Home on EPHhttps://ericphanson.com/Recent content in Home on EPHHugo -- gohugo.ioen-usMon, 20 Aug 2018 17:35:28 -0400Exercise 1https://ericphanson.com/teaching/qit/es3/exercise_1/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es3/exercise_1/Exercise 1. Given two qubit states \(\rho\) and \(\omega\) with Bloch vectors \(\vec r\) and \(\vec s\) respectively, show that \[ \|\rho-\omega\|_1 = \|\vec r - \vec s\|_2. \] The two norm of a vector \(\vec v = (v_x,v_y,v_z)\) is \(\|\vec v\|_2 = \sqrt{v_x^2 + v_y^2 + v_z^2}\).
We have \[ \rho = \frac{1}{2}(I+ \vec r \cdot \vec \sigma), \qquad \omega = \frac{1}{2}(I+ \vec s \cdot \vec \sigma), \] so \[ \|\rho - \sigma\|_1 = \frac{1}{2}\| (\vec r - \vec s)\cdot \vec \sigma\|_1 =\frac{1}{2} \left\| \begin{pmatrix} v_z & v_x - i v_y \\ v_x + i v_y & - v_z \end{pmatrix}\right\|_1 \] where \(\vec v = \vec r - \vec s\), just as from exercise 1 from example sheet 2.Exercise 1https://ericphanson.com/teaching/qit/es4/exercise_1/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es4/exercise_1/Exercise 1. The other Fuchs and van de Graaf inequality.
Let \(\{E_m\}_{m \in M}\) be a POVM, where \(M\) is a finite set. Given two states \(\rho_A\) and \(\sigma_A\), use the Cauchy-Schwarz inequality to show that \[ F(\rho_A,\sigma_A)\leq \sum_{m\in M}\sqrt{\operatorname{tr}[E_m \rho] \operatorname{tr}[E_m \sigma]} = F( \tilde \rho_M, \tilde \sigma_M ) \](1) where \[ \tilde \rho_M = \sum_{m\in M} \operatorname{tr}[E_m \rho] | m \rangle\langle m |, \qquad \tilde \sigma_M = \sum_{m\in M} \operatorname{tr}[E_m \sigma]| m \rangle\langle m | \](2) are diagonal states encoding probabilities of the measurement outcomes \(m\), in a Hilbert space \(\mathcal{H}_M\) of dimension \(M\) with orthonormal basis \(\{| m \rangle\}_{m\in M}\).Exercise 1https://ericphanson.com/teaching/qit/revision/exercise_1/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/revision/exercise_1/Exercise 1. Define the trace distance \(D(\rho,\sigma)\) between two states \(\rho,\sigma\in \mathcal{D}(\mathcal{H})\) and prove that it can be expressed in the form: \[ D(\rho,\sigma) = \frac{1}{2}(\operatorname{tr}Q + \operatorname{tr}R) \] where \(Q\) and \(R\) are suitably defined positive semi-definite operators in \(\mathcal{B}(\mathcal{H})\). Using the above identity, prove that \[ D(\rho,\sigma) = \max_P \operatorname{tr}(P(\rho -\sigma)) \] where the maximisation is over all projection operators \(P\in \mathcal{B}(\mathcal{H})\). Further, prove that \[ D(,\rho,\sigma) = \max_T \operatorname{tr}(T(\rho-\sigma)) \] where the maximisation is over all positive semi-definite operators \(T\in \mathcal{B}(\mathcal{H})\) with eigenvalues less than or equal to unity.Is QSEP-CIRCUIT QMA-hard?https://ericphanson.com/work/presentation/is-qsep-circuit-qma-hard/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/is-qsep-circuit-qma-hard/“QSEP-CIRCUIT, as defined in a recent paper by Patrick Hayden, Kevin Milner, and Mark Wilde (http://arxiv.org/abs/1211.6120)in the computer science department here at McGill, is the problem of determining whether or not a quantum state given as a circuit is separable or not. The problem has been shown to be in QIP(2), and to be NP-hard and QSZK-hard, which lends credence to the idea that the problem is QMA-hard. The goal of this talk is to provide background for and explain in detail what QSEP-CIRCUIT and QMA are, explain why we care whether or not QSEP-CIRCUIT is QMA-hard, and hopefully showcase quantum information theory as an exciting and dynamic field.QSEP-STATE in QMIPnehttps://ericphanson.com/work/poster/qsep-state-in-qmipne/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/poster/qsep-state-in-qmipne/A poster on my work with Chris Bahr, under the supervision of Patrick Hayden, on a quantum complexity theory problem about entanglement, presented at McGill University for an undergraduate computer science poster session. Abstract: The promise problem QSEP-STATE asks if a quantum state described by a circuit is close to a separable (not entangled) state across a given cut, or not. A quantum multiprover protocol was found to give an upper bound for the computational complexity problem of QMIPne, which is the class of problems that can be solved by a computationally bounded quantum verifier exchanging messages with unentangled, computationally unbounded, quantum provers.Question 1https://ericphanson.com/teaching/qit/questions/answer_1/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_1/Question 1. In Notes 8, why does equation (1.3) follow hold? We invert \(I_d \otimes R\) and \(\Lambda \otimes I_n\). Can you describe why please? Is it because the identity matrix commutes with everything?
Good question! The short answer is yes. In this part, we start with \[ \Lambda \otimes \operatorname{id}_k ( |\varphi_i\rangle \langle\varphi_i|), \] and use a relation from Notes 7 to write \(|\varphi_i\rangle = I_d\otimes R_i\, |\Omega\rangle\) for some matrix \(R_i\).Exercise 2https://ericphanson.com/teaching/qit/es3/exercise_2/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es3/exercise_2/Exercise 2. One Fuchs and van de Graaf inequality.
Show that for two pure qubit states \(\psi\) and \(\phi\), we have that \[ D(\psi,\phi) = \sqrt{1 - F(\psi,\phi)^2}. \] where \(D\) is the trace distance, and \(F\) the fidelity, defined in the lectures. Using this, show that for any two mixed qubit states \(\rho\) and \(\sigma\), \[ D(\rho,\sigma) \leq \sqrt{1- F(\rho,\sigma)^2}. \] For two pure states, we have \(F(\psi, \phi) = |\langle \psi|\phi \rangle|\).Exercise 2https://ericphanson.com/teaching/qit/es4/exercise_2/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es4/exercise_2/Exercise 2. Let \(|\psi\rangle_{ABE}\) be a pure state of a tripartite system \(ABE\). define the coherent information from \(A\) to \(B\) of \(\psi\) to be \[ I_c^{A>B}(\psi) = - S(A|B)_\psi. \] Here \(S(A|B)_\psi\) denotes the conditional entropy of the subsystem \(A\) with respect to subsystem \(B\), given that the composite system \(ABE\) is in the pure state \(|\psi\rangle_{ABE}\). Henceforth we shall omit \(\psi\).
