Questions tagged [bell-basis]

For questions related to Bell basis - i.e. converting a state to the Bell basis, working with such states and measuring in the basis.

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Density matrix after measuring Bell state in CHSH game

In these notes, the author says the following about the CHSH game Does Alice and Bob’s ability to succeed more than 75% of the time mean that they are communicating? Well, we know it’s not possible ...
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1answer
57 views

How to simulate the state of a system after partial measurement?

Based on the answers at here and here, I have a quantum simulator which implements the operations described in the classic Teleporting an Unknown Quantum State. Careful comparison of circuits with the ...
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63 views

Two-qubit Bell measurement matrix where the two qubits are not contiguouis

In the answer here, it is explained that where the measurement operates on only a subset of the qubits of the system (for example qubits 2 and 3 out of five), the matrix can be constructed using the ...
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What quantum gates admit a basis-independent interpretation of their action?

The SWAP gate swaps the state of the two qubits so that in the computational basis $|01\rangle \rightarrow |10 \rangle$ with a matrix representation given by: \begin{bmatrix} 1 & 0 & 0 & 0 ...
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1answer
43 views

Decompose bell measurement gate into combination of controlled-not gates and one-qubit gates

OPENQASM2.0 has only one two-qubit gate: controlled not. For a teleportation experiment, I need to perform a measurement in the Bell basis. That is, I need a two-qubit gate with matrix representation $...
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77 views

How to create a Bell state with asymmetric amplitudes using single-qubit and CNOT gates?

Is there a systematic way - in terms of a quantum circuit with single qubit and CNOT gates - to create a bell state with asymmetric amplitudes, e.g., $$ \alpha |00\rangle + \beta|11\rangle $$ where $\...
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1answer
69 views

Are $(|00\rangle-|11\rangle)/\sqrt2$ and $(|11\rangle-|00\rangle)/\sqrt2$ the same quantum state?

The state $(|00\rangle-|11\rangle)/\sqrt2$ is an entangled state. If we think about the state $(|11\rangle-|00\rangle)/\sqrt2$, is this also entangled, but with maybe a phase change? The above two can ...
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1answer
47 views

Are mixtures of pairs of Bell states perfectly distinguishable by local operations?

Consider the four Bell states $$ |\psi^{00}\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle), \hspace{2mm} |\psi^{01}\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle),\hspace{2mm} |\psi^{10}\...
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326 views

Why are Bell states the maximally entangled ones?

I just want to know why actually Bell states are examples of maximally entangled states and significance of that "maximal" term. Is there anything for proving that?
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Intuition for why $\frac{|00\rangle+|11\rangle}{\sqrt{2}}$ can be written as $\frac{|++\rangle+|--\rangle}{\sqrt{2}}$

In analyzing measurement of $\frac{|00\rangle+|11\rangle}{\sqrt{2}}$ in the local $|+\rangle$, $|−\rangle$ basis, through algebra manipulation, the initial state is first written as $\frac{|++\rangle+|...
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1answer
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What happens if a Pauli $X$ gate is applied to part of a Bell state?

I have started to learn about the mathematics behind ebits and I have a question. Assume $\color{red}{\text{Alice}}$ and $\color{blue}{\text{Bob}}$ share the following ebit: $\begin{align}\vert\Phi^+ \...
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What happens to the Bell state qubits after the Quantum Teleporation?

I'm reading on the Quantum Teleporation and I couldn't find anywhere what happens to the bell state qubits after the Quantum Teleporation.
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1answer
57 views

How to do error correction after encoded Bell measurement?

Need some help with the concepts of encoded/logical bell measurement. Please visualize the picture in your mind. Suppose I have a node with 7+7 qubits side by side, left 7 is $|0_{L}\rangle$ and right ...
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55 views

Calculating probability that two entangled qubits are the same when measured in different bases

Given the entangled state \begin{equation} |\Phi^+\rangle = \frac{1}{\sqrt 2} |00\rangle + \frac{1}{\sqrt 2} |11\rangle \end{equation} I am trying to calculate the probability that the two qubits end ...
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1answer
134 views

Does a basis of maximally entangled states exist for two-qubit or two-qutrit system so that the density matrices of the basis states don't commute?

I want to find a basis of maximally entangled states $|\Psi_i\rangle$, for $\mathcal{H}^{2} \otimes \mathcal{H}^{2}$ and, $\mathcal{H}^{3} \otimes \mathcal{H}^{3}$ such that the density matrices of ...
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2answers
79 views

Inconsistent result trying to simulate measurement of a 2 qubit system

Consider a 2 qubit system in the initial state: $$ \begin{pmatrix} 0 \\ \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} \\ 0 \end{pmatrix} $$ the so-called Bell-pair. Now let's measure the spin of the first ...
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Why does $(XZ\otimes I)|\Phi^+\rangle$ equal the Bell state $|\Psi^-\rangle$?

