Density Matrix of Multiple Qubits

  • #1
thatboi
133
18
Hey all,
I am having trouble relating probabilities with the density matrix of multiple qubits. Consider we have a system of 3 qubits: the first qubit is in the state ##\ket{\psi} = \frac{1}{\sqrt{2}}(\ket{0}+\ket{1})## and the remaining 2 qubits are prepared in the state described by the density matrix ##\rho = a_{1}\ket{\alpha_{1}}\bra{\alpha_{1}}\otimes\ket{\beta_{1}}\bra{\beta_{1}} + a_{2}\ket{\alpha_{2}}\bra{\alpha_{2}}\otimes\ket{\beta_{2}}\bra{\beta_{2}}##
where we only know that ##\braket{\alpha_{i}|\alpha_{i}} = \braket{\beta_{i}|\beta_{i}} = 1## for ##i = 1,2## and ##a_{1} + a_{2} = 1##.
Now suppose we form the density matrix ##\rho_{tot}## of all 3 qubits.

My question is: From ##\rho_{tot}##, how do I obtain the probability of the first qubit to be in state ##\ket{0}##, which we know is ##\frac{1}{2}##. Normally I would think of performing a partial trace but I'm not sure of what to take the partial trace over in this case since we do not have further information on the other 2 qubits.
 
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  • #2
thatboi said:
From ##\rho_{tot}##, how do I obtain the probability of the first qubit to be in state ##\ket{0}##, which we know is ##\frac{1}{2}##.
You don't have to obtain that probability from ##\rho_{tot}## because you already know it: you specified it when you specified the state of the first qubit. Since the state you specified is a product state, i.e., the first qubit is not entangled with the other two, the state of the other two qubits doesn't matter.
 
  • #3
PeterDonis said:
You don't have to obtain that probability from ##\rho_{tot}## because you already know it: you specified it when you specified the state of the first qubit. Since the state you specified is a product state, i.e., the first qubit is not entangled with the other two, the state of the other two qubits doesn't matter.
Hi,
I agree that we don't need ##\rho_{tot}##, but we should technically be able to obtain that probability from ##\rho_{tot}## still right. I was just wondering how I could get that value from ##\rho_{tot}##, such as from the diagonal values?
 
  • #4
thatboi said:
we should technically be able to obtain that probability from ##\rho_{tot}## still right.
You obtain it the way I just said: by recognizing that the total state is a product state, and therefore you don't need to know anything about the states of the other qubits. The "partial trace" operation for this case just means ignoring the other qubits and using the known state of the first qubit.

thatboi said:
I was just wondering how I could get that value from ##\rho_{tot}##, such as from the diagonal values?
You can of course view the "partial trace" operation, described above, as reading off the part of ##\rho_{tot}## that refers to the first qubit and using that and ignoring the rest. The answer is the same.
 
  • #5
PeterDonis said:
You obtain it the way I just said: by recognizing that the total state is a product state, and therefore you don't need to know anything about the states of the other qubits. The "partial trace" operation for this case just means ignoring the other qubits and using the known state of the first qubit.You can of course view the "partial trace" operation, described above, as reading off the part of ##\rho_{tot}## that refers to the first qubit and using that and ignoring the rest. The answer is the same.
Perhaps I misstated my problem, but my confusion originally arose from this post: https://quantumcomputing.stackexcha...ap-test-and-density-matrix-distinguishability
Specifically by the statement in the accepted answer: "We're interested by the diagonal coefficients of ##\rho_{3}## that can be written as ##\ket{0,i,j}\bra{0,i,j}##. Summing them would give us the probability of measuring |0⟩." I'm not sure what assumptions were made about ##\ket{i,j}##, but I wanted to know the reasoning behind this statement.
 
  • #6
thatboi said:
my confusion originally arose from this post
Which is not about the kind of state you described in the OP. It's about multi-qubit states that are not product states, with various quantum computing gates applied to them. Which kind of state do you really want to ask about?
 
