# Tag Info

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### 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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### 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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### 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 ...
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### What is a POVM?

TL;DR: POVM $M$ is not an operator. It's more like a probability distribution over the set of all possible measurement outcomes, but parametrized by quantum states. Positive operator-valued measures A ...
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### What are examples of extremal non-projective POVMs?

It is indeed possible to have extremal, non-projective POVMs. Examples can be drawn from SIC-POVMs, as suggested in a comment. For example, as mentioned in the Wiki page, the only possible SIC-POVM in ...
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### Why are POVMs useful? Are they just an axiomatic way to define measurement?

For me, generalised measurements cover everything (obviously, that's why they're generalised), with projective measurements being a simple case that covers what we usually want to be doing. So, yes, ...
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### Give an explicit example of a $d = 4$ SIC-POVM

As indicated by Danylo in his anwser, eq. (32) in arXiv: 1103.2030 presents the sixteen vectors ("ignoring overall phases and normalisation") \left( \begin{array}{cccc} x & 1 &...

### What's the POVM corresponding to single-qubit state tomography?

POVM for standard QST in the Pauli bases In standard single-qubit QST one measures in the Pauli bases, each with equal probability $\frac{1}{3}$. As @Rammus has pointed out, this corresponds to the ...
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### Are projective measurements the only optimal measurements to discriminate between two states?

Suppose you are given either $\rho_1$ or $\rho_2$, and you also know that the probabilities you got one or the other are $p_1$ and $p_2$, respectively. If you have no prior knowledge of the ...
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### How to distinguish between two very similar pure quantum states?

The claim does not specify what protocols for distinguishing quantum states are acceptable. In particular, it does not state whether we are allowed to err or reserve judgment. Below, we note success ...
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### How to find the POVM that optimally distinguishes between two given states?

The optimal probability of guessing correctly is $$\frac12 + \frac12 \Big\|\frac23 \rho_0 - \frac13 \rho_1 \Big\|_1$$ where $\| X \|_1 = \mathrm{Tr}[\sqrt{X^* X}]$ is the Schatten 1-norm. This ...
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### What are the matrices in the POVM for measuring the first $m$ qubits?

Well, since these are projective measurements on the subspace of the first $m$ qubits, we can just list all projectors on the computational basis of this first subspace and 'pad' them with $I$'s on ...
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### How do I efficiently implement a POVM using a fixed universal gate set and the ability to measure in the standard basis?

What is the guarantee this implementation is efficient? Is there any rule regarding when implementing such POVMs is efficient? The implementation of such a gate will only depend on the parameter $k$ (...
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In the case of projective measurements, we have a set of projectors $\{P_i\}$ satisfying the completeness relation $$\sum_iP_i=I.$$ Note that this also means they satisfy $\sum_iP_i^\dagger P_i$, ...