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Lecture on classical and quantum information

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Brief introduction to classical and quantum information is given with intuitive examples.

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Lecture on classical and quantum information

  1. 1. Lecture on classical and quantum information theory Krzysztof Pomorski University of Warsaw kdvpomorski@gmail.com 30 marca 2017 Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 1 / 66
  2. 2. Zakres wykładu: Omówione zastaną wybrane zagadnienia z klasycznej i kwantowej teorii informacji. Potwierdzona zostanie teza Landauera. Wskazane zostaną procesy utraty informacji w trakcie przekazywania danych na odległość. Nakreślone zostanie odwołanie do termodynamiki. Wskazana zostanie między innymi teza Żurka o nieklonowalności kubitu. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 2 / 66
  3. 3. 1 Motivation 2 Interlink between information and physics Maxwell’s demon The four laws of thermodynamics Thermodynamics of big systems 3 Classical information theory Shannon theory Classical measurement 4 Quantum information theory Analogy between QM and Statistical Physics Quantum measurement Qubit as Bloch sphere New properties from quantum mechanics Interaction of 2 quantum systems 5 Classical vs quantum information theory Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 3 / 66
  4. 4. Motivation 1. China has just launched the world’s first quantum communications satellite [600kg]. The satellite is both an extreme test of the weird properties of quantum mechanics, and a technology tested for what could be the start of a global, unhackable communications network. 2. Quantum Cryptography is like computing all over again. We cannot possibly tell what the implications may be. Andrew Hilton, director of CSFI 3. Quantum computer going beyond classical computer. 4. Construction of supersensitive devices. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 4 / 66
  5. 5. Information and physics [4] Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 5 / 66
  6. 6. Chaos vs order in classical systems [12] Detection of chaos in simple classical systems. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 6 / 66
  7. 7. Maxwell’s demon Maxwell’s demon is the name given to a thought experiment designed to question the possibility of violating the second law of thermodynamics. It was formulated and named after the Scottish physicist James Clerk Maxwell in 1867. Maxwell’s demon demonstration turns information into energy [if we accomodate all gas particle on one side thanks to information on position of particles]. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 7 / 66
  8. 8. Small quantum systems The Ohm law or Heat Flow law does not work properly in small systems since electron or phonon flow is not fully deterministic [it works on average]. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 8 / 66
  9. 9. The four laws of thermodynamics Zeroth law of thermodynamics: If two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other. This law helps define the notion of temperature. First law of thermodynamics: When energy passes, as work, as heat, or with matter, into or out from a system, the system’s internal energy changes in accord with the law of conservation of energy. Second law of thermodynamics: In a natural thermodynamic process, the sum of the entropies of the interacting thermodynamic systems increases. Third law of thermodynamics: The entropy of a system approaches a constant value as the temperature approaches absolute zero.With the exception of non-crystalline solids (glasses) the entropy of a system at absolute zero is typically close to zero, and is equal to the logarithm of the product of the quantum ground states. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 9 / 66
  10. 10. Thermodynamics of big systems [4] Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 10 / 66
  11. 11. Gibbs entropy [11] In statistical thermodynamics, entropy S is a measure of the number of microscopic configurations Ω that a thermodynamic system can have when in a state as specified by some macroscopic variables. Specifically, assuming that each of the microscopic configurations is equally probable, the entropy of the system is the natural logarithm of that number of configurations, multiplied by the Boltzmann constant kb. Let us consider the system with W degeneracies for given energy. For fixed internal energy U, volume V, number of particles N, S = −kb i pi log(pi ) = −kb i (1/wi )log(1/wi ) = kblogW (U, V , N) (1) We can view W = Ω as a measure of our lack of knowledge about a system [microcanonical ensemble]. 3 type of ensembles are given as: (1) The Microcanonical ensemble is an isolated system. (2) The Canonical ensemble is a system in contact with a heat bath. (3) The Grand Canonical ensemble is a system in contact with a heat and particle bath. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 11 / 66
