I recently attended a seminar where a professor of quantum cryptography told the audience that one quantum qubit can theoretically store "infinite information". I was very intrigued by this statement, and me being an absolute novice in this domain, do not have the means of verifying the validity of his statement. My questions are as follows:         Can we really compute the distance $|x - y| &lt; \epsilon$ using a quantum qubit? If so, can anyone throw light on how this is done?    Also, if not, how far along are we in actually computing this quantity?    Can we measure if quantum computers approximate this quantity better (or worse?) than classical computers?      Thanks a lot in advance!
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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 single qubit. I'm not sure I completely comprehend this.    Here's how I rationalized it: You take all the information you want to store, put it in binary form, and make it the real component of $\alpha $ or $\beta$ (the coefficients of the computational basis states).   Now I'm not sure if I've understood it right, but since it's still fuzzy in my head it would be great to get some kind of ELI5 explanation or possibly a more detailed picture of how this would, even theoretically, be possible.    Apologies if the question doesn't meet standards. I'm new to the forum and would be open to feedback regarding asking questions or answering them.
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In a three-qubit system, it's easy to derive the CNOT operator when the control &amp; target qubits are adjacent in significance - you just tensor the 2-bit CNOT operator with the identity matrix in the untouched qubit's position of significance: $$C_{10}|\phi_2\phi_1\phi_0\rangle = (\mathbb{I}_2 \otimes C_{10})|\phi_2\phi_1\phi_0\rangle.$$ However, it isn't obvious how to derive the CNOT operator when the control &amp; target qubits are not adjacent in significance: $C_{20}|\phi_2\phi_1\phi_0\rangle.$ How is this done?
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How many bits of information can be stored in a qubit? On making a measurement on a qubit, one can catch it in either $|0\rangle$ or $| 1\rangle$. Seems that measurement on a qubit can not reveal more than a single bit. But does it mean the a qubit can't store more than 1 bit of information?
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Would like to ask a question, but first i would like to say Hello Everybody in a way that plays the system, since some geniouses decided that one should not be able to say hello in a question.  The  in quantum mechanics is well known and considered one of most basic properties of natural reality. The  is also well known and also considered one of the most basic processes of natural reality.  The uncertainty principle uses and is related to . Planck's constant has the dimensions of action and in a statistical mechanics approach, also relates nicely with the partitioning of the phase-space providing the basic measure for the entropy functional ( provides a nice outline of this).  Apart from that, there are  which relate the Heisenberg Uncertainty Principle in quantum mechanics directly and intuitively to the 2nd Law of Thermodynamics.  Is this relation correct? And if so can we derive one from the other?  Thank you  PS. One can also check , which although not the same, is related in an interesting way.  UPDATE:  anna's answer is accepted since by mentioning the derivation of (part of) the 2nd law from unitary dynamics, answers the question at least in one way. Please consider this as still open so you can add another answer. There are more alternatives (and one of which is my stance, ie thermodynamics -> uncertainty)
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I was hoping you guys could recommend reading material on quantum information. I have a good background in quantum mechanics, including in foundational issues, as covered in Ballentine's Quantum Mechanics.  I am interested in the intersection of quantum information and quantum foundations, such as in the works of C. Fuchs and in the research of the Perimeter Institute. While I prefer a more rigorous text, I'm trying to be as open as possible and would enjoy any suggestions or recommended reads.  Currently, I am reading the book by Busch, Lahti and Grebowski, notes on ArXiv by Keyl and also the notes by Preskill.
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I'm having a discussion with someone. I said that it is -even theoretically- impossible to reach $0$ K, because that would imply that all molecules in the substance would stand perfectly still.  He said that this isn't true, because my theory violates energy-time uncertainty principle. He also told me to look up the Schrödinger equation and solve it for an oscillator approximating a molecule. See that it's lowest energy state is still non-zero.  Is he right in saying this and if so, can you explain me a bit better what he is talking about.
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can search an unordered list of length $N$ in time $\sqrt{N}$  on a quantum computer. Applied to cryptography, this means that it can recover n-bit keys and find preimages for n-bit hashes with a cost of $2^{n/2}$. But the basic version of Grover's algorithm is sequential. Each iteration uses the output of the previous iteration as input. I don't see an obvious way to parallelize the algorithm. Reducing the number of iterations significantly reduces the success probability more than linearly. If the algorithm is inherently sequential, building a bigger computer won't speed up the attack, limiting its impact. $2^{80}$ sequential cipher invocations looks pretty hard to achieve to me. On the other hand, $2^{80}$ parallel cipher invocations is already on the edge of feasibility today using classical computers. Assume an attacker using a quantum computer is limited to $T$ timesteps with $T&lt;\sqrt{N}$, so they can't run Grover's algorithm with the optimum number of sequential steps. What's the cost for finding a preimage/recovering a key? What is the average cost per-break if they have multiple targets?