Prove the following identities:
\(\frac{1}{2} I(A:B) + \frac{1}{2} I(A:E)= S(A)\)Exercise 2https://ericphanson.com/teaching/qit/revision/exercise_2/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/revision/exercise_2/Exercise 2. State the Holevo-Schumacher-Westmoreland theorem. Use it to obtain the product-state classical capacity of a qubit depolarizing channel \(\Lambda\) defined as follows: \[ \Lambda(\rho) = p \rho + \frac{1-p}{3}(\sigma_x\rho\sigma_x + \sigma_y \rho \sigma_y + \sigma_z \rho \sigma_z), \] where \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) are the Pauli matrices. Consider an ensemble of quantum states \(\mathcal{E}= \{p_x,\rho_x\}\) and let \(\chi(\mathcal{E})\) denote its Holevo quantity. Let \(\Lambda\) be a quantum channel. Prove that \[ \chi(\mathcal{E}') \leq \chi(\mathcal{E}) \] where \(\mathcal{E}' = \{p_x,\Lambda(\rho_x)\}\).Life on Phorcyshttps://ericphanson.com/work/misc/life-on-phorcys/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/misc/life-on-phorcys/2013 University Physics Competition submisssion. “In this article, we investigate the possible forms of life that may exist on planet Phorcys. Our approach is to determine key parameters describing the planet: temperature variation, ice cover, surface pressure and ocean presence and depth. We find that Phorcys is covered in a global ocean, with large ice caps and the possibility of an ice free equatorial region. The presence and size of this region is determined by a combination of the salinity of the ocean water, which affects the freezing point of water, and the day length of the planet, which affects how temperature varies with latitude.Question 2https://ericphanson.com/teaching/qit/questions/answer_2/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_2/Question 2. Hey! Could you also please help me with an exercise (Notes 5, eq 21): show that a separable bipartite state can always be expressed as convex combination of pure product states: \[ \rho_{AB} = \sum\limits_{i=1} p_i |\psi_i\rangle\langle\psi_i|\otimes |\phi_i\rangle\langle\phi_i|. \] I get confused when I expand the density matrices and try to confine the sum to only one variable.
Hi! Good question. The definition of a separable state is that we can write it in the form \[ \rho_{AB} = \sum_{i=1}^n p_i \omega_i^A \otimes \sigma_i^B \] for some probability distribution \(\{p_i\}_{i=1}^n\) and density matrices \(\omega_i^A\) and \(\sigma_i^B\).Exercise 3https://ericphanson.com/teaching/qit/es3/exercise_3/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es3/exercise_3/Exercise 3. Show that for any two states \(\rho\) and \(\sigma\) and any unitary \(U\), \[ F(U \rho U^\dagger,U \sigma U^\dagger) = F(\rho,\sigma) \] by using the fact that for any positive semi-definite operator \(A\) and unitary \(U\), we have \(\sqrt{U A U^\dagger} = U \sqrt{A} U^\dagger\).
We have \[\begin{aligned} F(U\rho U^\dagger, U\sigma U^\dagger) &= \| \sqrt{U \rho U^\dagger}\sqrt{U \sigma U^\dagger}\|_1 \\ &= \| U \sqrt{\rho} U^\dagger U \sqrt{\sigma}U^\dagger\|_1\\ &= \| U \sqrt{\rho}\sqrt{\sigma}U^\dagger\|_1\\ &= \max_{V} | \operatorname{tr}[ V U \sqrt{\rho}\sqrt{\sigma}U^\dagger ]| \\ &= \max_{V} | \operatorname{tr}[ U^\dagger V U \sqrt{\rho}\sqrt{\sigma} ]| \\ &= \max_{UVU^\dagger} | \operatorname{tr}[ U^\dagger U V U^\dagger U \sqrt{\rho}\sqrt{\sigma} ]| \\ &= \max_{UVU^\dagger} | \operatorname{tr}[\sqrt{\rho}\sqrt{\sigma} ]| \\ &= \|\sqrt{\rho}\sqrt{\sigma}\|_1 \\ &= F(\rho,\sigma).Exercise 3https://ericphanson.com/teaching/qit/es4/exercise_3/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es4/exercise_3/Exercise 3. A bipartite quantum state \(\rho_{AB}\) is said to be separable if it can be written as a convex combination of product states, i.e., if there exists an ensemble \(\{p_i, \sigma_A^{(i)} \otimes \tau_B^{(i)}\}\), with \(\sigma_A^{(i)} \in \mathcal{B}(\mathcal{H}_A)\) and \(\tau_B^{(i)}\in \mathcal{B}(\mathcal{H}_B)\), such that \[ \rho_{AB} = \sum_i p_i \sigma_A^{(i)} \otimes \tau_B^{(i)}. \] This allows us to extend the definition of entanglement to mixed states: a mixed state is entangled if it is not separable.Exercise 3https://ericphanson.com/teaching/qit/revision/exercise_3/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/revision/exercise_3/Exercise 3. The Bell states \(| \Phi^+_{AB} \rangle\), \(| \Phi^-_{AB} \rangle\), \(| \Psi^+_{AB} \rangle\), \(| \Psi^-_{AB} \rangle\), can be characterized by two classical bits, namely the parity bit and the phase bit. Show that the latter are eigenvalues of two commuting observables. The Bell states form an orthonormal basis of the two-qubit Hilbert space. It is referred to as the Bell basis. Let us denote it by \(B_1\). A sequence of two operations can be used to convert states of the computational basis \(B_2 := \{| ij \rangle: i,j\in \{0,1\}\}\) to the Bell states.Non-local games and communication complexityhttps://ericphanson.com/work/notes/non-local-games-and-communication-complexity/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/notes/non-local-games-and-communication-complexity/“Synthesizing some of the literature on non-local games and communication complexity scenarios provides a deeper understanding of the fundamental differences between classical mechanics and quantum mechanics, as exemplified by the quantum violation of Bell inequalities, which hold in classical mechanics. Quantum mechanical protocols are shown to provide significant advantages in certain tasks over classical protocols, and could create the first loophole-free demonstration of quantum non-locality. Non-local games are a scenario where two players attempt to perform a task without communication; communication complexity scenarios generalize non-local games by allowing but attempting to minimize communication.Question 3https://ericphanson.com/teaching/qit/questions/answer_3/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_3/Question 3. Could you please help me with an exercise given in the lectures (Notes 13, eq 1.18): show that there exist a set of unitary operators \(\{U_j\}_j\) and a probability distribution \(\{p_j\}_j\) such that \[ \rho_A \otimes \sigma_B = \sum\limits_j p_jU_j\rho_{AB}U_j^\dagger, \](1) where \(\sigma_B = I/d\) is a completely mixed state.
