24 votes

If all quantum gates must be unitary, what about measurement?

Unitary operations are only a special case of quantum operations, which are linear, completely positive maps ("channels") that map density operators to density operators. This becomes obvious in the ...
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17 votes

How does measurement of one qubit affect the others?

There are a lot of different ways of looking at qubits, and the state vector formalism is just one of them. In a general linear-algebraic sense a measurement is projection onto a basis. Here I will ...
17 votes
Accepted

What is the Helstrom measurement?

The Helstrom measurement is the measurement that has the minimum error probability when trying to distinguish between two states. For example, let's imagine you have two pure states $|\psi\rangle$ ...
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16 votes

What is the opposite of measurement, in a quantum circuit?

A measurement is basically a CNOT between the quantum computer and the external environment. The important distinction between this CNOT and the CNOTs entirely within the quantum computer is that the ...
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15 votes

If all quantum gates must be unitary, what about measurement?

Short Answer Quantum operations do not need to be unitary. In fact, many quantum algorithms and protocols make use of non-unitarity. Long Answer Measurements are arguably the most obvious example of ...
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13 votes
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Do multi-qubit measurements make a difference in quantum circuits?

Entangling measurements are powerful. In fact, they are so powerful that universal quantum computation can be performed by sequences of entangling measurements only (i.e., without extra need for ...
13 votes
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What makes quantum computations different from randomized classical computations?

The question is, how did you get to your final state? The magic is in the gate operations that transformed your initial state to your final state. If we knew the final state to begin with, we wouldn'...
12 votes

If all quantum gates must be unitary, what about measurement?

At risk of going off-topic from quantum computing and into physics, I'll answer what I think is a relevant subquestion of this topic, and use it to inform the discussion of unitary gates in quantum ...
12 votes

How can I build a circuit to generate an equal superposition of 3 outcomes for 2 qubits?

Here is how you might go about designing such a circuit.$\def\ket#1{\lvert#1\rangle}$ Suppose that you would like to produce the state $\ket{\psi} = \tfrac{1}{\sqrt 3} \bigl( \ket{00} + \ket{01} + \...
12 votes
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Density matrix after measurement on density matrix

So, Bob is given the following state (also called the maximally-mixed state): $\rho = \frac{1}{2}|0\rangle\langle 0| + \frac{1}{2}|1\rangle\langle 1| = \begin{bmatrix} \frac{1}{2} & 0 \\ 0 & \...
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11 votes

How can I build a circuit to generate an equal superposition of 3 outcomes for 2 qubits?

I'll tell you how to create any two qubit pure state you might ever be interested in. Hopefully you can use it to generate the state you want. Using a single qubit rotation followed by a cnot, it is ...
11 votes
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How can I build a circuit to generate an equal superposition of 3 outcomes for 2 qubits?

Break the problem in parts. Say we have already sent $\mid 00 \rangle$ to $\frac{1}{\sqrt{3}} \mid 00 \rangle + \frac{\sqrt{2}}{\sqrt{3}}\mid 01 \rangle$. We can send that to $\frac{1}{\sqrt{3}} \...
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11 votes
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How to perform quantum state tomography on two qubits?

Preliminary I would like to rewrite the equation that you have in a slightly different manner. Since a density matrix can be written as a matrix, we can also write it down as a linear combination of ...
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11 votes

What do the off-diagonal elements of a density matrix physically represent?

To put it very shortly, non-zero off-diagonal elements of the density matrix signify that your system features a quantum superposition between the elements of the basis that you chose to represent $\...
11 votes
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What is the relation between POVMs and observables (as Hermitian operators)?

One way of looking at the relationship between POVMs and observables arises from identifying their counterparts in the theory of probability of which quantum mechanics can be thought of as an ...
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10 votes

How does measurement of one qubit affect the others?

Less formally-stated than the other answers, but for beginners I like the intuitive method outlined by Prof. Vazirani in this video. Suppose you have a general two-qbit state: $|\psi\rangle = \begin{...
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10 votes
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Why can't a quantum computer strongly simulate itself?

Apart from the formal result about #P-hardness, there's something worth touching on, about the nature of strong simulation itself. I'll comment first on strong simulation, and then specifically on the ...
10 votes
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How to distinguish between collapsed and uncertain qubits in a quantum circuit?

Consider a quantum circuit on one qubit initialized in the state $|0\rangle$ and consisting of two Hadamard gates where we can insert a measurement between the Hadamards. The input state is $|0\rangle$...
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9 votes
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Representation of real numbers in quantum computers

There have been efforts to implement construct "floating point" representation of small rotations of qubit states, such as: Floating Point Representations in Quantum Circuit Synthesis. But there doesn'...
9 votes
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Measuring in standard basis meaning

A $1$-qubit system, in general, can be in a state $a|0\rangle+b|1\rangle$ where $|0\rangle$ and $|1\rangle$ are basis vectors of a two dimensional complex vector space. The standard basis for ...
9 votes
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Is there a classical limit to quantum computing?

In simpler terms your question is: if noise/decoherence keeps entering the computation, how can a big computation possibly survive? The key concept you're missing is quantum error correction, which ...
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9 votes
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Are three POVM measurements on a single qubit physically realizable?

Three outcomes amounts to more than one bit if the outcomes are all deterministic, and give you information about the original qubit. But suppose I have a coin (that is either heads or tails). I ...
9 votes
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Why does measuring one qubit after the other in this entangled system alter the result?

If the simulator is saying that state 00 occurs 75% of the time then the simulator has a bug. Reordering measurements can't make certain outcomes more likely in that way. It would violate the no ...
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9 votes
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Unentangling a qubit from a system: can we convert $\alpha|000\rangle+\beta|111\rangle$ into $\alpha|00\rangle+\beta|11\rangle$?

Yes, it is possible. To obtain the state $$\vert \phi \rangle = \alpha \vert 00 \rangle + \beta \vert 11 \rangle$$ from $$\vert \psi \rangle = \alpha \vert 000 \rangle + \beta \vert 111 \rangle,$$ ...
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9 votes
Accepted

What does it mean to "measure an operator"?

Any operator Hermitian $A$ can be described using its eigenvalue decomposition $$ A=\sum_\lambda\lambda P_\lambda, $$ where $\{\lambda\}$ are the distinct eigenvalues and $P_{\lambda}$ are the ...
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9 votes
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What's the 'physical consistency' in the partial trace scenario?

Measurement average Measurement average $\langle M \rangle_\rho$ of observable (a Hermitian operator) $M$ on the state $\rho$ is the average of measurement outcomes $m$ in the limit of infinite number ...
  • 14.7k
9 votes

What do the off-diagonal elements of a density matrix physically represent?

It can represent 'various' things depending upon the physical system and the context. For instance: (1) For a 'closed and isolated' system (let us say a qubit), it represents the 'coherence' between ...
8 votes

If all quantum gates must be unitary, what about measurement?

There are several misconceptions here, most of them originate from exposure to only the pure state formalism of quantum mechanics, so let's address them one by one: All quantum operations must be ...
8 votes

How does measurement of one qubit affect the others?

Suppose that, prior to measurement, your $n$-qubit system is in some state $\lvert \psi \rangle \in \mathcal H_2^{\otimes n}$, where $\mathcal H_2 \cong \mathbb C^2$ is the Hilbert space of a single ...
8 votes

Are true Projective Measurements possible experimentally?

Let's step back from QC for a moment and think about a textbook example: the projector onto position, $|x\rangle$. This projective measurement is obviously unphysical, as the eigenstates of $|x\...

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