Quantum information is the study of the informational content of quantum states. The most common object of study is the "qubit", the information in a two-state quantum system such as spin-1/2 or photon polarization.
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How do you come up with a POVM?
This is a made-up example, just to understand a concept. If changing the probability values aids your explanation, that's fine by me.
Say you have a physical quantity $E$ that can take values 1, 2, 3 ...
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Classical vs qubits: Superposition
Since a quantum information lecture today I have been wondering what does it really mean for a state to be in superposition? Is this something that is answerable?
This is what we learnt (or what I ...
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Creating matrix Hamiltonian for Feynman's CCNOT
I'm trying to read Quantum Mechanical Computer and to implement the CCNOT logical gate with Mathematica.
Since i wish to use the SWITCH implementation of the CNOT [Fig.8] i've realized that i need to ...
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4answers
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Application of non maximally entangled state
In quantum information and quantum computation, we generally use Bell type states which are maximally entangled. I find that the set of entangled states as interesting objects from a mathematical ...
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1answer
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Trying to understand mixed states
I took a basic quantum chemistry course (McQuarrie's "Quantum Chemistry"), but it never dealt with mixed states -- only pure states (or if it did, we never got to it in class).
So I'm trying to ...
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An entropy of the Wigner function
Is there an entropy that one can use for the Wigner quasi-probability distribution?
(In the sense of a phase-space probability distribution, not - just von Neumann entropy.)
One cannot simply use ...
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How many bits are needed to simulate the universe?
This is not the same as: How many bytes can the observable universe store?
The Bekenstein bound tells us how many bits of data can be stored in a space. Using this value, we can determine the ...
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Grover algorithm $R_D$ Circuit
I need sketch two circuits to understand Grover algorithm. The first is the operator $R_f$ and another is the operator $R_D = H^{\otimes n}(2|0\rangle\langle0|-I)H^{\otimes n}$. I get the first ...
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Is it possible to use quantum mechanics for an effective time based encryption?
This is for an application in cryptography. There is a concept called "time based cryptography", where a message can be decrypted only after a certain time, Say "12/12/2060, 12:30 GMT". There are some ...
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2answers
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Why does quantum cryptography give us uncrackable codes?
Why does quantum cryptography give us uncrackable codes? What makes it 'uncrackable'? Articles in for example pop science magazines always claim QC produces uncrackable coded, however I highly doubt ...
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Do quantum states contain exponentially more information than classical states?
Do quantum states contain exponentially more information than classical states? It might seem so at first sight, but what about in light of this talk?
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Many body quantum states analyzed as probabilistic sequences
Measurements of consecutive sites in a many body qudit system (e.q. a spin chain) can be interpreted as generating a probabilistic sequence of numbers $X_1 X_2 X_3 \ldots$, where $X_i\in ...
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Should it be obvious that independent quantum states are composed by taking the tensor product?
My text introduces multi-quibt quantum states with the example of a state that can be "factored" into two (non-entangled) substates. It then goes on to suggest that it should be obvious1 that the ...
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Qubit projections
Given the qubit:
$$\frac{|0\rangle+i|1\rangle}{\sqrt{2}}$$
What is the corresponding point on the extended complex plane and Bloch sphere?
How to perform calculations and get the point representing ...
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1answer
42 views
Statistical sum of physical quantities in a quantum system
Let $C = A + B$ (statistical sum, so $\mathbb{E}[C] = \mathbb{E}[A] + \mathbb{E}[B]$), and let $p(A = a) = 1$. Are the following true?
$\mathbb{E}[C^2] = a^2 + 2a\mathbb{E}[B] + \mathbb{E}[B^2]$
...
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Positivity in the Pauli/Bloch/coherence vector representation
Suppose $\rho$ is an $n$-qubit state and $\vec{x}$ is a vector of coefficients in the Pauli representation (also called the Bloch or coherence vector). That is
$$
x_k = {\rm Tr}(\rho \sigma_k),
$$
...
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projective measurement & POVM
Let us consider the following completely positive map $\mathcal{B}(\mathbb{C}^n)\ni\rho\mapsto L\rho L^\dagger$, where $L\in\mathcal{B}(\mathbb{C}^n)$ is any arbitrary operator (and can have rank ...