The exercise in (1.18) is a bit tricky— I wouldn’t know how to do it without having seen it before.Exercise 4https://ericphanson.com/teaching/qit/es3/exercise_4/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es3/exercise_4/Exercise 4. Show that the fidelity is jointly concave. That is, given any finite probability distribution \(\{p_i\}_{i=1}^n\) and quantum states \(\rho_i\) and \(\sigma_i\) for \(i=1,\ldots,n\), \[ F\left(\sum_{i=1}^n p_i \rho_i ,\sum_{i=1}^n p_i \sigma_i\right) \geq \sum_{i=1}^n p_i F(\rho_i,\sigma_i). \]
Hint: Choose purifications \(\phi_i\) of \(\rho_i\) and \(\psi_i\) of \(\sigma_i\) such that \(F(\rho_i,\sigma_i) = \langle \phi_i|\psi_i \rangle\), using Uhlmann’s theorem.
As the hint suggests, we will take purifications \(\phi_i\) of \(\rho_i\) and \(\psi_i\) of \(\sigma_i\) such that \(F(\rho_i,\sigma_i) = \langle \phi_i|\psi_i \rangle\).Exercise 4https://ericphanson.com/teaching/qit/es4/exercise_4/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es4/exercise_4/Exercise 4. Use the HSW theorem to find the product state capacity of the depolarizing channel, \(\Lambda\), defined by \[ \Lambda(\rho) = p\rho + (1-p) \frac{I}{2}. \]
Note we are only dealing with qubits (otherwise \(\Lambda\) would not be trace-preserving). First, note that for any unitary \(U\), we have that \[ \Lambda(U \rho U^*) = pU\rho U^* + (1-p) \frac{I}{2} \] and therefore \(\Lambda(U \rho U^*) = U \Lambda(\rho)U^*\).Exercise 4https://ericphanson.com/teaching/qit/revision/exercise_4/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/revision/exercise_4/Exercise 4. Prove that any completely positive trace-preserving map \(\Phi\), acting on states \(\rho\) in a Hilbert space \(\mathcal{H}_A\) can be written in the Kraus form: \[ \Phi(\rho) = \sum_k A_k \rho A_k^\dagger, \] where \(A_k\) are linear operators satisfying \(\sum_k A_k^\dagger A_k = I\) and \(I\) is the identity operator.
Hint: consider a maximally entangled state. Using a maximally entangled state and the properties of the swap operator, prove that the transposition operator \(T\) is positive but not completely positive.Math 354 Honours Analysis 3 Noteshttps://ericphanson.com/work/notes/math-354-honours-analysis-3-notes/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/notes/math-354-honours-analysis-3-notes/Introduction to metric spaces, topological spaces.Question 4https://ericphanson.com/teaching/qit/questions/answer_4/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_4/Question 4. How to prove superadditivity of Holevo capacity? It is given as Exercise 2 in notes 18: \[\chi^*(\Lambda_1 \otimes \Lambda_2) \geq \chi^*(\Lambda_1) + \chi^*(\Lambda_2).\]
The trick is that since the Holevo capacity is defined as a maximum over all ensembles, in particular it’s at least as large as its value on any tensor product ensemble.
P.S. for more on the additivity of the Holevo capacity, Section 20.5 of Mark Wilde’s book has an interesting discussion.Exercise 5https://ericphanson.com/teaching/qit/es3/exercise_5/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es3/exercise_5/Exercise 5. Show that the fidelity is multiplicative under tensor products. That is, \[ F(\rho_1 \otimes \rho_2 , \sigma_1\otimes \sigma_2) = F(\rho_1,\sigma_1) F(\rho_2, \sigma_2). \]
Hint: first show that \(\|A\otimes B\|_1 = \|A\|_1\|B\|_1\).
First, we have that \[\begin{aligned} \| A\otimes B\|_1 &= \operatorname{tr}[ \sqrt{(A^\dagger\otimes B^\dagger)(A\otimes B)} ]\\ &= \operatorname{tr}[ \sqrt{A^\dagger A \otimes B^\dagger B}]\\ &= \operatorname{tr}[ \sqrt{A^\dagger A} \otimes \sqrt{B^\dagger B}]\\ &= \operatorname{tr}[ \sqrt{A^\dagger A}] \operatorname{tr}[ \sqrt{B^\dagger B} ]\\ &= \|A\|_1 \|B\|_1.Exercise 5https://ericphanson.com/teaching/qit/es4/exercise_5/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es4/exercise_5/Exercise 5. Alice prepares a photon in one of two polarization states, given by the kets \(|a\rangle := |1\rangle\), and \(|b\rangle :=\sin \theta |0\rangle + \cos \theta |1\rangle\), depending on the outcome of a fair coin toss. If the outcome is heads, she prepares the state \(|a\rangle\). Otherwise she prepares the state \(|b\rangle\). Evaluate the Holevo \(\chi\) quantity for her ensemble of states. (Use the convention that \(|0\rangle =(1\,\,0)^T\) and \(|1\rangle = (0\,\,\,1)^T\)).Exercise 5https://ericphanson.com/teaching/qit/revision/exercise_5/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/revision/exercise_5/Exercise 5. Consider the decay of a two-level atom from its excited state to its ground state. Let the probability of this decay be \(p\). The spontaneous emission of a photon accompanies this decay.
Name the quantum channel that can be used to model this process and write its Kraus operators. What process does each of these Kraus operators correspond to? Give reasons for your answer. What is a unital channel?QSEP-State is in QMIPnehttps://ericphanson.com/work/presentation/qsep-state-is-in-qmipne/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/qsep-state-is-in-qmipne/“The promise problem QSEP-STATE asks if a quantum state described by a circuit is close to a separable(not entangled) state across a given cut, or not. A quantum multiprover protocol was found to give an upper bound for the computational complexity of QMIPne, which is the class of problems that can be solved by a computationally bounded quantum verifier exchanging messages with unentangled, computationally bounded, quantum provers.”Question 5https://ericphanson.com/teaching/qit/questions/answer_5/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_5/Question 5. Hi Eric, I’ve been getting myself a bit confused with the 2014 exam, Q3.iii; could you please tell me the best way to approach it?
Thanks
This question is about the “completely dephasing map,” \[ \Lambda(\rho) = \sum_y | y \rangle\langle y | \rho | y \rangle\langle y |. \](1) In part (ii) of the question, you’re asked to prove it is a CPTP map. Then part (iii) asks you to find an expression for the von Neumann entropy of a state \(\sigma =\Lambda(\rho)\).Exercise 6https://ericphanson.com/teaching/qit/es3/exercise_6/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es3/exercise_6/Exercise 6. Let \(\rho=\sum_ip_i\rho_i\) where \(\rho_i\) are density matrices which have support on orthogonal subspaces. Then prove that \[ S\bigg(\sum_ip_i\rho_i\bigg)=H(p)+\sum_ip_iS(\rho_i) \] where \(p=\{p_i\}\) is a probability distribution, and \(H(p)\) is the corresponding Shannon entropy.
