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Using higher dimension (d>2) quantum systems, or qudits, indeed provides an advantage through greater control of the Hilbert space. In quantum key distribution (QKD) for example, qudits can enhance the average raw key rates as one can encode more bits per symbol. Even more, they improve the robustness or the noise tolerance of the QKD protocol -- Alice and ...

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The trace defined as you did in the initial equation in your question is well defined, i.e. independent from the basis when the basis is orthonormal. Otherwise that formula gives rise to a number which depends on the basis (if non-orthonormal) and does not has much interest in physics. If you want to use non-orthonormal bases, you should adopt a different ...

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I think this will answer your question. How does the CNOT between qubits one and three work? $$\left|000\right\rangle \to \left|000\right\rangle$$ $$\left|001\right\rangle \to \left|001\right\rangle$$ $$\left|010\right\rangle \to \left|010\right\rangle$$ $$\left|011\right\rangle \to \left|011\right\rangle$$ $$\left|100\right\rangle \to ... 1 There aren't really many books on quantum cryptography. The only one I am aware of is a book titled Applied Quantum Cryptography. The Nielsen and Chuang book has a few pages dedicated to quantum cryptography in chapter 12.6. However, I would recommend the following review papers on quantum cryptography as opposed to textbooks: Gisin et. al (2001) ... 1 Closing the gap between lower and upper bounds for the tolerated errors in quantum key distribution protocols is a long standing problem in the field. In general, to give a lower bound on the key rate, one must provide a particular security proof of the protocol, but this proof may be suboptimal. For the BB84 protocol, the highest tolerable error rate I ... 1 You can compute the trace of an endomorphism using any basis (including non-orthogonal ones). In Dirac notation, you show this by inserting the identity expressed in the new basis and re-arranging:$$\begin{align*} \sum\limits_{|s\rangle \in B} \langle s^*| \rho |s\rangle &= \sum\limits_{|s\rangle \in B} \langle s^*| \left( \sum\limits_{|t\rangle \in ...

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Just because the entropy of the subsystems increases, that doesn't mean that the entropy of the whole system increases. This is possible here because of entanglement: an entangled pure state has zero overall entropy, but the subsystems have non zero entropy. A simple example is the state \frac{1}{\sqrt{2}}(|00\rangle+|11\rangle). ...

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No, you can still only read 1 value out - $a$ and $b$ tell you the probability of either being read; unless either $a$ or $b$ are zero, the readout is probabilistic. [Caveat: I assume by 'read' we are talking about non-quantum results, i.e. not entangling with the qubit(s)] We have a single qubit in the state: $a|0\rangle + b|1\rangle$ The probabilities ...

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To add to the answers above, an advantage to the quantum information theorist is that finite fields of ODD prime characteristic have nice properties, hence when using tools such as discrete phase spaces, qutrits can have properties that are otherwise hard to generalize to qubits. For example see arXiv:quant-ph/0602001.

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Presently there is no known general argument to say wehther some qudit dimensions are better than others for implementing fault tolerant quantum computation schemes. (I know of no paper showing something like that.) However, it is true that sometimes you can gain something by using qudits. (Meaning that some particular codes work better if you use qudits.) ...

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