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I have some difficulty with understanding no-cloning theorem. Simply speaking, according to the theorem, it is not possible to copy a quantum state.  On the other hand, CNOT gate can be used as so-called fan-out gate which purpose is to copy one qubit to another one, previously in state $|0\rangle$.  It seems that these two facts negate each other.   My question: How is no-cloning theorem compatible with the fact that fan-out gate works?
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I'm fascinated by the fundamental questions raised by the Double Slit Experiment at the quantum level. I found this  which seems like a great explanation. But is it scientifically accurate?
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I'm studying Shor's Algorithm. In the book, author explains QFT can be replaced by Hadamard gates? Why this process is possible??    Thank you everybody. This is QPE. I attach part of book!!
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Consider a free electron in space. Let us suppose we measure its position to be at point A with a high degree of accuracy at time 0. If I recall my QM correctly, as time passes the wave function spreads out, and there is a small but finite chance of finding it pretty much anywhere in the universe. Suppose it's measured one second later by a different observer more than one light second away and, although extremely unlikely, this observer discovers that electron. I.e. the electron appears to have traversed the intervening distance faster than light speed. What's going on here?  I can think of several, not necessarily contradictory, possibilities:   I'm misremembering how wave functions work, and in particular the wave function has zero (not just very small) amplitude beyond the light speed cone. Since we can't control this travel, no information is transmitted and therefore special relativity is preserved (similar to how non-local correlations from EPR type experiments don't transmit information) Although the difference between positions is greater than could have been traversed by the electron traveling at c, had we measured the momentum instead, we would have always found it to be less than $m_e c$ and it's really the instantaneous momentum that special relativity restricts; not the distance divided by time. My question is ill-posed, and somehow meaningless.   Would anyone care to explain how this issue is resolved?
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Could someone experienced in the field tell me what the minimal math knowledge one must obtain in order to grasp the introductory Quantum Mechanics book/course?  I do have math knowledge but I must say, currently, kind of a poor one. I did a basic introductory course in Calculus, Linear algebra and Probability Theory. Perhaps you could suggest some books I have to go through before I can start with QM?
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Please don't sign this as a duplicate of , they aren't. I am interested about the actual algorithm, not a proof of the existence.  Does a such mapping exists? I think, it must, because both of them have the same cardinality ($\aleph_1$).  My actual suggestion for a such mapping were simply merge the digits of the real numbers. For example, from $\pi$ and $e$ we could got  $3.1415...$ and $2.7182...$ would lead to $32.17411852...$ .  Although I am not sure it were ok. Maybe a better, clearer solution also exists?
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Consider the  of Bennett et. al. How does one prove that this protocol would never violate the conservation of energy? At the face of it, it doesn't seem to be something obvious, as the various measurements, communications, rotations etc. don't seem to be able to account for the differences in the Hamiltonians of Alice's and Bob's labs. The state $\psi$ might have an expected energy $E_1$ in Alice's lab and a completely different expected energy $E_2$ in Bob's. Where does the difference come from?
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I've been searching for a quantum algorithm to compute the the inner product between two $n$-qubit quantum states, namely $\langle\phi|\psi\rangle$, which is in general a complex number. One can get $|\langle\phi|\psi\rangle|^2$ through thte SWAP test for multiple qubits, but this is not really the inner product as the information of the real and imaginary parts are lost. I came across this , which claims that the task can be done by simply using CZ gates. I'm not sure if I am mistaken, but it seems to me that such a circuit works for computational basis states, but not for general quantum states. Could somebody help me confirm if this is actually the case? If the above algorithm doesn't work for general quantum states, what algorithm would you suggest?
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I was very inspired by Michio Kaku's  on the possibilities of quantum computing and also listening to Talia Gershon's  on it.   As I come from a business &amp; analytics background, what materials can I begin exploring to prepare to learn more about the field of quantum computing? Eg. what would I need to study to understand Peter Shor's algorithm,  on physics, etc.
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From what I understand, information is communicated instantly between two quantum-entangled particles regardless of the spatial distance between them. However, does this necessarily imply superluminal data transfer? If the "distance" between the particles is simply assumed to be measured in our common 3D euclidean space then that would seem to imply superluminal communication. However, could there be other paths between the two particles that lie beyond our observable 3D euclidean space, other paths that involve additional dimensions in which the particles remain very close or some sort of folding of space on other dimensions that provide an information pathway? Anybody aware of any such research/findings? Thnx.