Let us write the total number of states as \(n\). Recall \(S(\rho) := - \operatorname{tr}[\rho \log \rho]\). In fact, we don’t need to restrict \(S\) to act on states; for any \(X\geq 0\), we can define \[ S(X) = - \operatorname{tr}[X\log X].Exercise 6https://ericphanson.com/teaching/qit/es4/exercise_6/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es4/exercise_6/Exercise 6. Alice encodes classical information into \(n\) photons which she sends to Bob through a quantum channel. What is the maximum number of bits of information that Bob can infer from the output of the channel by doing measurements on it?
Hint: Use the Holevo bound.
Let \(X\) denote Alice’s classical information, and \(Y\) the random variable obtained by Bob from measurements on what he obtains from the quantum channel.Question 6https://ericphanson.com/teaching/qit/questions/answer_6/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_6/Question 6. On the 2011 exam Q3.iii, do you act on each term in the MES product individually as separable states with the channel acting on the second qubit and the identity leaving the first, and then adding the 4 terms together? Or is there a better way without writing everything out?
This question is about the qubit depolarizing channel1, which can be written as \[ \Phi(\rho) = \frac{q}{2}I + (1-q)\rho \](1) where \(q = 4p/3\).The black hole information paradoxhttps://ericphanson.com/work/notes/the-black-hole-information-paradox/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/notes/the-black-hole-information-paradox/“The black hole information paradox concerns the intersection of quantum mechanics and general relativity,so its resolution could point the way towards a quantum theory of gravity. The paradox is a contradiction between our understanding of the limits of our physical theories and the consequences of them. With curvature on the scales of that in our solar system, current physical theories have been very successful experimentally; thus, there seems to exist a “solar system limit” in which the laws of physics reduce to those of quantum field theory.Exercise 7https://ericphanson.com/teaching/qit/es3/exercise_7/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es3/exercise_7/Exercise 7. Prove that the von Neumann entropy is a concave function of its inputs, i.e., given probabilities \(p_i \ge 0\), \({\displaystyle{\sum_{i=1}^r p_i =1}}\), and corresponding density operators \(\rho_i\): \[ S\left( \sum_{i=1}^r p_i \rho_i \right) \ge \sum_{i=1}^r p_i S\left(\rho_i\right). \](1) Show that if each \(p_i > 0\) and \(\rho_i\neq \rho_j\) for \(i\neq j\), then the inequality (1) is strict. Note: means the von Neumann entropy is strictly concave.
Hint: Consider the state \(\sigma_{AB} := \sum_i p_i \rho_i \otimes |i\rangle \langle i|\), its reduced states \(\sigma_A\), \(\sigma_B\) and use subadditivity.Exercise 7https://ericphanson.com/teaching/qit/es4/exercise_7/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es4/exercise_7/Exercise 7. Entropy exchange. The entropy exchange for a state \(\rho\) and a quantum channel \(\Lambda\) is defined as follows: \[ S(\rho, \Lambda):= S(\rho_{RQ}^\prime), \] where \(\rho_{RQ}^\prime= ({\rm id}_R \otimes \Lambda) \psi^\rho_{RQ}\), with \(\psi^\rho_{RQ}\) being a purification of \(\rho\).
Prove that \(S(\rho, \Lambda) = S(\rho_E^\prime)\), where \(\rho_E^\prime = \operatorname{tr}_{RQ} (\rho_{RQE}^\prime)\) with \[ \rho_{RQE}^\prime = (I_R \otimes U_{QE}) (\psi^\rho_{RQ} \otimes |0_E\rangle \langle 0_E|) (I_R \otimes U_{QE}^\dagger), \] with \(U_{QE}\) being the Stinespring dilation of the channel \(\Lambda\).Landauer's Principlehttps://ericphanson.com/work/presentation/landauers-principle/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/landauers-principle/“Landauer’s principle states that the energy cost to erase one bit of information bythe action of a thermal reservoir at equilibrium temperature $T$ is always at least $kT$, and provides an interesting link between information theory and physics. My goal is to discuss Landauer’s principle generally, and for the case of a 2 level system coupled to a thermal reservoir modelled by an XY spin chain (or sequence of spin-1⁄2 particles), using the tools of quantum statistical mechanics.Question 7https://ericphanson.com/teaching/qit/questions/answer_7/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_7/Question 7. Regarding Q2 of the 2012 exam: Is there a better way to calculate part iv) entanglement fidelity than doing purification directly?
This question asks to to compute the entanglement fidelity \(F_\text{e}(\rho,\Lambda)\) where \(\rho\) is a qubit and \(\Lambda\) is the amplitude damping channel. The entanglement fidelity is \[ F_\text{e}(\rho,\Lambda) = \langle \psi_{RA}^\rho | \operatorname{id}\otimes \Lambda(\psi^\rho_{RA}) | \psi^\rho_{RA} \rangle \] where \(\psi^\rho_\text{RA}\) is a purification of \(\rho\). Is there a clever trick we can use to compute \(F_\text{e}(\rho,\Lambda)\)?Exercise 8https://ericphanson.com/teaching/qit/es3/exercise_8/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es3/exercise_8/Exercise 8. Consider the following classical-quantum or cq-states on the Hilbert space \(\mathbb{C}^n\otimes \mathcal{H}\): \[ \rho = \sum_{i=1}^np_i|i\rangle\langle i|\otimes \rho_i, \qquad \qquad \sigma = \sum_{i=1}^np_i|i\rangle\langle i|\otimes \sigma_i \] where \(\rho_i,\sigma_i\in\mathcal{D}(\mathcal{H})\) and \(\{p_i\}\) is a probability distribution. Evaluate the quantum relative entropy \(D(\rho\|\sigma)\) and use the result to prove that \[ D\bigg(\sum_ip_i\rho_i\|\sum_ip_i\sigma_i\bigg)\leq \sum_ip_iD(\rho_i\|\sigma_i), \] i.e. joint convexity of the quantum relative entropy.