  • #7
PeterDonis said:
Which is not about the kind of state you described in the OP. It's about multi-qubit states that are not product states, with various quantum computing gates applied to them. Which kind of state do you really want to ask about?
I would like a clarification on my question regarding the SWAP gate that I mentioned in my previous reply.
 
  • #8
thatboi said:
"We're interested by the diagonal coefficients of ##\rho_{3}## that can be written as ##\ket{0,i,j}\bra{0,i,j}##. Summing them would give us the probability of measuring |0⟩." I'm not sure what assumptions were made about ##\ket{i,j}##, but I wanted to know the reasoning behind this statement.
It's an obvious consequence of the definition of a density matrix. The coefficients described are by definition the ones whose sum gives the probability of measuring ##\ket{0}##.
 
  • #9
thatboi said:
Normally I would think of performing a partial trace but I'm not sure of what to take the partial trace over in this case since we do not have further information on the other 2 qubits.
You know that the other two qubits (which I will call "the rest") have a density matrix, and trace of any density matrix is 1. That's all what you need. More explicitly, your full system has a density matrix of the form
$$\rho=\rho_{\rm first}\otimes\rho_{\rm rest}$$
where ##\rho_{\rm first}=|\psi\rangle\langle\psi|## and ##\rho_{\rm rest}## describes the other two qubits. The reduced density matrix of the first subsystem is therefore
$$\rho_{\rm first \; (reduced)}={\rm Tr}_{\rm rest}\,\rho
= \rho_{\rm first} ({\rm Tr}_{\rm rest}\rho_{\rm rest}) = \rho_{\rm first}$$
The result is trivial because the full state is a product, i.e. there is no entanglement between the first subsystem and the rest.
 
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  • #10
The total density matrix will be ##0.5 \begin{pmatrix}
\rho & \rho \\
\rho & \rho
\end{pmatrix}##
 

FAQ: Density Matrix of Multiple Qubits

What is a density matrix in the context of multiple qubits?

A density matrix is a mathematical representation that describes the statistical state of a quantum system, especially when the system is in a mixed state. For multiple qubits, the density matrix encapsulates all the information about the probabilities and coherences of the various possible states of the qubits, allowing us to describe entangled and mixed states.

How is the density matrix of multiple qubits constructed?

The density matrix for multiple qubits is constructed by taking the outer product of the state vector with itself if the system is in a pure state. For a mixed state, it is constructed as a weighted sum of the outer products of the state vectors, where the weights represent the probabilities of each state. Mathematically, for a pure state |ψ⟩, the density matrix ρ is given by ρ = |ψ⟩⟨ψ|. For a mixed state, ρ = Σ p_i |ψ_i⟩⟨ψ_i|, where p_i are the probabilities of the states |ψ_i⟩.

What is the significance of the trace of a density matrix?

The trace of a density matrix is always equal to 1, which reflects the fact that the total probability of all possible states must sum to 1. This property is crucial for ensuring that the density matrix correctly represents a valid quantum state. Mathematically, Tr(ρ) = 1.

How do you determine if a density matrix represents a pure or mixed state?

A density matrix represents a pure state if and only if its trace of the square is equal to 1, i.e., Tr(ρ^2) = 1. For a mixed state, Tr(ρ^2) < 1. This criterion helps distinguish between pure states, which have no uncertainty, and mixed states, which represent a statistical ensemble of different possible states.

What is the partial trace and how is it used with density matrices of multiple qubits?

The partial trace is a mathematical operation used to trace out (ignore) parts of a quantum system, effectively reducing the density matrix to a subsystem of interest. For example, if we have a density matrix ρ_AB for a system of two qubits A and B, the partial trace over qubit B, denoted Tr_B(ρ_AB), gives the reduced density matrix for qubit A. This operation is essential for studying subsystems and their properties independently of the rest of the system.

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