I'm slightly confused by the solution provided below by a suggested solution online to convert |$\phi^+$⟩ to |$\psi^-$⟩. I tried doing the operation XZ but I got $\frac{1}{\sqrt2}$(|10⟩-|01⟩) instead ...
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617 views

Measure a state in Hadamard basis [closed]

I can't understand how a state measure on the Hadmard basis collapses to $$ |+\rangle =\dfrac{|0\rangle+|1\rangle}{\sqrt{2}} $$ or $$ |-\rangle =\dfrac{|0\rangle-|1\rangle}{\sqrt{2}} $$ as they dont ...
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2answers
300 views

Show that the Bell states form a basis

I can't seem to understand how to show that the Bell states for a basis. Should I explain that through the circuit and what gates are used or by the basic proof behind proving a set as a basis?
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716 views

Are entangled and Bell states the same thing?

I am a little confused whether the entangled state and Bell state are the same thing? If they have a bit of contrast, what is the difference between them?
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What gate combinations create entangled two-qubit states?

I know that in order to make a two quit entangled state, this quantum circuit is used: But I was wondering if there are any other gate combinations which also create entangled two quit states. If ...
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1answer
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Bell's experiment: checking the probabilities

I have some doubts on the calculations behind Bell's experiment, which goes as follows: a pair of entangled photons is shared between two scientists, Alice and Bob, who possess a bunch of polarizers ...
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63 views

Show algebraically that $U|0\rangle\otimes |0\rangle+U|1\rangle\otimes|1\rangle=|0\rangle\otimes U^T|0\rangle+|1\rangle\otimes U^T|1\rangle$

Suppose that Alice applies a unitary operator $U$ with real entries to her qubit in an EPR pair $|\beta\rangle=\frac{1}{\sqrt 2}(|00\rangle+|11\rangle)$. Is this the same as having Bob apply $U^T$ to ...
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EPR pair and individual operations [closed]

I have created an EPR pair. Let's suppose it's $(|00\rangle + |11\rangle)/\sqrt{2}$. We both of our halves and then we move into our places. If I ask you how does your half look like (mathematical ...
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149 views

Q-Sphere representation of Bell States

I'm currently going through Lab1 of the Qiskit Quantum Computing Course, where one task is to create the Bell State $$\frac{1}{\sqrt{2}}\left(\vert01\rangle + \vert10\rangle\right)$$ Technically it ...
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1answer
258 views

Probability on measuring bell state in x-basis with pauli operator sigma x [duplicate]

I am confused about how to calculate the probabilities of getting a certain result when measuring a Pauli observable on a Bell state. When you measure an observable the state is projected onto an ...
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1answer
144 views

Show that Quantum Fourier Transform maps Bell states to Bell states

Given Bell states $\mathcal{B} = \{\left\vert \phi^{\pm} \right\rangle, \left\vert \psi^{\pm} \right\rangle\}$, show that Quantum Fourier Transform(QFT) maps $\mathcal{B}\rightarrow \mathcal{B}$ by ...
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Trouble understanding the EPR Experiment

I fail to understand the EPR experiment. From Wikipedia: Alice now measures the spin along the z-axis. She can obtain one of two possible outcomes: +z or −z. Suppose she gets +z. Informally speaking, ...
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671 views

How to show that Bell states are orthonormal

I was reading some material on QC online and I found some material that explains how to show that Bell states are orthonormal but without details. I understand that we need to check $\langle state1|...
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1answer
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EPR Experiment: What does it mean for Alice to measure $\vec{v} \cdot \vec{\sigma}$ on her qubit?

I am trying to understand Box 2.7 on page 113 of Quantum Computation and Quantum Information book by Nielsen and Chuang. They start out with following wave function: \begin{equation} \psi = \frac{|01\...
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How can I measure a qubit on a generic axis on Qiskit?

I'm trying to prove Bell's inequality by following the one made by Wigner, that is: $P(+_a,+_b) \le P(+_a,+_c) + P(+_c,+_b)$ And that this proof is NOT satisfied for angles between the various axes $\...
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Output of Bell State Measurement

I was trying to see the output of Bell State Measurement in IBM Quantum Experience but in the simulation down the circuit in the histogram, it showed me that the possible qubit will be $|00\rangle$ ...
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1answer
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How do get from $|0\rangle=\alpha|a\rangle+\beta|b\rangle$, $|1\rangle=\gamma|a\rangle+\delta|b\rangle$ to an expression for $|01\rangle-|10\rangle$?