The solution I presented during the example class used the monotonicity of the relative entropy under partial trace, which was proven in the lectures using the joint convexity of the entropy.Exercise 8https://ericphanson.com/teaching/qit/es4/exercise_8/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es4/exercise_8/Exercise 8. Quantum Fano inequality. Prove the quantum Fano inequality: \[ S(\rho,\Lambda)\le h(F_e(\rho,\Lambda))+(1-F_e(\rho,\Lambda))\,\log(d^2-1) \] where
\(\Lambda\) denotes a quantum operation with Kraus representation \[ \Lambda(\rho)=\sum_{i=0}^{d^2} V_i \rho V_i^\dagger, \] with \(\rho\) being the state of a quantum system \(Q\) with Hilbert space of dimension \(d\). \(h(\cdotp)\) denotes the binary Shannon entropy, i.e., for any \(0<p<1\), \[ h(p) = - p \log p - (1-p) \log (1-p). \] \(S(\rho, \Lambda)=-\operatorname{tr}(W \log W)\), with \(W\) being a matrix with elements \(W_{ij}=\operatorname{tr}(V_i\rho V_j)\).Question 8https://ericphanson.com/teaching/qit/questions/answer_8/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_8/Question 8. Regarding Q3 of the 2009 exam: I don’t know how to show part b), that \[ S(A|B) \geq - \log \dim \mathcal{H}_A. \]
This one’s a little tricky. This question is Theorem 11.5.1 of Mark Wilde’s book, and he uses a purification to solve it, which seems like a good way to go. The way that comes more naturally to me is as follows. Since the quantum conditional entropy is concave, it’s minimum over the set of states must occur on a pure state.The existence and uniqueness of the Haar Measurehttps://ericphanson.com/work/notes/the-existence-and-uniqueness-of-the-haar-measure/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/notes/the-existence-and-uniqueness-of-the-haar-measure/“The Haar measure allows integration over topological groups,which has many applications; for example, the use of the Haar measure in proving the Fully Quantum Slepian Wolf (or state transfer) protocol fundamental to quantum information theory is briefly discussed in section VI. But first we build the definitions necessary to define the Haar measure, and state some fundamental theorems. Next, we show a left Haar measure exists on locally compact Hausdorff topological groups, and is unique (up to scale) on such groups which are also $\sigma$-compact.Exercise 9https://ericphanson.com/teaching/qit/es3/exercise_9/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es3/exercise_9/Exercise 9. Let \(\rho_{AB}= \sum_i p_i \rho_{AB}^i\) be the state of a bipartite quantum system \(AB\). Using the joint convexity of the relative entropy, prove that the quantum conditional entropy is concave in the state \(\rho_{AB}\).
We want to show that for each \(t\in (0,1)\) and pair of quantum states \(\rho_{AB}\) and \(\sigma_{AB}\) on some bipartite Hilbert space \(\mathcal{H}_A\otimes \mathcal{H}_B\), show that \[ S(A|B)_{\omega} \geq t S(A|B)_{\rho} + (1-t) S(A|B)_{\sigma}.Exercise 9https://ericphanson.com/teaching/qit/es4/exercise_9/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es4/exercise_9/Exercise 9. Continuity of the von Neumann entropy (Fannes’ inequality): Suppose \(\rho, \sigma \in \mathcal{D}( \mathcal{H})\) are states such that their trace distance \(D(\rho, \sigma)\) satisfies the bound \[ 2 D(\rho, \sigma) = \|\rho - \sigma\|_1 \le 1/e. \] Then \[ |S(\rho) - S(\sigma)| \le \|\rho - \sigma\|_1 \log d + \eta\left( \|\rho - \sigma\|_1\right), \](1) where \(d = \dim \mathcal{H}\), and \(\eta(x) := - x \log x.\)
Let us prove this theorem in steps:Landauer's Principle in Repeated Interation Systemshttps://ericphanson.com/work/paper/landauers-principle-in-repeated-interation-systems/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/paper/landauers-principle-in-repeated-interation-systems/“We study Landauer’s Principle for Repeated Interaction Systems (RIS) consisting ofa reference quantum system $\mathcal{S}$ in contact with a structured environment $\mathcal{E}$ made of a chain of independent quantum probes; $\mathcal{S}$ interacts with each probe, for a fixed duration, in sequence. We first adapt Landauer’s lower bound, which relates the energy variation of the environment $\mathcal{E}$ to a decrease of entropy of the system $\mathcal{S}$ during the evolution, to the peculiar discrete time dynamics of RIS.Question 9https://ericphanson.com/teaching/qit/questions/answer_9/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_9/Question 9. Regarding Q2.i of the 2014 exam: For part (a) I don’t understand why bistochastic is important; my solution is simply \(H(Y|X=x) = H(q)\) for each \(x\), so \(H(Y|X) = H(q)\).
For part (b) I got an answer \(\log(3) - H(q)\) but my argument is not that convincing: solve \(p(y|x)p(x) = \{1/3, 1/3, 1/3\}\) but \(p(y|x)\) may not be invertible.
For part (a): I agree, you only need that the rows are permutations of each other for this part.Exercise 10https://ericphanson.com/teaching/qit/es3/exercise_10/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es3/exercise_10/Exercise 10. Let \(\mathcal{H}_A, \mathcal{H}_B\) and \(\mathcal{H}_{B'}\) be Hilbert spaces and \(\rho\in\mathcal{D}(\mathcal{H}_{AB})\) be a state. Further, let \(\Lambda:\mathcal{H}_B\rightarrow\mathcal{H}_{B'}\) be a CPTP map and define \[ \sigma_{AB'}= ( \operatorname{id}_A \otimes \Lambda)\rho_{AB}. \] Prove that \[ S(A|B)_{\rho_{AB}} \leq S(A|B')_{\sigma_{AB'}}. \] Hint: Use strong subadditivity.
To use subadditivity, we need three systems. We’ll introduce a third: by Stinespring’s dilation theorem, there is a Hilbert space \(\mathcal{H}_C\), an isometry \(U : \mathcal{H}_B\otimes \mathcal{H}_C \to \mathcal{H}_{B'}\otimes \mathcal{H}_{C}\) and a pure state \(\varphi\in\mathcal{D}(\mathcal{H}_C)\) such that \(\Lambda(\rho)=\operatorname{tr}_C(U(\rho\otimes\varphi)U^\dagger)\) for all \(\rho\in\mathcal{D}(\mathcal{H}_B)\).Exercise 10https://ericphanson.com/teaching/qit/es4/exercise_10/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es4/exercise_10/Exercise 10. An interesting class of quantum channels are the entanglement-breaking (EB) channels. An EB channel \(\Lambda\) is one for which \((\operatorname{id}\otimes \Lambda)(\omega)\) is separable, even for entangled \(\omega\)1. The Holevo capacity has been proved to be additive for EB channels.