  12. 12. Schematic representation of communication channel Rysunek: From Shannon ’s A Mathematical Theory of Communication, page 3. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 12 / 66
  13. 13. The Shannon information content h(x) of an outcome x is defined to be h(x) = log2( 1 P(x) ). (2) It is measured in bits. [The word ’bit’ is also used to denote a variable whose value is 0 or 1]. The entropy of an ensemble X is defined to be the average Shannon information content of an outcome: H(x) = 1 P(x) log2( 1 P(x) ), (3) with the convention for P(x)=0 that 0 × log(1/0) = 0, since limΘ→0+ Θlog(1/Theta) = 0. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 13 / 66
  14. 14. Definition of channel capacity C The capacity C of a discrete channel is given by C = LimT→∞ logN(T) T , (4) where N(T) is the number of allowed signals of duration T. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 14 / 66
  15. 15. The first theorem deals with communication over a noiseless channel. Let a source have entropy H(bits per symbol) and a channel have a capacity C (bits per transmit at the average rate C/H - symbols per second over the channel where is arbitrarily small. It is not possible to transmit at an average rate greater than C/H. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 15 / 66
  16. 16. Shannon-Hartley theorem It states that the channel capacity C, meaning the theoretical tightest upper bound on the information rate of data that can be communicated at an arbitrarily low error rate using an average received signal power S through an analog communication channel subject to additive white Gaussian noise of power N: C = B log2 1 + S N (5) where: C - the channel capacity in bits per second, a theoretical upper bound on the net bit rate (information rate, sometimes denoted I) excluding error-correction codes; B - the bandwidth of the channel in hertz (passband bandwidth in case of a bandpass signal); S - the average received signal power over the bandwidth (in case of a carrier-modulated passband transmission, often denoted C), measured in watts (or volts squared); N - the average power of the noise and interference over the bandwidth, measured in watts (or volts squared). Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 16 / 66
  17. 17. Classical channel with white noise The capacity of a channel of band W perturbed by white thermal noise power N when the average transmitter power is limited to P is given by C = W × log(P + S/N) (6) Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 17 / 66
  18. 18. Landauer principle: .Each time a single bit of information is erased it the amount of energy dissipated to environment is kBTln2. where T is the temperature of enviroment and kB is Boltzmann constant. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 18 / 66
  19. 19. Derivation of Landauer principle [4] Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 19 / 66
  20. 20. Classical measurement-example Imagine two wooden balls of Radius R1 and R2 in space [as far from Pluton ”planet”] that are charged with Q1 and Q2 where Q1 and Q2 have opposite signs and that they have bounded state. They are analogical to Newton gravitational problem of 2 bodies. We can photograph two moving balls by shining radiation on this system. In such case we perturbed the already existing [because of photon pressure] state by changing balls trajectory, moving some of charge to vacuum and so on. It is impossible to copy dynamical behavior of the system without changing it by small perturbation!!!! In very real sense it is example of classical non-cloning theorem since we cannot copy the exact classical state [only with certain approximation]. It is desirable to refer to [17]. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 20 / 66
  21. 21. Analogy between Quantum Mechanics and Statistical Physics Superfluid liquid helium flowing out of container against gravitational field. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 21 / 66
  22. 22. Hamiltonian and Lagrangian formalism Hamiltonian equations of motion: d dt pk = − dH dqk , d dt qk = − dH dpk .L = k d dt (q)p − H (7) Lagrangian equations of motions: dL dqk = d dt dL d(dqk dt ) . (8) H = Ek + Ep = 1 2 m( d dt y)2 + mgy, L = Ek − Ep = 1 2 m( d dt y)2 − mgy. (9) Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 22 / 66
