Questions tagged [quantum-state]
Quantum systems can mathematically be described by their 'quantum state'. When the system is closed/isolated, the state is 'pure' and can be written as a sum (i.e. 'superposition') of basis vectors. When the system is a subsystem of an open system, the state is instead usually 'mixed' and cannot be written as a pure state, so has to be written as a density matrix. Consider using the density-matrix tag when relevant
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Does anyone has some code for Mathematica or Python to convert a diagonal matrix into Dirac (bra-ket) notation?
I have the following matrix which I have to translate into Dirac's notations.
\begin{array}{cccccccc}
\frac{1}{2} \left(q_0+q_3\right){}^2 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{...
3
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1answer
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Nielsen & Chuang Excercise Question on CSS code
I was reading the CSS ( Steane Code) from the Nielsen & Chuang book. It asked in Ex. 10.27 to prove that $$|x+C_2\rangle\equiv \dfrac{1}{\sqrt{|C_2|}}\sum_{y\in C_2}(-1)^{u.y}|x+y+v\rangle $$ and ...
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How to distinguish two states with same density matrix using a quantum state tomography?
I tried to measure quantum state with a quantum state tomography. However, I encountered a situation when two different quantum states had the same density matrix. In particular, these states were $\...
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0answers
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Implementing B92 protocol on IBM Qx
I wanted to implement the B92 protocol on IBM Qx. Implementing the protocol would require that a particular qubit should not be measured when using a particular base i.e. there should be no output at ...
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2answers
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Why is the probability vector of a uniformly random state $\sum_i\alpha_i|i\rangle$ uniformly random only if $\alpha_i\in\mathbb C$?
In these lecture notes by Scott Aaronson, the author states the following (towards the end of the document, just before the Linearity section):
There's actually another phenomenon with the same "...
6
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1answer
147 views
Does the trace distance have a geometrical interpretation?
Consider the trace distance between two quantum states $\rho,\sigma$, defined via
$$D(\rho,\sigma)=\frac12\operatorname{Tr}|\rho-\sigma|,$$
where $|A|\equiv\sqrt{A^\dagger A}$.
When $\rho$ and $\...
3
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1answer
74 views
Secret sharing though quantum operations
I have a secret say $s$. I have a dealer $D$ and three participants $A, B, C$. I want to share this secret $s$ in such a way that the participation of all $3$ is essential to reconstruct the secret. ...
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The Elevator Problem [closed]
Today I(z) walked into an elevator and did not press the key to the floor to which I wanted to go.
As there were 2 other people x and y who had pressed the keys 2 and 3 respectively.
I wanted to go ...
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How to generate quantum circuit for the following transformation? [on hold]
$$
|j\rangle|k\rangle\mapsto|j\rangle|(j+k+1\,)\text{ mod }m\rangle.
$$
How to extend the basic qubit quantum gates to get the above transformation?
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How to generate the following $n$-level $n$-particle singlet state?
$$\boxed{|S_{n}\rangle = \frac{1} {\sqrt{n!}} \sum_{S\in P_n^{n}} ( \,-1) \,^{\Gamma(S)}|s_{0}\rangle |s_{1}\rangle ....|s_{n-1}\rangle}$$
Here $P_n^{n}$ is the set of all permutations of $Z_n := \{0,...
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How can infinite information be theoretically encoded or stored in a single qubit?
I've just gotten started with Nielsen and Chuang's text, and I'm a little stuck. They mention that theoretically, it would be possible to store an infinite amount of information in the state of a ...
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How to generate a superposition of m-level n-particle states $|j_{0} ,j_{1}, …,j_{n-1}\rangle$ with $\sum_{k=0}^{n-1} j_k \mathrm{mod}\ m \ = \ 0$?
The m level n-particle state $|X_{N}\rangle$ is defined as
$$\boxed{|X_{N}\rangle = \frac{1} {m^\frac{n-1}{2}}\sum_{\sum_{k=0}^{n-1} j_k \mathrm{mod}\ m \ = \ 0}|j_{0}\rangle |j_{1}\rangle ....|j_{...
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If a state is only “close to” an eigenstate of an operator, how many applications of the operator does it take to scramble the state?
Suppose we have an operator $U$, and a register $|\lambda\rangle$ in an eigenstate of $U$ with eigenvalue $\lambda=1$. Repeatedly applying $U$ to $|\lambda\rangle$ does not affect $|\lambda\rangle$ - ...
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Clarification of bra-ket notation [duplicate]
How do I get from equation 1.31 to equation 1.32? It seems like some terms are changing.
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1answer
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Decomposing a controlled phase gate into CNOTs
I'm trying to understand the following derivation of decomposing a controlled $R_k$ (phase) gate into a combination of CNOTs and single qubit gates, but there's one main thing about the process that ...
5
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1answer
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What does W stand for in the W entangled state?
