Title: All Decks / ETH_Sem6_old / QIT

Q: Definition Probability space:
A: \((\Omega,\mathcal{E}, P)\)
With the sample space \((\Omega,\mathcal{E})\)
Probability measure \(P\)
\(\Omega\) a set and \(\mathcal E\) a \(\sigma\)-algebra of subsets of \(\Omega\) called events
\(P: \mathcal{E} \to \R^+\)

Q: Axioms of probability:
A: \(P[\Omega] = 1\)
\(P[\cup_{i\in \N}E_i] = \sum_{i\in\N}P[E_i]\)
for \(E_i\) disjoint events

Q: Definition Random Variable:
A: \((\Omega,\mathcal{E},P)\) a probability space
\((\chi,\mathcal F)\) a measurable space
\(X: \Omega \to \chi\)
with \(\chi\) measurable wrt. \(\mathcal E, \mathcal F\)
 -> (\(\forall F \in \mathcal F: X^-1 (F) \in \mathcal{E})\)

Q: Induced probability measure
A: \(P_X[F] := P[X^{-1}(F)]\) \(\forall F\in \mathcal{F}\)
(The probability for finding a \(F\) which would map onto \(X\) via \(P\))

Q: Def: Marginal distribution
A: Given \(P_{XY}\) we call \(P_X\) and \(P_Y\) the marginal distributions.

Q: Functions of random variables:
A: Same as induced probability measure

Q: Jensen Ineq
A: For real convex functions

Classical trace distance:

Definition Channel

Def Entropy:

Def Shannon Entropy:

Def min-entropy:

Def Mutual information:

Smooth entropy:

Q: Def: Rate (QIT)
A: Where \(l^\varepsilon\) denotes the maximal length of a message that can still be encoded with error below \(\varepsilon\)

Q: Channel coding theorem
A: Actually the bound is tight

Information processing inequality

Def: Trace of an operator

Spectral decomposition

Polar decomposition:

SVD

Schmidt decomposition:

Tracenorm:

Q: Definition density operator:
A: \(\rho = \sum_x P_X(x) \ket{e_x}\bra{e_x}\)
with \(\rho \geq 0\) and \(tr(\rho) = 1\)

Q: Def pure density operator:
A: \(\rho = \ket \phi \bra \phi\)

Q: Def fully mixed density operator
A: \(\rho = \frac{1}{d} \mathbb{1}_d\)

Q: Characterisation of \(\rho\) pure:
A: \(tr\rho^2 = 1\)

Q: def: Purification of \(\rho_A\):
A: \(\rho_{AB}\) such that.
\(\rho_A = tr_B (\rho_{AB})\)

Classical density operator:

Def: Trace distance

Trace distance for QM states given classical measurements

Trace distance of subsystems:

Def fidelity:

Ulman equation:

Q: Relation fidelity and trace distance
A: Only for pure states

Q: Evolution of Density operator:
A: \(\rho_{A B}^{\prime}\,=\,(U_{A}\otimes U_{B})(\rho_{A B})(U_{A}^{*}\otimes U_{B}^{*})\)

Q: Measurement density operator formalism + post measurement state:
A: \({\cal P}_{X}(x)=\mathrm{tr}({\cal P}_{x}\otimes\mathrm{id}_{B}\rho_{A B})\)
\(\rho_{A B,x}^{\prime}=\frac{1}{P_{X}(x)}(P_{x}\otimes\mathrm{id}_{B})\rho_{A B}(P_{x}\otimes\mathrm{id}_{B})\)

Q: Definition Bell States/ EPR pairs:
A: \(|\beta_{\mu}\rangle:=\sum_{a,b\in\{0,1\}}{\frac{1}{\sqrt{2}}}(\sigma_{\mu})_{a b}|a,b\rangle\)

Q: Meaning CPM and POVM?
A: CPM: Completeley Positive Map
POVM: Positive Operator Valued Measure

Q: Convexity of physical maps:
A: Needs to be convex under probabilistic mixture:
\({\mathcal{E}}\left(p\rho_{0}+(1-p)\rho_{1}\right)=p{\mathcal{E}}(\rho_{0})+(1-p){\mathcal{E}}(\rho_{1})\)
With \(p\) the mixing parameter
\(\rho_{1,2}\): States
\(\mathcal E\): The map

Q: Def: Positive map:
A: \({\mathcal{E}}\in{\mathrm{Hom}}({\mathrm{End}}(\mathcal H_{A}),{\mathrm{End}}(\mathcal H_{B}))\)
\(\mathcal E(S) \ge 0 \forall S\ge0\)

Q: Are tensorproducts of positive maps positive?
A: No!
We need complete positivity!

Q: Def: Complete Positivity:
A: \(\forall \mathcal H_R: \mathcal E \otimes\mathcal I_R\) is positive

Q: What is the fundamental idea of the Choi-Jamiolkowsky isomorphism?
A: Isomorphism between maps and states. Fundamentally it relates a map \(\mathcal E_{B | A}\) to a conditional state \(\rho_{A',B}\) (B given A) which is basically an entanglement between an arbitrary input and the corresponding output

Q: Mathematical definition Choi-Jamiolkowsky mapping
A: \(\tau:{\mathcal{E}}\mapsto(\mathcal{I}_{A^{\prime}}\otimes\mathcal{E})(|\Psi\rangle\langle\Psi|)\)
With \(\ket \Psi\)
\(\Psi=\frac{1}{\sqrt{d}}\sum_{i=1}^{d}e_{i}\otimes e_{i}\)
We 'clone' our state and then let it be (A') on one side and transform it under \(\mathcal E\) on the other side

Q: What spaces does the CJ mapping map to and from?
A: \(\tau: Hom(End(\mathcal H_A), End(\mathcal H_B)) \to End(\mathcal{H}_{A'} \otimes \mathcal H _B )\)

Q: What is the idea of the inverse CJ mapping?
A: Transform the density matrix \(S_A \) to \(A' \) then connect it with an auxilary system \(B\).
Use this as the "given state" for the conditional state \(\rho_{A',B}\)
Then trace out \(A'\) and correct for the fact that using the fully mixed state by multiplying with \(d\)

Inverse of Choi-Jamiolkovsky mapping

Definition \(\mathcal T\) in CJ reverse map

How do we charakterize a CPTP map under the CJ iso?