  23. 23. Moving from classical to quantum mechanics We have given equations of motion in Hamiltonian formalism. Suppose that f (p, q, t) is a function on the manifold. Then d dt f (p, q, t) = ∂f ∂q dq dt + ∂f ∂p dp dt + ∂f ∂t . Further, one may take p = p(t) and q = q(t) to be solutions to Hamilton’s equations; that is, ˙q = ∂H ∂p = {q, H}    ˙q = ∂H ∂p = {q, H} ˙p = −∂H ∂q = {p, H} Then d dt f (p, q, t) = ∂f ∂q ∂H ∂p − ∂f ∂p ∂H ∂q + ∂f ∂t = {f , H} + ∂f ∂t . Quantization is by replacement Poisson bracket {A, H} with commutator[A, H] = AH − HA. All observables in Quantum Mechanics as x, p or any other are operators and they do not need to commute and they cannot be fully determined at the same time [as it is in classical mechanics] in general!!! [For example ˆx = x and ˆpx = i d dx ] Since [ˆx, ˆp] = i I they do not commute and cannot be measured at the same time. For non-commuting variables x and p we have∆x∆p >= /2. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 23 / 66
  24. 24. Quantum harmonic oscillator (H0(x) + V (x))ψ(x) = ( 1 2m ˆp2 + ˆV (x))ψ(x) = Eψ(x) (10) (− 2 2m d2 dx2 + 1 2 kx2 )ψ(x) = Eψ(x) (11) Here ψ(x) is eigenfunction and E is eigenvalue. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 24 / 66
  25. 25. Quantum measurement-example Particle is at certain measurement with output A1 with certain probability. Once the measurement says A1 value the whole wavefunctions collapses and particle in only this state with probability 1. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 25 / 66
  26. 26. Qubit and quantum-mechanics Rysunek: Qubit in hydrogen atom. ˆH = ( ˆH0 + V (x))|ψ >= E|ψ > (12) |ψ(x) >=       ψ1(x) ψ2(x) ... ψm(x)       (13) Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 26 / 66
  27. 27. |ψ >= cos(Θ/2)|0 > +sin(Θ/2)eiϕ |1 >, (14) |ψ >= cos(Θ/2) 0 1 + sin(Θ/2)eiϕ 1 0 , (15) Bloch sphere representation of qubit, where Θ is the ’latitude’ and ϕ the ’longitude’ angles of ϕ on the Bloch sphere. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 27 / 66
  28. 28. Physical implementation of qubit Rysunek: Josephson junction as system of two coupling superconductors. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 28 / 66
  29. 29. T1 is the average time that the system takes for its excited state |1 > to decay to the ground state |0 >. T2 represents the average time over which the qubit energy-level difference does not vary [10]. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 29 / 66
  30. 30. Quantum interference Electron [particle] behaves as wave !!! Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 30 / 66
  31. 31. Quantum parallelism Rysunek: Photon distribution in system. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 31 / 66
  32. 32. Quantum entanglement |ψ >= 1 2 (|0 > |1 > −|1 > |0 >) (16) Entanglement is the strange phenomenon in which two quantum particles become so deeply linked that they share the same existence. When this happens, a measurement on one particle immediately influences the other, regardless of the distance between them. Measurement with 1 in first qubit gives state (|1 >< 1|×I)|ψ >= (|1 >< 1|×I) 1 2 (|0 > |1 > −|1 > |0 >) = − 1 2 |1 > |0 > (17) while the measurement of 0 in first qubit gives the state (|0 >< 0|×I)|ψ >= (|0 >< 0|×I) 1 2 (|0 > |1 > −|1 > |0 >) = + 1 2 |0 > |1 > . (18) Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 32 / 66
  33. 33. Cooper pair splitter Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 33 / 66
  34. 34. Interaction of two quantum systems A and B [as qubits] ˆH = ˆHa + ˆHb + ˆHab = Ha(2times2) × I2by2 + I2by2 × Hb(2times2) + Hab(4times4) (19) Here × denotes tensor product and I is identity matrix. The eigenstate of two isolated (non-interacting systems ˆHab = 0) A and B is given as |ψ >= |ψA > |ψB >=      ψ1A ψ2A ψ1B ψ2B      . (20) Consequently we have (Ha + Hb)|ψ >=      HA(1, 1) HA(1, 2) 0 0 HA(2, 1) HA(2, 2) 0 0 0 0 HB(1, 1) HB(1, 2) 0 0 HB(1, 1) HB(1, 2)      = E|ψ > . (21) Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 34 / 66
  35. 35. Idea of quantum computer The DiVincenzo criteria is a list of conditions that are necessary for constructing a quantum computer proposed by the theoretical physicist David P. DiVincenzo in his 2000 paper ”The Physical Implementation of Quantum Computation”. Quantum computation was first proposed by Richard Feynman (1982) as a means to efficiently simulate quantum systems. There have been many proposals of how to construct a quantum computer, all of which have varying degrees of success against the different challenges of constructing quantum devices. Some of these proposals involve using superconducting qubits, trapped ions, liquid and solid state nuclear magnetic resonance or optical cluster states all of which have remarkable prospects, however, they all have issues that prevent practical implementation. The DiVincenzo criteria are a list of conditions that are necessary for constructing the quantum computer as proposed by Feynman. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 35 / 66