For the $|W_3\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$, what does W stand for? Does it refer to an author name? Anyone knows a reference? Thanks
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1answer
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Transforming the first Bell state into the other Bell states
As I understand it, you can transform the different Bell states into one another by applying various gates. Wikipedia has the Bell states written out as follows:
And says that you can generate bell ...
3
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0answers
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Understanding shared Bell states from quantum entanglement
I'm trying to understand an entanglement swapping derivation provided in this PDF (pages 2 - 3)
I have several things about this process that I don't understand, and I was hoping someone could ...
5
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2answers
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How and why does swap test works?
I am having some trouble understanding why a SWAP test would work. I meant I read that and understood the concepts as follows:
If the two input states are equal, the output register always results
in ...
3
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1answer
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How to prepare an initial state for variational quantum algorithms?
I would like to prepare an initial state for variational quantum algorithms.
The initial state should include the following states: $|000\rangle, |010\rangle, |100\rangle$, and $|001\rangle$.
How ...
4
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2answers
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Physical Interpretation of Pauli Matrices as Polarization Check
We know that the Pauli matrices are:
$$\sigma_x = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \sigma_y = \begin{bmatrix}0 & -i \\ i & 0\end{bmatrix}, \sigma_z = \begin{bmatrix}1 & ...
4
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2answers
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Input and output qubit notation in quantum gates
I am new to quantum computing and I am having trouble understanding the notation used for input/output qubits in quantum gates. I will use the CNOT gate as an example.
In several (most) references I'...
2
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1answer
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Is IBM Q composer using little-endian representation for multi-qubit states?
I am still confused when I am trying to use the IBM Q composer to reproduce some quantum circuits I found in different papers, and I am wondering if it is because there are two ways to represent a ...
2
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0answers
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How to define initial state $\rvert \Psi(0) \rangle \equiv \rvert 1, -1 \rangle \otimes \rvert 0 \rangle_{\text{cav}} $ of a system in QuTiP?
Say, we have a $\require{mhchem}\ce{^87Rb}$ atom having an electric dipole transition on the $D_{1}$ line and we have two hyperfine ground states, one on $F = 1$ and one on $F = 2$ level. So, we take ...
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1answer
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Do pure states lie on a hypersphere in the Bloch representation?
It is known that every state $\rho$ of a $d$-level system (or if you prefer, qudits living in a $d$-dimensional Hilbert space) can be mapped into elements of $\mathbb R^{d^2-1}$ through the mapping ...
2
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1answer
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How to design a measurement that distinguishes the following pair of two-qubit states?
A source constantly produces a stream of photons in one of the following states
$$|\varphi_1\rangle=\dfrac{1}{\sqrt2}(a|00\rangle+ b|01\rangle+c|10\rangle+d|11\rangle)$$
$$|\varphi_2\rangle=\dfrac{1}...
2
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1answer
65 views
The solution when we transmit a qubit through a Pauli channel?
A Pauli channel is defined as a convex combination of Pauli operators, i.e. $\epsilon_{\text{Pauli}} (\rho)=\sum_{j} q_j\sigma_j\rho \sigma_j$, where $0 \leq q_j \leq 1$ and $\sum_j q_j=1$. Now, I ...
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3answers
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Can everything in QM be described with degrees instead of matrices and vectors?
I found this explanation.
"The Hadamard gate can also be expressed as a 90º rotation around the Y-axis, followed by a 180º rotation around the X-axis. So $H=XY^{1/2}H = X Y^{1/2}H=XY^{1/2}$."
Can ...
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1answer
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What is the mean value of displacement operator for the coherent state?
Can anyone help me to find the mean value of the displacement operator $$D(\alpha) = \exp( \alpha a^\dagger -\alpha^* a)$$ for a Coherent State $\left|\beta\right> = D\left(\beta\right)\left|0\...
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4answers
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How many combinations can 'n' qubits have?
With $4$ bits there are $2^4=16$ combinations. How many combinations can you have with $4$ qubits (assuming they are all superimposed)?
EDIT:
As per here, the argument is that if a qubit can hold ...
3
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1answer
50 views
How to show whether two states are indistinguishable or not by measuring in a different basis?
I'm struggling with understanding a bit of basic quantum mechanics math that I was hoping someone could clarify.
If I have two states such as these:
$$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$$
and ...
2
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1answer
63 views
In Qiskit, when creating multi-bit quantum and classical registers, what is the ordering of MSB to LSB?
When creating multi-bit quantum and classical registers, what is the ordering of MSB to LSB? For instance, I created a quantum register via
...
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1answer
55 views
Can a mixed state be represented on a Bloch sphere?
I have a hard time getting the exact difference between concepts of superposed and mixed states. Is it possible to represent the second one on the Bloch sphere, to show the main difference?
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1answer
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Is their a way to perfectly clone coherent state and squeezed states?