Prove that any channel of the following form is EB: \[ \Lambda(\rho) = \sum_k \sigma_k \operatorname{tr}(E_k\rho), \](1) where \(\sigma_k\) are density matrices and \(\{E_k\}\) is a POVM. The above form has the following physical interpretation.Landauer's Principle in RIShttps://ericphanson.com/work/presentation/landauers-principle-in-ris/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/landauers-principle-in-ris/Slides for a talk I gave to the Physics of Information lab at Waterloo, February 2016, about Landauer’s Principle in Repeated Interaction Systems. Based on the preprint arxiv/1510.00533.Question 10https://ericphanson.com/teaching/qit/questions/answer_10/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_10/Question 10. This may be a very stupid question, but in the notes we talk about a Bell measurement which can determine which Bell state two qubits are in, and these are given by the projection operators: \[P_{00} = | \phi^{+} \rangle \langle \phi^{+} | \] …etc. In the notes it claims that a Bell Measurement leaves the state unchanged, but surely that’s only the case if the Bell state of the two qubits matches that in the projector, otherwise you have two orthogonal states and the result of applying the projection operator is zero?Combinatorics Noteshttps://ericphanson.com/work/notes/combinatorics-notes/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/notes/combinatorics-notes/Extremal combinatorics: Sperner systems, the Littlewood-Offord problem, intersecting hypergraphics, compression, Turan type problems, Ramsey theory, convexity, incidence problems, and algebraic methods. Contribute typo fixes here.Exercise 11https://ericphanson.com/teaching/qit/es3/exercise_11/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es3/exercise_11/Exercise 11. Projective measurements do not decrease von Neumann entropy
Suppose a projective measurement described by a set of projection operators \(\{P_i\}\) is performed on a quantum system, but we never learn the result of the measurement. If the state of the system before the measurement was \(\rho\) then the state after the measurement is given by \[\rho^\prime = \sum_i P_i \rho P_i.\] Prove that the entropy of this final state is at least as great as the original entropy: \[S(\rho^\prime) \ge S(\rho),\] with equality if and only if \(\rho = \rho^\prime\).Exercise 11https://ericphanson.com/teaching/qit/es4/exercise_11/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/es4/exercise_11/Exercise 11. Projective measurements do not decrease von Neumann entropy
Suppose a projective measurement described by a set of projection operators \(\{P_i\}\) is performed on a quantum system, but we never learn the result of the measurement. If the state of the system before the measurement was \(\rho\) then the state after the measurement is given by \[\rho^\prime = \sum_i P_i \rho P_i.\] Prove that the entropy of this final state is at least as great as the original entropy: \[S(\rho^\prime) \ge S(\rho),\] with equality if and only if \(\rho = \rho^\prime\).Question 11https://ericphanson.com/teaching/qit/questions/answer_11/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_11/Question 11. Dear Eric,
Our definition of the quantum relative entropy (equation (1.6) of Notes 13) is for \(\rho\) a state and \(\sigma\) a positive definite operator, but during the proof of Klein’s inequality in the same notes, we seem to make use that \(\sigma\) has trace 1 although it was not one of the assumptions! (This is when showing that \(r_i\) is a probabililty distribution).
Good point! Klein’s inequality only holds when both arguments of the quantum relative entropy are states.Landauer's Principle in Repeated Interaction Systemshttps://ericphanson.com/work/essay/landauers-principle-in-repeated-interaction-systems/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/essay/landauers-principle-in-repeated-interaction-systems/My masters thesis; essentially, an expanded version of most of arXiv:1510.00533, in my own words. The abstract: “Landauer’s Principle states that there is a lower bound on the energy required to change the state of a small system from an initial state to a final state by interacting with a thermodynamic reservoir;of particular interest is when the bound is saturated and the minimal energy cost obtained for a given state transformation.Landauer’s Principle in Repeated Interaction Systemshttps://ericphanson.com/work/presentation/landauers-principle-in-repeated-interaction-systems/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/landauers-principle-in-repeated-interaction-systems/The slides for the talk I gave at the Autrans summer school Stochastic Methods in Quantum Mechanics on my work with Alain Joye, Yan Pautrat, and Renaud Raquepas on Landauer’s Principle in repeated interaction systems.Question 12https://ericphanson.com/teaching/qit/questions/answer_12/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_12/Question 12. When we talk about capacities in a completely classical context, Shannon’s Source Coding Theorem gives that any rate \(R>H(U)\) is reliable. Whereas in the quantum case the HSW Theorem states that a rate \(R\) is reliable if \(R< \chi^*(\Lambda)\) . But in both contexts means roughly the same thing. For the classical case: the number of bits of codeword/number of uses of the source. And for quantum: ‘number of bits of classical message that is transmitted per use of the channel’.Landauer's Principle in Repeated Interation Systemshttps://ericphanson.com/work/poster/landauers-principle-in-repeated-interation-systems/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/poster/landauers-principle-in-repeated-interation-systems/A poster on my work with Alain Joye, Yan Pautrat, and Renaud Raquepas on Landauer’s Principle in repeated interaction systems, presented at the one-day event CCIMI New Directions in the Mathematics of Information.Question 13https://ericphanson.com/teaching/qit/questions/answer_13/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_13/Question 13. In Notes 11, on encoding two classical bits as a Bell state it says: ‘We can encode two classical bits in the state of the two qubit system. This information can be recovered by performing a joint measurement (on the two qubits) which projects onto the Bell basis. This is called a Bell measurement and is a measurement in which the measurement operators are the Bell state projectors’.Local continuity bounds for entropies of finite probability distributionshttps://ericphanson.com/work/poster/local-continuity-bounds-for-entropies-of-finite-probability-distributions/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/poster/local-continuity-bounds-for-entropies-of-finite-probability-distributions/A poster on my work with my supervisor Nilanjana Datta on local continuity bounds for the entropies of finite distributions, presented at the one-day event High Dimensional Mathematics. I gave a short “elevator pitch” to advertise the poster, the video recording of which is posted here.Question 14https://ericphanson.com/teaching/qit/questions/answer_14/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/teaching/qit/questions/answer_14/Question 14. Uhlmann’s theorem says that \(F(\rho, \sigma) = \max_{\text{all purifications}} | \langle \phi_\rho | \phi_\sigma \rangle |\). Why is the absolute value there? Is it just to make complex numbers into reals, so that we can compare them? So technically the formula would be correct even if there was \(\text{Re} \langle \phi_\rho | \phi_\sigma \rangle\) or \(\text{Im} \langle \phi_\rho | \phi_\sigma \rangle\) instead? Am I right to think that we can always adjust the complex phase of \(\langle \phi_\rho | \phi_\sigma \rangle\) by considering \(|\phi_\rho \rangle \rightarrow |\psi_\rho\rangle = |\phi_\rho \rangle |0\rangle\) and \(|\phi_\sigma \rangle \rightarrow |\psi_\sigma\rangle = |\phi_\sigma \rangle[\cos(\theta) + i\sin(\theta)] |0\rangle\)?Local continuity bounds for entropies of finite probability distributionshttps://ericphanson.com/work/presentation/local-continuity-bounds-for-entropies-of-finite-probability-distributions/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/local-continuity-bounds-for-entropies-of-finite-probability-distributions/A