  36. 36. Idea of quantum computer and DiVincenzo criteria In order to construct a quantum computer the following conditions must be met by the experimental setup. The first five are necessary for quantum computation and the remaining two are necessary for quantum communication. 1. A scalable physical system with well characterised qubits. 2. The ability to initialise the state of the qubits to a simple fiducial state. 3. Long relevant decoherence times. 4. A universal set of quantum gates. 5. A qubit-specific measurement capability. 6. The ability to interconvert stationary and flying qubits. 7. The ability to faithfully transmit flying qubits between specified locations. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 36 / 66
  37. 37. Classical annealing Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 37 / 66
  38. 38. Quantum annealing Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 38 / 66
  39. 39. Quantum memory Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 39 / 66
  40. 40. Fidelity Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 40 / 66
  41. 41. Quantum memory performance Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 41 / 66
  42. 42. Physical system implementing quantum annealing Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 42 / 66
  43. 43. Entanglement in Josephson junction system Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 43 / 66
  44. 44. Unitary quantum gates The diagram representing the action of a unitary matrix U corresponding to a quantum gate on a qubit in a state UU† = 1. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 44 / 66
  45. 45. CNOT gate Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 45 / 66
  46. 46. Basic quantum gates Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 46 / 66
  47. 47. Zurek no-cloning reasoning Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 47 / 66
  48. 48. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 48 / 66
  49. 49. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 49 / 66
  50. 50. Quantum teleportation Quantum reexportation is a process by which quantum information (e.g. the exact state of an atom or photon) can be transmitted (exactly, in principle) from one location to another, with the help of classical communication and previously shared quantum entanglement between the sending and receiving location. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 50 / 66
  51. 51. Example of weak measurement Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 51 / 66
  52. 52. Concept of density matrix The density matrix is generalization of description given by Schrodinger equation and formally is defined as the outer product of the wavefunction and its conjugate so it is matrix of the form as ρ(t) = |ψ(t) >< ψ(t)|. We have H|ψ(t) >= E|ψ(t) > and < ψ(t)|H† =< ψ(t)|E. Liouville-Von Neumann equation describes the dynamics of density matrix with time: d dt ρ = i [ρ, H]. (22) Average of observable A is given as < A >= tr(Aρ). (23) In case of thermal ensemble in equilibrium we have density matrix given as Z = tr(e−H/(kT))-partition function, ρequilibrium = e−En/kT /Z [thermodynamical ensemble]. ρ2 = 1 for pure state < 1 for mixed state (24) Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 52 / 66
  53. 53. Tensor product of matrices a11 a12 a21 a22 × b11 b12 b21 b22 = =       a11 b11 b12 b21 b22 a12 b11 b12 b21 b22 a21 b11 b12 b21 b22 a22 b11 b12 b21 b22       Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 53 / 66
  54. 54. Quantum Entropy (Von Neumann entropy) Definition of Entanglement in density matrix picture Entangled state is when its density matrix cannot be written as tensor product of two density matrices that is ρ = ρA × ρB. Quantum entropy is given by formula H(ρ) = Tr[ρlog(1/ρ)] = −Tr[ρlog(ρ)]. (25) If ρ is the joint state of two quantum systems A and B then the quantum mutual information is I(A, B) = H(ρA) + H(ρB) − H(ρ). (26) Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 54 / 66
  55. 55. QM equation of motion with presence of dissipation Decoherence and equations of motion. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 55 / 66
  56. 56. QM equation of motion with use of Wigner functions W (x, p) = 1 π +∞ −∞ ψ† (x + y)ψ(x − y)e 2iyp dy (27) Equations of motion for Wigner function. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 56 / 66
