# Tag Info

21

The church of the larger (or higher, or greater) Hilbert space is just a trick that some people like (myself included) for rewriting some operations. The most general operations that you can write down for a system are described by completely positive maps, while we like describing things with unitaries, which you can always do by moving from the original ...

19

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 Kraus-representation of the channel, $$\Phi(\rho)=\sum_{i=1}^n K_i \rho K_i^\dagger,$$ where the so-called Kraus operators $K_i$ fulfill $\sum_{i=1}^n K_i^\... 12 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 computing. The question here is: Why do we want unitarity in quantum gates? The less specific answer is as above, it gives us 'reversibility', or as ... 12 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 non-unitary transitions being a fundamental component of algorithms (in the sense that a "measurement" is equivalent to sampling from the probability ... 10 Let's start by finding a complementary channel for any channel given by a Kraus representation $$\Phi(X) = \sum_{k=1}^n A_k X A_k^{\dagger}.$$ To make the necessary equations clear, let us assume that the channel has the form$\Phi:\mathrm{L}(\mathcal{X})\rightarrow \mathrm{L}(\mathcal{Y})$for finite-dimensional Hilbert spaces$\mathcal{X}$and$\mathcal{...

9

This question is posed, and answered positively, in Nielsen & Chuang in a subsection of chapter 8 entitled "System-environment models for and operator-sum representation". In my version, it can be found on page 365. Imagine $|\psi\rangle$ is an arbitrary pure state on the space upon which you wish to enact the operators. Let $|e_0\rangle$ be some fixed ...

9

Matrix inequalities of the form $A\ge B$ should be read as $$A-B\ge 0\ ,$$ which in turn means that all eigenvalues of $A-B$ are larger or equal than zero. In the given case, $M\le I$ means that all eigenvalues of $M$ are smaller or equal than one. (Note that this convention for $\ge$ used on matrices depends on the field. In other fields, "$\ge0$" ...

8

You cannot always find such a Kraus decomposition. Notice that any CPTP map $\mathcal E$ which does have a decomposition as a probabilistic mixture unitaries is unital, which is to say that it maps the identity to the identity, and in particular it maps the maximally mixed state to the maximally mixed state: $$\mathcal E(\tfrac{1}{d} \mathbf 1) = \tfrac{1}{... 8 For any controlled-U, if the input state is |+\rangle|\phi\rangle where |\phi\rangle is not an eigenstate of U, then the output state is entangled. This immediately deals with trivial cases such as U=I because in that case all states |\phi\rangle are eigenstates, and so it is not entangling. For any other U, there is an input state that is ... 7 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 unitary to allow reversibility, but what about measurement? This is false. In general, the states of a quantum system are not just vectors in a Hilbert space ... 7 One way to understand the relationship between the Choi representation of a channel and its possible Kraus representations is to use the vectorization map. Suppose that we have two finite-dimensional Hilbert spaces \mathcal{X} and \mathcal{Y}, and that we have fixed a standard basis \{|1\rangle,\ldots,|n\rangle\} of \mathcal{X} and a standard basis ... 7 Effectively, the Z operation (represented by the Pauli Z matrix) applies a rotation about the Z-axis. As you note, rotations can also be written in the form e^{-i Z t}. To see that, you can use a trick pretty similar to the one used to derive Euler's identity (e^{i \theta} = \cos(\theta) + i \sin(\theta)) to rewrite the Taylor series that you quoted ... 6 Any map which is not Completely Positive, Trace Preserving (CPTP), is not possible as an "allowed operation" (a more-or-less complete account of how some system transforms) in quantum mechanics, regardless of what states it is meant to act upon. The constraint of maps being CPTP comes from the physics itself. Physical transformations on closed systems are ... 6 Basically, it means that the correlations could be used to send a message. Or simply that Bob’s measurement outcomes can reveal some details of Alice’s actions. This is impossible when Alice and Bob each hold one qubit of a Bell pair. Despite the entanglement present, as well as contextuality, signaling in this case would result faster than light ... 6 This really depends where you want to start from. For instance, you can construct the Choi state of \mathcal E, i.e.,$$ \sigma = (\mathcal E \otimes \mathbb I)(|\Omega\rangle\langle\Omega|)\ , $$with \Omega = \tfrac{1}{\sqrt{D}}\sum_{i=1}^D |i,i\rangle, and then extract the Kraus operators of \mathcal E(\rho)=\sum M_i\rho M_i^\dagger by taking any ... 6 There are several ways that you could realise the depolarising map  \mathcal N_p(\rho) = (1\!-\!p)\:\!\rho + p \!\!\:\cdot\!\tfrac{1}{2}\mathbf 1 map on a quantum computer — including an idealised quantum computer, in which waiting around for the noise to do the work for you would not be an available method.\def\ket#1{\lvert#1\rangle} We start ... 6 It suffices to prove that if P and Q are positive semidefinite operators, then$$ \operatorname{im}(P) \subseteq \operatorname{im}(P+Q). $$Once you have this, the statement follows by taking P = \eta(a) and Q = \rho - \eta(a). Suppose that u is a vector with u \perp \operatorname{im}(P+Q). This implies that$$ 0 = u^{\ast} (P + Q) u = u^{\ast} ...

