Copyright 2001 S.D. Personick, All rights reserved Quantum Information Science and Technology Dr. Stewart (“Stu”) Personick Professor, Drexel University ECE Department Feb 16, 2001

Copyright 2001 S.D. Personick, All rights reserved Quantum Information Science and Technology Using the “weird” properties of the physical world, that are predicted by the quantum theory, to: -make computers that can perform computations, in reasonable amounts of time, that appear to require impractically large amounts of time using classical computers (e.g., find the factors of a large number) -construct communication systems with remarkable and useful properties (e.g., provably secure)

Copyright 2001 S.D. Personick, All rights reserved Quantum Information Science and Technology Some history: -1920’s…scientists invent the quantum formalism to provide a mathematical framework for “predicting” (retroactively) certain puzzling observations…e.g., why don’t electrons spiral into the nucleus of an atom? -this quantum formalism predicts all sorts of weird and non-intuitive things… e.g., the EPR thought experiment -1990’s… when technology evolves to the point where experiments can be conducted…the weird predictions of the quantum theory turn out to be consistent with experimental results

Copyright 2001 S.D. Personick, All rights reserved Quantum Information Science and Technology The more you think about, and understand quantum theory and its implications… the more uneasy you become: e.g., Anybody who is not shocked by quantum theory has not understood it ---Niels Bohr

Copyright 2001 S.D. Personick, All rights reserved Quantum Computation The simplest physical system, such as an electron in spin state “up” or “down”, can be modeled as having a “state” that is a vector in a two-dimensional space |0> |1> |s>= a|1> + b |0>

Copyright 2001 S.D. Personick, All rights reserved Quantum Computation The quantum state of a physical system represents the entire physical reality of the physical system (no “hidden variables”). |0> |1> |s>= a|1> + b|0>

Copyright 2001 S.D. Personick, All rights reserved Quantum Computation The quantum state of a physical system evolves, in time, according to Schrodinger’s equation: [ ih/2pi] d/dt|s> = H|s> |0> |1> |s>= a|1> + b|0>

Copyright 2001 S.D. Personick, All rights reserved Quantum Computation The quantum state of a physical system evolves, in time, according to Schrodinger’s equation: [ ih/2pi] d/dt|s> = H|s> We can perform a computation by preparing an initial state |s>, allowing it to interact with a physical system (I.e., select H) of our choice, and then performing a measurement on the evolved state |s’> |0> |1> |s>= a|1> + b|0>

Copyright 2001 S.D. Personick, All rights reserved Quantum Computation Is it real? Several possible answers, including: -yes, it is real…you can make real systems that successfully implement quantum computation -no, there is a “showstopper” problem that prevents us from making a system that successfully implements quantum computation…but we just haven’t thought of it yet Answering this question is what the QuIST program and related research efforts are trying to accomplish

Copyright 2001 S.D. Personick, All rights reserved Example of a quantum logic operation Quantum Controlled X Operator |u> |s> |u> |s’> |s> = a|1> +b|0> |u>= |1> or |0> If |u>=|0>, then |s’> = |s>; If |u> =|1>, then |s’>= a|0> + b|1>

Copyright 2001 S.D. Personick, All rights reserved Example of a quantum logic operation (this stuff is weird) Quantum Controlled X Operator |u> |s> |u> |s’> |s> = a|1> +b|0> |u>= c|1>+d|0> Output state is c[|1>(b|1>+a|0>)] +d[|0>( a|1> + b|0>)] …the output QuBits are “entangled” |s> and |u> are “quantum bits” (QuBits)

Copyright 2001 S.D. Personick, All rights reserved Example of a quantum logic operation (this stuff is weird) Quantum Controlled X Operator |u> |s> |u> |s’> |s> = |0> |u>= [ |1>+|0> ] / (2**0.5) The output state is [|1>|1>] +[|0>|0>)]/ (2**0.5) …one of the four “Bell States” or “EPR” pairs (Einstein, Podolsky, Rosen) |s> and |u> are “quantum bits” (QuBits)

Copyright 2001 S.D. Personick, All rights reserved Schrodinger’s Cat (When I hear about Schrodinger’s Cat…I reach for my gun --- Stephen Hawking) Quantum Controlled Cat Killer Operator |u> |alive > |u> |s’> |s> = |alive> |u>= [ |1>+|0> ] / (2**0.5) The output state is [|0>|alive>] +[|1>|dead>)]/ (2**0.5) |s> and |u> are “quantum bits” (QuBits)

Copyright 2001 S.D. Personick, All rights reserved Example of “communication” at faster than the speed of light (this stuff is weird) Quantum Controlled X Operator |u> |s> |u> |s’> |s> = |0> |u>= [ |1>+|0> ] / (2**0.5) The output state is [|1>|1>] +[|0>|0>)]/ (2**0.5); If you measure QuBit 1, to “determine” whether it is in state |0> or state |1>, then you know the state of QuBit 2 |s> and |u> are “quantum bits” (QuBits)

Copyright 2001 S.D. Personick, All rights reserved Quantum Teleportation EPR pair: |b> = [ |1>|1> + |0>|0> ] / (2**0.5) H |b> |x> Measure XZ |x> Alice Bob

Copyright 2001 S.D. Personick, All rights reserved Quick Review of Shannon’s Formula Channel CoderDecoder XY A B Source information rate = H(A) I (X:Y)= Mutual information between X and Y= H(X) - H(X|Y) where: h(u) = -p(u) log p(u); H (U)=sum {h(u)}

Copyright 2001 S.D. Personick, All rights reserved Quick Review of Shannon’s Formula Channel CoderDecoder Channel input symbols {x} Channel output symbols {y} C (channel capacity) = H(X) - H(X|Y) maximized over all apriori probability distributions of X XY

Copyright 2001 S.D. Personick, All rights reserved Applying Shannon’s formula to an optical fiber link with specific types physical transmitters and receivers Laser Receiver Fibe r Classical channel capacity: C= B log[1 + E/N] Homodyne receiver case: N= hf Heterodyne receiver case: N=2hf Optical preamplifier: N=2hf E/N ~P/hfB; for B=20 GHz, hfB ~3 x 10**-9 watts ~ - 55 dBm C (20 GHz, -25dBm ) ~ 20 GHz x 10 bps/Hz = 200 Gbps C (20 GHz, -8 dBm ) ~ 20 GHz x 16 bps/Hz = 320 Gbps

Copyright 2001 S.D. Personick, All rights reserved Extending Shannon’s formula to an optical fiber link viewed from the perspective of quantum theory Coder Any physical transmitting subsystem consistent with the laws of physics fiber Any physical receiver consistent with quantum measurement theory Quantu m state Decoder

Copyright 2001 S.D. Personick, All rights reserved Special Case Ref: Yuen and Ozawa Physical Review Letters Jan 25, 1993 Assumption: the received signal is in a single electromagnetic field mode, subject to a constraint that the average received number of photons (energy constraint), as defined by a measurement of the “number operator”, is less than N photons c (capacity per use of the channel)= (n+1) log (n+1) - n log n where n= E/hf, and E is the average energy per received symbol

Copyright 2001 S.D. Personick, All rights reserved Special Case: Yuen and Ozawa Physical Review Letters Jan c (capacity per use of the channel)= (n+1) log (n+1) - n log n = log (n+1) + n log(1 + 1/n) where n= E/hf, and E is the average energy per received symbol. n infinity log (1+n) infinity n log (1+ 1/n) (bits)