My question is linked to the Nielsen Chuang book. Particularly equation 2.216 on basis change from $|0\rangle$, $|1\rangle$ to orthonormal $|a\rangle$ and $|b\rangle$. How do we get the equation from ...
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1answer
76 views

Two qubit state + Depolarizing channel = Bell diagonal state?

In multiple sources, e.g. RGK, KGR, it is stated (without proof) that if you take any two qubit state and send it through a depolarizing channel, the resulting state would be a Bell-diagonal state. ...
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2answers
262 views

What are the possible results of measuring $X$ and $Z$ on the state $|01\rangle+|10\rangle$?

When calculating the probability of getting +1 on X-basis on the first qubit of Bell's state $|01\rangle+|10\rangle$, the result is 1/2 with the state after measurement |++⟩ while the probability of ...
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Can the following Bell states have probability amplitudes other than 1/2 and still be entangled?

From my understanding, a qubit is entangled when the state of one qubit depends on the other, and vice versa. Can the following bell states have probability amplitudes other than 1/2 and still be ...
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4answers
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Bell state preparation

I was watching some lectures on qubits. They were talking about how to generate a Bell state. They described it as follows: Prepare state 00: $$\left |0 \right> \otimes \left |0 \right>$$ Apply ...
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How does Bell measurement work in the teleportation?

I'm a complete beginner and one of the first things I was taught was the teleportation protocol. In the protocol, the party sending its state (which we call say $|\phi\rangle$) makes a Bell ...
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1answer
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What is the representative matrix for a measurement in the Bell-state basis?

I have a few questions about measurement in Bell-state basis. In particular, if $Z = \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}$ is for a measurement on the computational basis, then what ...
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1answer
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Problem about entanglement swapping

I have been learning about the concept of entanglement swapping and found an equation mentioned in the textbook, Mathematics Of Quantum Computing: An Introduction written by Wolfgang Scherer. At ...
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539 views

How to apply the Schmidt Decomposition to a Bell state?

I am trying to understand the Schmidt Decomposition, currently in my QC class. We had a tutorial where we were told if $|\psi\rangle$ is a pure state of a composite system A then there exists $|i_A\...
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What happens when you send a Bell state through depolarizing channel?

For noise parameter $Q$ and a density matrix $\rho$, we know that the depolarization channel $\mathcal{E}$ would act like: $$ \mathcal{E}(\rho) = (1 - Q)\rho +Q\frac{I}{2}, $$ where $I$ is the ...
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What does the minus sign in the four bell states represent?

I am in grade 11, so answers as simple as possible. I understand that in quantum teleportation, the bell measurement must be made on the teleportee and the sender, and I understand that yields one of ...
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Creating Bell States with CNOT conditioned to the control qubit being set to zero

I was using qiskit to generate all of the Bell States. I created the following circuits for each state: $|\Phi^+\rangle :$ $|\Phi^-\rangle :$ $|\Psi^+\rangle :$ $|\Psi^-\rangle :$ The $|\Psi^\pm\...
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Fidelity With Bell State Calculation

Let's say I have the following state: $$ |\psi\rangle = \sqrt{\frac{2}{3}} |0000\rangle_{a_1b_1a_2b_2} + \sqrt{\frac{1}{6}} \big( |0011\rangle_{a_1b_1a_2b_2} + |1100\rangle_{a_1b_1a_2b_2} \big). $$ I ...
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1answer
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Creating Bell states (a question from tutorial) [closed]

I was going through a Quantum Computation lecture series on YouTube given by Umesh Vazirani. There in Lecture 6.3 he talks about creating a Bell state using a Hadamard and CNOT gate. He seems to be ...
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Constructing and Measuring in an Arbitrary 3-qubit basis

As part of a Quantum Theory project I have "constructed" an arbitrary 3-qubit basis: $\left|B_0\right> =\left|000\right>$ $\left|B_1\right> = \frac{1}{\sqrt{2}}\cos(x)(\left|100\right> +...
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646 views

How to measure in the bell basis?

The question is quite straightforward, I have some two qubit state and I want to measure it in the Bell basis. This question answers me partly, but I still have some doubts because, I thought that ...
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Prove that there exist tripartite $|\psi\rangle$ which cannot be written as $|\psi\rangle=\sum_i\lambda_i|i_A\rangle|i_B\rangle|i_C\rangle$

Exercise 2.77 in Nielsen and Chuang asks to show by example that there exist tripartite states $| \psi \rangle_{ABC} $ which cannot be written as $$| \psi \rangle = \sum_i \lambda_i | i_A \rangle | ...
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Is back in time (classical) information transfer possible (based on delayed choice entanglement swapping)?

Delayed choice entanglement swapping. Two pairs of entangled photons are produced, and one photon from each pair is sent to a party called Victor. Of the two remaining photons, one photon is sent to ...