presentation on my work with my supervisor Nilanjana Datta on local continuity bounds for the entropies of finite distributions, presented at the one-day event High Dimensional Mathematics. These are the slides I gave accompanying my short talk, the video of which is here.Landauer's Principle for Trajectories of Repeated Interaction Systemshttps://ericphanson.com/work/paper/landauers-principle-for-trajectories-of-repeated-interaction-systems/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/paper/landauers-principle-for-trajectories-of-repeated-interaction-systems/A refinement and generalization of our previous work on Landauer’s Principle in repeated interaction systems. We consider a two-time measurement protocol of the energy of the quantum probes, and recover a large deviations principle and a central limit theorem in the adiabatic limit of a repeated interaction system. Abstract: “We analyze Landauer’s principle for repeated interaction systems consisting of a reference quantum system $\mathcal{S}$ in contact with a environment $\mathcal{E}$ consisting of a chain of independent quantum probes.A universal construction of tight continuity bounds for entropieshttps://ericphanson.com/work/poster/a-universal-construction-of-tight-continuity-bounds-for-entropies/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/poster/a-universal-construction-of-tight-continuity-bounds-for-entropies/A poster on my work with my supervisor Nilanjana Datta on uniform continuity bounds for the single-partite entropies, presented at the conference Beyond IID 2017 in Singapore in August 2017, and the main conference of the thematic semester Analysis in Quantum Information Theory in Paris in December 2017.Maximum and minimum entropy states yielding local continuity boundshttps://ericphanson.com/work/paper/maximum-and-minimum-entropy-states-yielding-local-continuity-bounds/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/paper/maximum-and-minimum-entropy-states-yielding-local-continuity-bounds/We consider the geometry of the trace-ball of quantum states, find maximal and minimal states in a particular partial order called majorization, and use these states to construct local continuity bounds for quantum entropies. We also apply the theory of convex optimization to motivate the construction of the maximal state, and to find general optimality conditions for a particular class of functions subject to a trace-ball constraint. Abstract: “Given an arbitrary quantum state ($\sigma$), we obtain an explicit construction of a state $\rho^\varepsilon(\sigma)$ (resp.Quantum Information Theoryhttps://ericphanson.com/work/presentation/quantum-information-theory/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/quantum-information-theory/Winning entry for the 2017 CCIMI video contest. Video:
Many thanks to falcxne for allowing use of his songs “Said You Wouldn’t Change” (from the album Timbits III), “I Hate Homework (Interlude)” (from Timbits II) and “NYC (Interlude)” (from Timbits III); you can find these and more at https://falcxne.bandcamp.com/. And thanks to Phillipe Faist for creating the entropy zoo and allowing me to use it in the video. For more details on the continuity bounds mentioned, you can find a preprint here, which is joint work with my supervisor, Dr.Extrema in majorization order with applications to entropic continuity boundshttps://ericphanson.com/work/essay/extrema-in-majorization-order-with-applications-to-entropic-continuity-bounds/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/essay/extrema-in-majorization-order-with-applications-to-entropic-continuity-bounds/An essay submitted in the competition for a Smith-Knight or Rayleigh-Knight prize. This essay combines my work on uniform continuity bounds (arXiv:1707.04249) with that on local continuity bounds (arXiv:1706.02212), and unifies the notation. Abstract: “Majorization is a pre-order of vectors, giving a sense in which one vector can be said to be more disordered than another. Two given vectors, however, may be incomparable. This concept has been extendedto quantum mechanical states, to particular success in the theory of entanglement manipulation.Fast small random density matriceshttps://ericphanson.com/blog/2018/fast-small-random-density-matrices/Sun, 23 Sep 2018 20:07:42 +0000https://ericphanson.com/blog/2018/fast-small-random-density-matrices/Sometimes, I find it useful to generate small random density matrices to test some statement I hope is true. When it’s not true, often this can quickly surface a counterexample. I’ve been using the scientific programming language Julia which recently hit version 1.0! That means there won’t be any “breaking changes” to the language (until 2.0). Pre-1.0, every new version (0.5, 0.6, etc) would change the language and you’d have to update your code.Website rewritehttps://ericphanson.com/blog/2018/website-rewrite/Sat, 25 Aug 2018 19:26:42 -0400https://ericphanson.com/blog/2018/website-rewrite/I decided to redo my website using Hugo (before I was using Jekyll). Both are static site generators: you write in Markdown, a simple clean formatting language, and it generates HTML webpages following a consistent format. It’s great because the generation only has to occur once after the Markdown is written, and nothing active has to happen for each user of the website (unlike a so-called dynamic website). This reduces costs, increases reliability, and improves securitythe user needs very little access to the server hosting the webpages in a static siteLocally maximizing the Rényi entropieshttps://ericphanson.com/blog/2018/locally-maximizing-the-r%C3%A9nyi-entropies/Sat, 25 Aug 2018 00:00:00 +0000https://ericphanson.com/blog/2018/locally-maximizing-the-r%C3%A9nyi-entropies/As I was rewriting my website, I found some visualizations I had stored on my old website to show a collaborator, and I figured it was worth writing a little to have a more proper place to put them; hence this post 😊.
Probability distributions on three letters consist just of three non-negative numbers which add up to 1, which we can see as a vector in \(\mathbb{R}^3\). The set of all such distributions form a simplex, which looks like a 2D triangle laying in \(\mathbb{R}^3\):How to make an index in LaTeXhttps://ericphanson.com/blog/2018/how-to-make-an-index-in-latex/Wed, 20 Jun 2018 00:00:00 +0000https://ericphanson.com/blog/2018/how-to-make-an-index-in-latex/I’m sure there are many ways to make an index in LaTeX, but I was asked this question recently and thought I’d put my response here, which is based on what I did in my combinatorics notes.
In the preamble, make a command \defw (short for “define word”), use the package imakeidx, and call \makeindex:
\usepackage{imakeidx} \usepackage{xparse} \NewDocumentCommand{\defw}{m o}{% {\emph{#1}}% \IfNoValueTF{#2} {\index{#1}} {\index{#2}}% } \makeindex Generate the index at the end of the file with \printindex, like one does with a bibliography.arXiv-search (the sad goodbye-for-now post)https://ericphanson.com/blog/2018/arxiv-search/Mon, 14 May 2018 00:00:00 +0000https://ericphanson.com/blog/2018/arxiv-search/For a few months, I and a few others were working on a project we called “arxiv-search”, an attempt to search and sort all of the arxiv (~1 million papers). We were inspired by Andrej Karpathy’s arxiv sanity preserver which is an excellent tool for a limited set of papers (~50,000). Starting from that project, we ended up writing a new backend and frontend. Our backend used elasticsearch which is a large scale search engine which runs constantly on a big server, indexing metadata and responding to search requestsIn fact, the arxiv itself very recently started using elasticsearch to improve their own search results.A synchronized dance of eigenvalueshttps://ericphanson.com/blog/2017/a-synchronized-dance-of-eigenvalues/Tue, 26 Sep 2017 00:00:00 +0000https://ericphanson.com/blog/2017/a-synchronized-dance-of-eigenvalues/The motivation behind this post is to show some off some nice gifs. But I thought maybe they aren’t actually interesting without any context, so below I’ll try to explain what the gifs are about.