  57. 57. Quantum information theory Quantum information theory deals with four main topics: (1) Transmission of classical information over quantum channels. (2) The tradeoff between acquisition of information about a quantum state and disturbance of the state (briefly included in quantum cryptography). (3) Quantifying quantum entanglement. (4) Transmission of quantum information over quantum channels. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 57 / 66
  58. 58. Properties of the Isolated Quantum System with Finite Volume and a Finite Number of Particles [7] 1) These quantum systems evolve for reversible equations of motion (Schrodinger’s equation) 2) Poincare’s theorem is also accurate for these systems. 3) For quantum systems, it is also possible to define the entropy of ensemble. Entropy is a measure of uncertainty about the state of a system. A pure state provides a maximally complete description of a quantum system. Therefore, any pure state entropy is zero by definition. For the mixed-state case, the system corresponds to a set of pure states. Therefore, entropy already exceeds zero. Assume that the probability of a pure state is near 1. This mixed state is almost pure and its entropy is almost zero. On the other hand, when all pure states of the mixed state have equivalent probabilities, entropy reaches a maximum. 4) During the evolution of a quantum system, the pure state can evolve to a pure state only. In the mixed state, the probabilities of pure states also remain unchanged. Thus, the entropy of ensemble does not change during the evolution of a quantum system. 5) We can represent a large quantum system by a small number of parameters named macroscopic parameters. A large set of pure states defined by microscopic parameters corresponds to this mixed macroscopic state. The entropy of a macroscopic state can be calculated based on this pure set. We define this entropy as macroscopic entropy. In contrast with the entropy of ensemble, the macroscopic entropy should not be conserved during the evolution of a qsystem. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 58 / 66
  59. 59. 6) A quantum system will not be considered to be an isolated system due to its interaction with the measuring device. Its initial pure state evolves to a mixed state and its microscopic entropy increases. This evolution cannot be reversed by inversion of the measured system as inversion of the measuring device is also necessary. Poincare recurrence theorem [ from Wikipedia ] Any dynamical system defined by an ordinary differential equation determines a flow map f t mapping phase space on itself. The system is said to be volume-preserving if the volume of a set in phase space is invariant under the flow. For instance, all Hamiltonian systems are volume-preserving because of Liouville’s theorem. The theorem is then: If a flow preserves volume and has only bounded orbits, then for each open set there exist orbits that intersect the set infinitely often. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 59 / 66
  60. 60. Quantum vs classical information Quantum information differs strongly from classical information, epitomized by the bit, in many striking and unfamiliar ways. Among these are the following: A unit of quantum information is the qubit. Unlike classical digital states (which are discrete), a qubit is continuous-valued, describable by a direction on the Bloch sphere. Despite being continuously valued in this way, a qubit is the smallest possible unit of quantum information. The reason for this indivisibility is due to the Heisenberg uncertainty principle: despite the qubit state being continuously-valued, it is impossible to measure the value precisely. A qubit cannot be (wholly) converted into classical bits; that is, it cannot be ’read’. This is the no-teleportation theorem. Despite the awkwardly-named no-teleportation theorem, qubits can be moved from one physical particle to another, by means of quantum teleportation. That is, qubits can be transported, independently of the underlying physical particle. An arbitrary qubit can neither be copied, nor destroyed. This is the content of the no cloning theorem and the no-deleting theorem. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 60 / 66