Coherent state and squeezed states can be swapped why cloning coherent states can't be perfect to clone we obtain $$|\cos(|z|)b\rangle \otimes |\sin(|z|)b\rangle,$$
If we set $|z| = \pi/4$ then
$$|...
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1answer
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How to decode 2-bit message from 2 entangled qubits?
I'm trying to do exercise 3 of this quantum course.
Alice and Bob prepare an EPR pair in the Bell + state. They each take one qubit home.
Suddenly, Alice decides she wishes to convey one of 4
...
4
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1answer
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Extensions of product states
Given a product state $\rho_{AC} = \rho_A\otimes \rho_C$, what can we say about the structure of states $\rho_{ABC}$ that are extensions of $\rho_{A}\otimes \rho_C$? By extension I mean that $\text{tr}...
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1answer
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Help in understanding an exercise on observable / measurement
I'm working through Quantum Computing for Computer Scientists (Yanofsky & Mannucci, 2008), and am getting a little confused about Observables. From what I understand an observable is a question ...
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2answers
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Are all operations simply matrix multiplication?
In the most simple example I can think of, we have:
a linear operator $A$, which is also a 2 x 2 matrix.
a vector $|v_i⟩$, which can be considered a 2 x 1 matrix.
If we see an example of an ...
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2answers
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Is $\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}e^{2\pi ixj \left/\phantom\vert\!{N} \right.}|j\rangle |j\rangle$ a valid entangled quantum state?
$$\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}e^{2\pi i x j \left/ {\phantom\vert\!\!} N \right.}\,\left|j\right\rangle_h\left|j\right\rangle_t$$
For a valid state we should have sum of probabilities = 1. ...
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1answer
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How are measurements on $Z$ and $X$ axes interpreted in the Bloch sphere?
I'm having trouble understanding how the measurement on $z$ and $x$ axes can be interpreted in terms of the Bloch sphere representation.
.
I know that the state can be written as
$$∣𝜓⟩=\cos(𝜃/2)|...
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1answer
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What is the probability of measuring $|j\rangle$ with $j\in\{0, 1, 2, … N-1\}$?
Consider the state
$$\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}e^{2\pi i\frac{x_1}{N}j}\left|j\right>_h\left|j\right>_t.$$
In the above entangled state, what is the probability of getting $|j\rangle_t$ ...
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1answer
60 views
Find the reduced density matrix for a four-qubit system
I have the state vector $|p\rangle$ made up of 4 qubits. Say system A is made up of the first and second qubits while system B is made up of qubits 3 and 4. I want to find the reduced density matrix ...
3
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1answer
85 views
(Quantum Computing/Machine Learning) Estimate of the absolute value of the probability amplitude of |0⟩ in the superposition?
You are given a source of an unknown qubit state. You make measurements in the computational basis, that is, your measurement is $\{|0\rangle \langle0|,|1\rangle \langle 1|\}$. You observe that you ...
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1answer
31 views
Find the local unitary that takes the bell state to a state phi that has an extractable bell state
I have a state $|p\rangle$ that has an extractable Bell state and I want to write it as a Bell state, $|b\rangle$, with a local unitary acting on one side. Basically I am trying to find a local ...
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1answer
51 views
Is there any meaning for a density operator if we omit the j-th row and column in quantum mechanics?
Assume we have a density operator (Hermitian, PSD, with trace 1, where PSD means positive semi-definite) called A for a particle. $v_i$ shows the i-th eigenvector of A and $\lambda_i$ shows the i-th ...
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1answer
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What is the average value of $|c_i\bar c_j|$ for a random state $|\psi\rangle=\sum_i c_i|i\rangle$?
Consider the density matrix $\rho=|\psi\rangle\!\langle\psi|$ of a random pure state in an $N$-dimensional space (in other words, an $N$-dimensional qudit, $|\psi\rangle\in\mathbb C^N$), $\rho_{ij}=...
2
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1answer
52 views
Bloch Sphere - Rotation Matrix
A qubit is given in the following form:
$\left|\psi\right\rangle = \cos\left(\dfrac{\theta}{2}\right)\left|0\right\rangle +
e^{i\phi}\sin\left(\dfrac{\theta}{2}\right)\left|1\right\rangle$.
Let's ...
4
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1answer
34 views
Non maximally entangled states for QKD
Why aren't non maximally entangled states produced and used in quantum key distribution schemes? What would be the advantage/disadvantage to use such states rather than maximally entangled ones?
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2answers
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The correct set of measurement operators on a mutiple qubit system
I was wondering if the complete set of measurement operators for a state :
$|\phi \rangle=c_{00}|00\rangle+c_{01}|01\rangle+c_{10}|10\rangle+c_{11}|11\rangle$
Would be given by :
$P_0\otimes I=|00\...
1
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1answer
143 views
How do I interpret the readout error for a quantum computer?
For example, the ibmqx2 computer has the most recent readout error of $4.40 \cdot 10^{-2}$. Does this mean per 1000 measurements, 44 faulty results exist?