Brief mathematical introduction to Perron-Frobenius theory Consider the set of \(n\times n\) matrices \(M_n\), with complex entries.Perceptron demohttps://ericphanson.com/blog/2017/perceptron-demo/Wed, 19 Jul 2017 00:00:00 +0000https://ericphanson.com/blog/2017/perceptron-demo/A javascript demonstration of the perceptron algorithm, written for a reading group session. This was originally a separate webpage, but when I rewrote my website I decided to make it a blog post. This post can still be reached from /perceptron-demo, though.
Perceptron Perceptron is a very simple binary classification online learning algorithm, dating from the 1950s1. Such an algorithm tries to classify input (here, points in the plane) into one of two categories.The arrow of time in RIShttps://ericphanson.com/blog/2017/the-arrow-of-time-in-ris/Wed, 18 Jan 2017 21:09:03 +0000https://ericphanson.com/blog/2017/the-arrow-of-time-in-ris/Repeated interaction systems (RIS) Introduction to arrow of time A more careful description of the forward and backward processes Hypothesis testing on the arrow of time Connection to Landauer’s Principle I gave an informal talk today on the arrow of time in repeated interaction systems and I thought I’d write about it here.
Repeated interaction systems (RIS) For a recent review of RIS, see arxiv/1305.2472. We also introduce them in my and my coauthor’s work Landauer’s Principle in Repeated Interaction Systems.The traveling salesman and 10 lines of Pythonhttps://ericphanson.com/blog/2016/the-traveling-salesman-and-10-lines-of-python/Tue, 25 Oct 2016 00:00:00 +0000https://ericphanson.com/blog/2016/the-traveling-salesman-and-10-lines-of-python/Update (21 May 18): It turns out this post is one of the top hits on google for “python travelling salesmen”! That means a lot of people who want to solve the travelling salesmen problem in python end up here. While I tried to do a good job explaining a simple algorithm for this, it was for a challenge to make a progam in 10 lines of code or fewer. That constraint means it’s definitely not the best code around for using for a demanding application, and not the best for learning to write good, readable code.Setting up SublimeTexthttps://ericphanson.com/blog/2016/setting-up-sublimetext/Fri, 14 Oct 2016 00:00:00 +0000https://ericphanson.com/blog/2016/setting-up-sublimetext/I’m setting up a new laptop and figured it was a good chance to document setting up SublimeText for LaTeX, as quasi follow up to my recent post about live LaTeXing. However, installing LaTeX can be hard, and configuring the SublimeText package LaTeXTools to work with LaTeX has the potential to be hard (often just works, but if not, can be confusing), and I won’t write about those parts, because they are better documented elsewhere.Live notetaking with LaTeXhttps://ericphanson.com/blog/2016/live-notetaking-with-latex/Sat, 24 Sep 2016 00:00:00 +0000https://ericphanson.com/blog/2016/live-notetaking-with-latex/A friend suggested I write a guide about live notetaking in LaTeX, since it’s pretty useful and something I have a fair amount of experience with. I had been making attempts at live-LaTeXing course notes for a year or two before I was able to make it through a semester long class; the first course I fully LaTeX’d was Vojkan Jaksic’s excellent Analysis 3 (introduction to metric spaces and topology)PDF. Not my best work, but it did the job.Landauer's Principle and the balance equationhttps://ericphanson.com/blog/2016/landauers-principle-and-the-balance-equation/Mon, 29 Feb 2016 21:50:00 +0000https://ericphanson.com/blog/2016/landauers-principle-and-the-balance-equation/I’ve been working on my thesisEdit: this was my master’s thesis
over reading week, and I think I’ve finished my introduction to Landauer’s Principle. I ended up writing a pretty detailed derivation of the balance equation, and thus Landauer’s bound, so I thought it might be useful to post here.
Landauer's principle states that there is a minimal energetic cost for a state transformation \(\rho^\text{i}\to \rho^\text{f}\) on a system \(\mathcal{S}\) via the action of a thermal reservoir \(\mathcal{E}\) at temperature \((k_B\beta)^{-1}\)⊕\(k_B \approx 1.Cantor's set and functionhttps://ericphanson.com/blog/2016/cantors-set-and-function/Tue, 09 Feb 2016 00:00:00 +0000https://ericphanson.com/blog/2016/cantors-set-and-function/Cantor’s Set Construction Properties Cantor’s function I wrote these notes in February 2016 for an Analysis 2 tutorial when I was a teaching assistant at McGill, and always intended to put them here eventually; before August 2018 though, I hadn’t translated them to something web-friendly and only had posted a PDFThe web version has slightly improved wording in some parts.
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Cantor’s Set Cantor’s set is an interesting subset of \([0,1]\), with properties that help illuminate concepts in analysis.Completeness Ihttps://ericphanson.com/blog/2016/completeness-i/Fri, 01 Jan 2016 00:00:00 +0000https://ericphanson.com/blog/2016/completeness-i/Last semester, I helped a friend review McGill’s Analyis 3 course by trying to provide a better feel for completeness; this post will be a slightly edited version of that. Originally, I wanted to write about both completeness and compactness, and their connections, but I ended up only getting to completeness, and actually not everything I wanted to talk about. So I’ll call this Completeness I, leaving open the possibility for more of these in the future.New bloghttps://ericphanson.com/blog/2015/new-blog/Mon, 14 Dec 2015 00:00:00 +0000https://ericphanson.com/blog/2015/new-blog/I’ve made a new blog!
I started a “coffee blog” a month or so ago but only posted once (I think I’ll bring that post over here, too). I do like the idea though, and want to write about math and my research. So this should become a place for me to do that.
Part of my impetus is that journals and traditional publication methods don’t provide a pathway for discussing failed attempts.Abouthttps://ericphanson.com/about/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/about/I’m a PhD student in the Department of Applied Math and Theoretical Physics at Cambridge University, in my second year.
I’m interested in quantum statistical mechanics and quantum information theory; see here for my recent work.
I also made a video introducing quantum information theory (though it’s mostly just about information theory):
Many thanks to falcxne for allowing use of his songs “Said You Wouldn’t Change” (from the album Timbits III), “I Hate Homework (Interlude)” (from Timbits II) and “NYC (Interlude)” (from Timbits III); you can find these and more at https://falcxne.Contact mehttps://ericphanson.com/contact/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/contact/This is an attempt to have a LaTeX-friendly way to contact me.
Write your message in the box below. The “Editor (popout)” button should pop-out a more full-featured editor (via StackEdit), but it might not work on mobile. When you’re done, click the “x” in the top-left to leave the editor, and press “Submit”. That will email me the message via a service called formspree. I am not affiliated with them or with StackEdit, so please don’t type in anything secret.RIS full dipole codehttps://ericphanson.com/code/ris_full_dipole/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/code/ris_full_dipole/Files Mathematica file: RIS_full_dipole.nb
PDF: RIS_full_dipole.pdf
Last updated 8 April 2016.
Back to code.Search Resultshttps://ericphanson.com/search/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/search/Thankshttps://ericphanson.com/thanks/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/thanks/Thanks for sending me a message! I’ll try to get back to you soon.