  61. 61. Although a single qubit can be transported from place to place (e.g. via quantum teleportation), it cannot be delivered to multiple recipients; this is the no-broadcast theorem, and is essentially implied by the no-cloning theorem. Qubits can be changed, by applying linear transformations or quantum gates to them, to alter their state. Classical bits may be combined with and extracted from configurations of multiple qubits, through the use of quantum gates. That is, two or more qubits can be arranged in such a way as to convey classical bits. The simplest such configuration is the Bell state, which consists of two qubits and four classical bits (i.e. requires two qubits and four classical bits to fully describe). Quantum information can be moved about, in a quantum channel, analogous to the concept of a classical communications channel. Quantum messages have a finite size, measured in qubits; quantum channels have a finite channel capacity, measured in qubits per second. Multiple qubits can be used to carry classical bits. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 61 / 66
  62. 62. Although n qubits can carry more than n classical bits of information, the greatest amount of classical information that can be retrieved is n. This is Holevo’s theorem. Quantum information, and changes in quantum information, can be quantitatively measured by using an analogue of Shannon entropy, called the von Neumann entropy. Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 62 / 66
  63. 63. Many of the same entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy and the conditional quantum entropy. Quantum algorithms have a different computational complexity than classical algorithms. The most famous example of this is Shor’s factoring algorithm, which is not known to have a polynomial time classical algorithm, but does have a polynomial time quantum algorithm. Other examples include Grover’s search algorithm, where the quantum algorithm gives a quadratic speed-up over the best possible classical algorithm. Quantum key distribution allows unconditionally secure transmission of classical information, unlike classical encryption, which can always be broken in principle, if not in practice. (Note that certain subtle points are hotly debated). Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 63 / 66
  64. 64. Holevo bound Initially the sender, Alice, holds a long classical message. She encodes letter i (which appears with probability pi ) of this message into a pure state which, during the transmission, is turned into a possibly mixed quantum state q i due to the incomplete knowledge of the environment or of Eve’s actions. These quantum states are then passed on to the receiver, Bob, who then has the task to infer Alice’s classical message from these quantum states. The upper bound for the capacity for such a transmission, i.e. the information I that Bob can obtain about Alice’ s message per sent quantum state, is known as the Holevo bound I <= S(ρ) − i pi S(ρi ) (28) Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 64 / 66
  65. 65. References [1]. http://www.ieee.ca/millennium/radio/radio_differences.html [2]. Jakob Foerster, Lecture 1 of the Course on Information Theory, Pattern Recognition, and Neural Networks. Produced by: David Mac Kay (University of Cambridge) https://www.youtube.com/watch?v=BCiZc0n6COY [2]. Elementary gates for quantum computation, Adriano Barenco et al. , https://arxiv.org/abs/quant-ph/9503016 [3]. Entanglement in a Quantum Annealing Processor, PRX 4, 2014 [4]. Presentation: The physics of information: from MaxwellŹs demon to Landauer by Eric Lutz from University of Erlangen-Nrnberg [5]. Physics of Information F. Alexander Bais J. Doyne Farmer http://samoa.santafe.edu/media/workingpapers/07-08-029.pdf. [6]. https://ru.coursera.org/learn/quantum-optics-single-photon/ lecture/eo9Ym/7-3-one-photon-polarization-as-a-qubit [7]. Basic Paradoxes of Statistical Classical Physics and Quantum Mechanics by Oleg Kupervasser. [8]. http://web.physics.ucsb.edu/~quniverse/dhqm-exprob.html [9]. Quantum memories in atomic ensembles-G.Braunbeck Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 65 / 66
  66. 66. [10]. Superconducting circuits and quantum information-Physics today, J.You and F.Nori, 2005 [11]. Wikipedia [12]. Deterministic chaos-Shuster [13]. Landauer and noise [14]. Anne Hillebrand, PhD thesis: Quantum Protocols involving Multiparticle Entanglement and their Representations in the zx-calculus, 2011 [15]. Lecture on statistical physics, http://www.physics.udel.edu/~glyde/PHYS813/Lectures/chapter_6.pdf [16]. A single quantum cannot be cloned, W.K.Wootters, W.H.Zurek, Nature, Vol.299, 1982 [17]. Physical Review Letters, Vol. 88., No.21.,2002, Classical No-Cloning Theorem, A. Daffertshofer et al. [18]. Teleportation breakthrough made, Paul Rincon, 2004 http://news.bbc.co.uk/2/hi/science/nature/3811785.stm [19]. Quantum information theory and Holevo bound https://www.cs.cmu.edu/~odonnell/quantum15/lecture18.pdf Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 66 / 66

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