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How are qubits better than classical bit?
WHAT I KNOW:
classical computers store information in bits which can either be 0 or 1, but in quantum computer the qubit can store 0 , 1 or a state that is the superposition of these two states.
Now ...
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Why are the equal probabilities for Bell state measurement outcomes essential for “quantum teleportation”?
I've recently been introduced to the basics of finite-dimensional quantum mechanics from a purely mathematical point of view (with a quantum-information theme to it). When discussing quantum ...
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Toric Code and Random Bond Ising Model
It was established by Dennis, Kitaev et al. that the 2D Toric Code
can be mapped to a 2D Random Bond Ising Model. The original derivation
was given in the paper "Topological quantum memory" which ...
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0answers
38 views
Deutsch-Jozsa algorithm [closed]
How many calls are required to determine is the function balanced or not on the classical computer with probability of error < 50%.
Ref: Deutsch-Jozsa algorithm.
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1answer
59 views
Two qubits problem [closed]
Given the 2 qubit state:
(a/b) |00> + (c/b) |01> + (c/b) |10> + (d/b) |11>
What is the probability that 2 qubits are equal?
Thanks much!
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Constructing a Toffoli gate from CNOT and single-qubit gates [closed]
Toffoli gate in terms of CNOT and single-qubit gates.
Thanks much!
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2answers
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What does the sum of two qubits tell about their correlations?
How much can I learn about correlations between two quits by measuring
the sum of their values? What is the best way to formalize such a
question?
Below is my original, longer formulation of ...
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2answers
69 views
Purpose of Grover's algorithm?
How is the output of Grover's algorithm useful if the result is required to use the oracle? If we already know the desired state, what's the point of using the algorithm?
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Hamiltonian matrix propertu
A professor made an statement to prove the variational theorem:
Because the Hamiltonian (H operator of quantum physics) is diagonal in its own eigenfunction, the terms in $\left \langle \Phi _{m} ...
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1answer
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Violation of the Normalization Constraint?
Say we have two qubits $|a\rangle$ and $|b\rangle$ both initialized to $|0\rangle$. We then apply the rotation gate $R_{x}(\frac{\pi}{2})$ of matrix representation
$\left( \begin{array}{}
...
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4answers
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Is this statement about quantum mechanics valid?
In Philosophy of Language by William G. Lycan, there are the lines:
Even apparent truths of logic, such as truths of the form "Either P or not P", might be abandoned in light of suitably weird ...
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Entangled or unentangled?
I got a little puzzled when thinking about two entangled fermions.
Say that we have a Hilbert space in which we have two fermionic orbitals $a$ and $b$. Then the Hilbert space $H$'s dimension is just ...
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1answer
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Question on hadamard gate and cnot gate circuit tables
I'm trying to solve this problem for homework:
Now show that if the CNOT gate is applied in the Hadamard basis - i.e. apply the Hadamard gate to
the inputs and outputs of the CNOT gate - then ...
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8answers
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Why quantum entanglement is considered to be active link between particles?
From everything I've read about quantum mechanics and quantum entanglement phenomena it's unobvious for me, why quantum entanglement is considered to be active link. I.e. it's stated every time that ...
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2answers
666 views
Recent breakthroughs in quantum computing?
Can anyone explain to me why we have had no major breakthroughs in the theory of quantum computation in the past 15 years?
Shor's algorithm set the standard, since then we've had Grover's algorithm ...
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Circuit identities HTH [closed]
Using this circuit indetities $HXH=Z, HYH=-Y, HZH = X$ prove $HTH=R_x(\pi/4)$. here $H$ is Hadamard matrix, $X,Y$ and $Z$ are Pauli matrix, $R_x$ is a rotation matrix and $T=\left[ \begin{array}{cc}
1 ...
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Partial Measure Probability
Let be a
$$|\psi\rangle = \dfrac{3}{5\sqrt{2}}|00 \rangle- \dfrac{3i}{5\sqrt{2}}|01 \rangle+ \dfrac{2\sqrt{2}}{5}|10 \rangle - \dfrac{2\sqrt{2} i}{5}|11 \rangle$$
state with two qubits. ...
