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Computations in quantum information processing are implemented by means of unitary operations. Sometimes, we need to think not about a specific unitary operation required to execute a specific computation, but about the whole space of unitary transformations. (Examples will be given below). For a single $n$-dimensional qudit, (which can also be a tensor ...


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In the paper that you refer to, they are essentially asking "when can we implement the partial transpose map $\Theta=I_2\otimes\Lambda$?". So, that means the SPA of this map must be positive. What you have calculated, by comparison, is to ask when the SPA of the transpose map $\Lambda$ can be made positive. It might sound like these ought to be the same ...


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I just recently have been watching a series great YouTube lectures by Ryan O'Donnell at Carnegie Mellon. The last one in particular has some answers to the above question - especially the last 10 minutes or so. I will summarize my limited understanding. Misunderstandings are my own... A "closed timelike curve" (CTC) may be something akin to a wormhole in ...


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Measuring an observable does not mean applying the observable operator to a quantum state but rather measuring the state in the eigenbasis of the operator. A measurement will basically produce an eigenvalue of that observable operator and the system will collapse to a subspace corresponding to all states having that eigenvalue. For example, if you're ...


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You can use the same tools you used to get this output to check that it is correct: the state $|000\rangle$ would be represented as tensor product $|0\rangle \otimes |0\rangle \otimes |0\rangle$, which in your Python notation would be np.kron(np.kron(state_0, state_0), state_0). This should give you the same column vector you got from running your code, with ...


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Superposition is a linear combination of vectors which is a single vector. It may not be equal to some basis vector (from computational basis, for example), but this is not a big deal in general. As for question, I suppose operators are normal (so we can apply the spectral theorem). In the simplest case if operator $A$ has eigenvalue $\lambda$ with ...


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The general expression for the fidelity is $$ F(\rho,\sigma)=\left(\text{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^2=(\text{Tr}|\sqrt{\rho\sigma}|)^2. $$ Assume $\rho$ and $\sigma$ are $2\times 2$ matrices. Then $\sqrt{\rho\sigma}$ is also a $2\times 2$ matrix which we assume to have eigenvalues $\lambda_1$ and $\lambda_2$. Thus, $$ F=(|\lambda_1|+|\...


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Yes, unitarity preserves eigenvalues. This is because the definition of eigenvalues is that any Hermitian matrix $H$ can be brought into diagonal form by a unitary $V$, $$ VHV^\dagger=D, $$ and the diagonal elements of $D$ are the eigenvalues. So, now consider $UHU^\dagger$. This can be diagonalised by a unitary $VU^\dagger$ into the same matrix $D$: $$ (VU^\...


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If you are looking for a place to start, I probably start by asking you a few clarifying questions. Quantum Computation & Information is a broad field. I would say that it can primarily be viewed as a spectrum, similar to classical computing, from hardware up to algorithms. To get an idea of the hardware side, a great reference is Jerry Chow's thesis. ...


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It means that if you lose information from your system, that information must have been transferred to the system's surroundings. This shows up as an increase in the entropy in the surroundings. This is directly related to the 2nd law of thermodynamics which says the entropy of an isolated system is always increasing. See Wikipedia: Entropy in thermodynamics ...


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I've never thought about this before, and certainly not done any in-depth calculations, but see if this gets you started.... Obviously you want to do two different things: show that there are $n$-sharable states, and show that these states are not $m$-shareable for some $m>n$ (and hopefully get $n$ and $m$ as close as possible). Let $$ |\psi\rangle=\...


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