epr_product_2(measurement))
    epr_products.append(epr_product_3(measurement))

    epr_products.append(epr_product_4(measurement))
    epr_products.append(epr_product_1_2(measurement))
    epr_products.append(epr_product_1_3(measurement))

    epr_products.append(epr_product_1_4(measurement))
    epr_products.append(epr_product_2_1(measurement))
    epr_products.append(epr_product_2_3(measurement))

    epr_products.append(epr_product_2_4(measurement))
    epr_products.append(epr_product_3_1(measurement))
    epr_products.append(epr_product_3_2(measurement))

    epr_products.append(epr_product_3_4(measurement))
    epr_products.append(epr_product_4_1(measurement))
    epr_products.append(epr_product_4_2(measurement))
    epr_products.append(epr_product_4_3(measurement))

    for ep in epr_products:
        #Check if statevectors are equivalent
        if my_state_vec.equiv(ep):
            print("Statevec equivalent with", ep)

I have theoretical reason to believe that my_state_vec should be such that qubits 0,1 and 2,3 are EPR pairs. But I find that it is not equivalent to any of the EPR product statevectors.

The statevector produced by the qubits 0,1,2,3 is: \begin{bmatrix}-0.25\\ -0.25\\ 0.25\\ 0.25\\ 0.25\\ 0.25\\ 0.25\\ 0.25\\ 0.25\\ -0.25\\ -0.25\\ 0.25\\ -0.25\\ 0.25\\ -0.25\\ 0.25\end{bmatrix}

Is there a better way of checking for EPR pair products?

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ A high-level suggestion. To see if qubits A and B form an EPR pair (1) compute density matrix of the two-qubit system AB. It should have rank 1, i.e. be a pure state. (2) compute the density matrix of A alone, it should be an identity matrix if A and B form a Bell pair. $\endgroup$
    – Nikita Nemkov
    Jul 26, 2022 at 19:49
  • $\begingroup$ @NikitaNemkov I tried this suggestion. Step (1) checked out and AB is a pure state. Step (2), I get the density matrix of A as a diagonal matrix (all diagonal entries = 0.5). So this suggests, AB don't form a bell pair? $\endgroup$
    – Yash Sharma
    Aug 18, 2022 at 0:44
  • $\begingroup$ I also updated the question (added the resultant statevector of the 4 qubits) $\endgroup$
    – Yash Sharma
    Aug 18, 2022 at 0:53
  • 1
    $\begingroup$ I didn't look at your statevector, but diagonal matrix with entries 0.5 sounds exactly right. Actually, a density matrix can not be an identity because it must have unit trace. So, saying that $\rho$ is an identity is often a jargon for $\rho=\mathbb{I}/2^n$. $\endgroup$
    – Nikita Nemkov
    Aug 18, 2022 at 17:35

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.