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1answer
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Is it ever necessary to extend an analysis of Grover's algorithm beyond $k/N = 1/2$?
Is it ever necessary to extend an analysis of Grover's algorithm beyond $k/N = 1/2$, where $k$ is the number of "hits" in a total of $N$ possible values for $|\,x\rangle$?
If we know $k$, and know ...
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Is there a symmetry associated to the conservation of information?
Conservation of information seems to be a deep physical principle.
For instance, Unitarity is a key concept in Quantum Mechanics and Quantum Field Theory.
We may wonder if there is an underlying ...
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Quantum information science references
I was hoping you guys could recommend reading material on Quantum Information Science. First off, here's my background.
Personally, I started reading Ballentine's Quantum Mechanics and I found it be ...
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2answers
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Dealing with environment in a CHSH game
I am reading arxiv:1209.0448. I understand that my questions
could be highly trivial. I would appreciate if anyone helps me to resolve
my confusions.
In a CHSH game, Alice and Bob cannot have ...
3
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1answer
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What is quantum discord?
What is quantum discord? I stumbled upon this term on Quantum Computing: The power of discord, but have never heard of it before. Can you give a bit more mathematical explanation of the term here?
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1answer
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I am interested in learning Quantum Computing what should I do? [closed]
I wish to learn about quantum computing which seems to be a topic of hot research and overall just intrigues me. I have a strong background in discrete mathematics and number theory. And am a pretty ...
3
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1answer
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Bloch sphere representation
Suppose you know that a qubit is either is in state $|+\rangle$ with probability $p$ or in state $|-\rangle$ with probability $1-p$. If this is the best you know about the qubit's state, where in the ...
9
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2answers
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Entropy of a state subject to the action of a set of random unitaries
Suppose that we have a known set of unitaries $U_1,...,U_n$ randomly selected from the Haar measure and suppose that each unitary is applied with probability $\frac{1}{n}$ to some input state $\rho$ ...
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1answer
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Types of photon qubit encoding
How many types of qubit encoding on photons exist nowadays? I know only two:
Encoding on polarization:
$$ \lvert \Psi \rangle = \alpha \lvert H \rangle + \beta \lvert V \rangle $$
$$ \lvert H ...
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1answer
66 views
Information bearing degrees of freedom of a quantum simple harmonic oscillator
I am trying to make sense of arXiv:physics/0210005. I am confused with the concept of information bearing degrees of freedom of a system mentioned at the very beginning. To verify the arguments of the ...
0
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1answer
58 views
Landauer's principle vs Wien's displacement law
Can we argue based on Landauer's principle that if one bit information is changed inside a blackbody, the total radiated energy should be at least or in order of kTln2? If it is so, can we also argue ...
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Landauer's principle vs Rayleigh–Jeans law
Can we argue based on Landauer's principle that if one bit information is changed inside a blackbody, the total radiated energy should be at least or in order of $kTln2$? If it is so, can we also ...
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2 following gates, inverse circuit
I have a circuit that has 4 wires and 2 following each other Toffoli gates.
The first Toffoli gate occupies 3 wires from above, the following Toffoli gate occupies 3 wires from below.
What will look ...
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1answer
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2 following gates, permutation matrix
I have a circuit that has 4 wires and 2 following each other Toffoli gates.
I have permutation matrix for each Toffoli gate (A and B).
Do I have to multiply that 2 matrices to get the entire ...
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How large must the Quantum teleportation fidelity have to be in order for it to be useful?
This question relates and stems from my original question. Please read this one and the comments before answering this question.
Quantum Teleportation Fidelity
I know that for discrete variables ...
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How is the energy/eigenvalue gap plot drawn for adiabatic quantum computation?
I was going through arXiv:quant-ph/0001106v1, the first paper by Farhi on adiabatic quantum computation.
Equation 2.24 says, $$\tilde{H}(s) = (1-s)H_B + sH_P$$ which means the adiabatic evolution ...