5

This is not the unitary that you have to implement: you need a two-qubit unitary $$\frac{1}{\sqrt{3}}\left(\begin{array}{cccc} 1 & 1 & 1 & 0 \\ 1 & \omega & \omega^2 & 0 \\ 1 & \omega^2 & \omega & 0 \\ 0 & 0 & 0 & \sqrt{3} \end{array}\right),$$ where $\omega=e^{2i\pi/3}$, the point being that if you introduce ...

5

Let $\mathcal{N}$ be the channels which subscripts for which conventions. $$\mathcal{N}_{N.C.} (\rho) = \begin{pmatrix} \rho_{00} & \rho_{01} \sqrt{1-\lambda}\\ \rho_{10} \sqrt{1-\lambda} & \rho_{11} \end{pmatrix}$$ As compared to $$\mathcal{N}_{P} (\rho) = \begin{pmatrix} \rho_{00} & \rho_{01} (1-\lambda)\\ \rho_{10} (1-\lambda) & \... 5 "Church of the higher hilbert space" is a term coined by John Smolin. According to quantiki it is: for the dilation constructions of channels and states, which [...] provide a neat characterization of the set of permissible quantum operations and to quote wikipedia, it: describe[s] the habit of regarding every mixed state of a quantum system as a pure ... 5 First let me mention a minor point concerning terminology. The type of channel you are suggesting is often called a Pauli channel; the term depolarizing channel usually refers to the case where p_x = p_y = p_z. Anyway, it is not really correct to say that Pauli channels are the channel model considered for quantum error correction. Standard quantum error ... 5 Not exactly sure what you find confusing, but the ultimate need for Stinespring dilation theorem is that in quantum mechanics the dynamics is in general defined by a completely positive trace preserving map (CPTP) \rho \mapsto \Lambda(\rho). Now, we have a belief (rightly or wrongly) that all there is is a unitary evolution governed by Schrodinger's ... 5 There is an ambiguity in the choice of Kraus operators: If \{E_a\} is a set of Kraus operators for a channel \mathcal E, so is \{F_b\} with F_b=\sum_a v_{ab} E_a, with (v_{ab}) an isometry. In particular, you can choose a (v) which diagonalizes the matrix X_{ac}=\mathrm{tr}[E_a^\dagger E_b], in which case \{F_b\} satisfies your ... 5 The shift operator takes his name from the fact that it shifts the position of its input, as in, it sends 1\to2, 2\to3 etc, with the last computational basis element being sent back to the first one: d\to 1 (or the same thing starting with 0, depending on notation). As per the "boost" operator Z, I have usually seen those referred to as "clock ... 5 The orthonormal basis |j\rangle of the d dimensional finite Hilbert space corresponds to a configuration space of equally spaced clockwise ordered d points on a circle S^1 or equivalently, the vertices of a d-dimensional regular polygon. One may think of a point as a discrete location of a particle, then the shift operator X shifts the particle ... 4 I'll add a small bit complementing the other answers, just about the idea of measurement. Measurement is usually taken as a postulate of quantum mechanics. There's usually some preceding postulates about hilbert spaces, but following that Every measurable physical quantity A is described by an operator \hat{A} acting on a Hilbert space \mathcal{H}. ... 4 Let's recap a bit: In classical information theory, the analogous formula is the Shannon noisy channel coding theorem. It's charming, because it is basically just a very simple optimization of the mutual information. The quantum channel capacity is that it is given by$$ \lim\limits_{n\to\infty} \frac{1}{n}Q(T^{\otimes n})  where $T$ is the quantum ...

4

The partial transpose is not the only positive but not completely positive operation that is possible on 2x2 and 2x3 systems. Trivially, any completely positive operation (such as a local unitary) combined with the partial transpose is a different positive operation. The point is that, as wikipedia puts it every such map $\Lambda$ can be written as $... 4 I agree with the main points that Niel makes: not all operators are observables, and the purpose of the ones you list is typically to transform states, not to be measured. However, the operators you list happen to be hermitian (allowing them also to represent observables) as well as unitary (allowing them to represent transformations), so in this case we ... 4 In the paper that you refer to, they are essentially asking "when can we implement the partial transpose map$\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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