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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Differences between pure/mixed/entangled/separable/superposed states

I am currently trying to establish a clear picture of pure/mixed/entangled/separable/superposed states. In the following I will always assume a basis of $|1\rangle$ and $|0\rangle$ for my quantum ...
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On Bell inequality and bound entangled states

I have recently seen some presentation slides of Michał Horodecki (slide number 77) in which he discussed the following conjecture. Bound entangled states satisfy all Bell inequalities The ...
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Invariance of states under local unitary transformations [closed]

How can I show explicitly that the bell state $$|\psi^{-}>=\frac{1}{\sqrt{2}}(|0>|1>-|1>|0>)$$ is invariant under local unitary transformations $U_{1}\otimes U_{2}$ ?
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Can Werner states have bound entanglement?

Let us consider the maximally entangled state \begin{equation} |\psi\rangle=\frac{1}{\sqrt{n}}(|0,0\rangle+\cdots+|n-1,n-1\rangle) \end{equation} and construct the pseudo-pure state \begin{equation} ...
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Can the entangelement of basis vectors increase under local operations?

Say I have a bipartite state $\rho = \sum_ip_i|\psi_{i}\rangle \langle \psi_{i}|_{AB}$ Where $\{|\psi_{i}\rangle_{AB}\}$ forms an orthonormal basis. I now perform some local quantum operation on ...
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Do the states forming an orthonormal basis have the same amount of entanglement?

If $\{|\psi_{i}\rangle\}$ is an orthonormal basis for a bipartite system, will $E(|\psi_i\rangle) = E(|\psi_j\rangle)$ for all $i, j$, where $E$ is some entanglement measure?
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Spin Transition Energies

I am reading a paper: http://arxiv.org/ftp/arxiv/papers/1305/1305.2445.pdf On p. 22, the following Hamiltonian is given: $$ H = \mu_B g \mathbf{B} \cdot \mathbf{S} + D(S_Z^2+\frac{1}{3}S(S+1)) + ...
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Question about entangled states

I have a question about entangled state. Suppose I consider the entangled state $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$. I saw an argument for how measurement of the first bit is affected by ...
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Limits of superdense coding

Holevo's theorem says that no more than n bits can be stored (and retrieved) in n qubits. Indeed, allowing error can't improve this either -- the probability of retrieving the correct information is ...
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Composition of squeeze operators?

I'm wondering if it exists a composition law for the squeezing operation ? I guess so for geometric reason, since they are (generalized, and the phase is annoying of course) hyperbolic rotations of ...
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Why does this state have a Schmidt rank of 1?

A system is entangled if and only if the Schmidt rank is greater than 1. Why does this ...
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The role of state space composition in quantum computation

In a paper by Richard Josza and Noah Linden they argue that the way state spaces of composite systems are formed is a key aspect in the benefits of quantum computers. In (classical) phase space, two ...
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Is a communications channel based on quantum mechanics as effective as one based on any other physics?

What I mean by effective in the question refers to the time and space requirements for sending information over a quantum communications channel. Having read "Mike and Ike" but doing no independent ...
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Ising spin vs Pauli spin matrices

Are Ising spins scalar or operators? I am not a condensed matter physicist hence having some confusion. I have learnt about Ising models from adiabatic quantum algorithm papers. For example this ...
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Why can a qbit be used as a classical bit if information about the measurement axis is needed?

If Alice wants to send one bit of classical information she can use a qbit. Then Bob needs to know which axis to measure to get the information. This needs an extra agreement between Alice and Bob ...
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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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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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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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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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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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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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Creating matrix Hamiltonian for Feynman's CCNOT [closed]

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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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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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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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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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? So can you give me a ...
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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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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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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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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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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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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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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 ...
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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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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 ...
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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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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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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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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 ...
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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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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 ...
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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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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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QFT in Quantum Computing and Control Theory?

Is QFT being applied to quantum computing and control theory? I took yesteryear a basic course on quantum computing and if I remember correctly we didn't touch on any QFT (though I think that if it ...
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How to measure a qubit in a random basis

Let a two dimensional system be in the state $\phi=|0\rangle\langle0|$, for any basis $M$ spanned by the orthogonal vectors $|\psi_0\rangle,|\psi_1\rangle$, we can measure $\phi$ in basis $M$ and ...
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Quantum gate: Phase shift

I dont undestand how to apply a phase shift gate to a qubit. By example how to map $|\psi_0\rangle = \cos (30^\circ) |0\rangle + \sin (30^\circ) |1\rangle$ to $|\psi_1\rangle = \cos(-15^\circ) ...