What can we say if the choi state is pure?

Q: Idea and intuition stinespring dilation:
A: Intuition: Any CPTPM can be represented as unitary evolution on a larger system
Idea: Purifying the Choi State and then taking it back to a map purifies the map.

Q: Mathematical formulation Stinespring dilation:
A: \(\forall \mathcal E \in TPCPM(\mathcal H_A, \mathcal H_B): \exists_U \in Hom(\mathcal H_A, \mathcal H_B \otimes \mathcal H_R) \space\space\space \mathcal E: S_A \mapsto tr_R(US_AU^*)\)

Diagram stinespring dilation

Q: Krauss/ Operator-sum representation: Formula
A: For \(\mathcal E\) a CPTPM
Note the converse also holds

Q: Idea Krauss representation
A: We can 'remap' state vectors to the original state (a bit like interaction picture)

Q: How do we model measurements in QIT?
A: We consider the post measurement state tensored with the classical measurement results. This gives us a classical state:

Q: Mathematical definition POVM:
A: With \(M_x = E^*_xE_x\)

Definition diamond norm:

Q: Idea diamond norm:
A: To distinguishe CPMs we might want to send in arbitrarily entangled states to get more info.
Diamond norm extends the space of a map \(\mathcal E\) to allow entangled inputs.
We then maximize the trace norm of an arbitrary state pushed though this extended \(\mathcal E\)

Definition trace norm for resources:

Q: Definition Effect in GPT
A: Convex function from statespace to \([0,1]\) which exactly identify a state \(E_x(\omega_1) = E_x(\omega_2) \implies \omega_1 = \omega_2\)

Q: Def: Fiducial effects
A: List of effects that fully and minimaly describe the state.
Note we can then write states as:

Q: Intuition Measurement in GPT's
A: A measurement is something where we have quantified all possible outputs, that is the probability of someting happening is 1

Mathematical definition Measurement in GPT

Q: Postulates of QM in QIT
A: 1) Local tomography
Combined states can be characterized by individual measurements on subspaces
2) Equivalence of subspaces
Removing one effect from a system makes it equivalent to systems with smaller capacity
3) Symmetry
All pure states can be reversibly transformed into eachother
4) All measurements allowed
Every linear map that can be interpreted as a probability is an effect

Q: Definition capacity GPT
A: Largest number of perfectly distinguishable states

Q: How do the dimensions and number of effects of a state space combine?
A: \(d_{AB} + 1 = (d_A + 1)(d_B + 1)\)
With \(d_A \) the number of fiducial effects

Q: How do capacities of systems combine?
A: They are multiplicative!

Q: Dimension of a quantum system from hilbert space:
A: \(dim(\mathcal H_A)^2 -1\)
Example: Qbit
\(\mathcal H_A\) ist \(2D\) -> \(3D\) system (bloch vektor + amplitude)
With capacity \(dim \mathcal H_A = 2\)

Q: Pure state GPT
A: Cannot be written as non-trivial mixture of two states.
Easyer:
Non pure state can allways be written as non-trivial mixture of  states

Bell states:

Last chapter (not inclusive 7) QIT

Def von Neumann entropy:

Q: QIT min-entropy:
A: with \(||\rho||_\infty\) the largest eigenvalue of \(\rho\)

Q: QIT max-entropy
A: With \(|supp(\rho)|\) the cardinality of the support of rho

What can we tell about a states Entropy?

Q: Entropy for pure states on \(AB\):
A: Entropy on the subsystems is equal

Entropy of tensor product of states

Entropy of probabilistic mixture:

Entropy of classical state:

Conditional entropy and chain rule

Q: When can the conditional entropy be negative?
A: Iff the state is entangled

Cyclicity of entropy on pure states

Strong subadditivity Entropy

Upper bound to quantum entropy

Def: Mutual information

Q: Properties of mutual info:
A: Positivity
Information processing inequality
Knowledge addition

Use of min-entropy:

Def conditional min-entropy

Q: Lower bound for conditional min-entropy
A: This is an upper bound to how good we can guess

Q: Definition separable state:
A: With \(\tau_k\) on A and \(\sigma_k\) on B

Q: Quantum channel notation:
[...]
A: Authentic, insecure quantum, secure

Schema for realizability of perfect quantum cryptography

Q: Meaning correctness in QKD:
A: Probability of transfering the key correctly is \(\varepsilon\) small

Q: Meaning robustness QKD:
A: Probability of a passive listener disturbing a key gen is \(\varepsilon\) small

Q: Meaning secrecy QKD:
A: The difference between the resulting state that Emily can have \(\rho_{S_AE}\) and a completeley mixed state with probability \(p\) or an arbitrary state with \(p-1\) is \(\varepsilon\) small
aka. Emily can only get uniform or uncorrelated or no key at all

Still missing: prococols in 7 and BB86 QIT