Picturing Quantum Processes,
|
|
|
- Leona Whitehead
- 6 years ago
- Views:
Transcription
1 Bob Coecke & Aleks Kissinger, Picturing Quantum Processes, Cambridge University Press, to appear.
2 Bob: Ch. 01 Processes as diagrams Ch. 02 String diagrams Ch. 03 Hilbert space from diagrams Ch. 04 Quantum processes Ch. 05 Quantum measurement Ch. 06 Picturing classical processes Aleks: Ch. 07 Picturing phases and complementarity Ch. 08 Quantum theory: the full picture Ch. 09 Quantum computing Ch. 10 Quantum foundations
3 Ch. 1 Processes as diagrams Philosophy [i.e. physics] is written in this grand book I mean the universe which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth. Galileo Galilei, Il Saggiatore, Here we introduce: process theories diagrammatic language
4 Ch. 1 Processes as diagrams processes as boxes and systems as wires
5 Ch. 1 Processes as diagrams processes as boxes and systems as wires
6 Ch. 1 Processes as diagrams processes as boxes and systems as wires
7 Ch. 1 Processes as diagrams processes as boxes and systems as wires
8 Ch. 1 Processes as diagrams processes as boxes and systems as wires
9 Ch. 1 Processes as diagrams processes as boxes and systems as wires
10 Ch. 1 Processes as diagrams composing processes
11 Ch. 1 Processes as diagrams composing processes
12 Ch. 1 Processes as diagrams composing processes
13 Ch. 1 Processes as diagrams composing processes
14 Ch. 1 Processes as diagrams composing processes
15 Ch. 1 Processes as diagrams composing processes
16 Ch. 1 Processes as diagrams process theories
17 Ch. 1 Processes as diagrams process theories... consist of: set of systems S set of processes P, with ins and outs in S,
18 Ch. 1 Processes as diagrams process theories... consist of: set of systems S set of processes P, with ins and outs in S, which are: closed under plugging.
19 Ch. 1 Processes as diagrams process theories... consist of: set of systems S set of processes P, with ins and outs in S, which are: closed under plugging. They tell us: how to interpret boxes and wires, and hence, when two diagrams are equal.
20 Ch. 1 Processes as diagrams process theories
21 Ch. 1 Processes as diagrams process theories
22 Ch. 1 Processes as diagrams process theories
23 Ch. 1 Processes as diagrams process theories
24 Ch. 1 Processes as diagrams diagrams symbolically
25 Ch. 1 Processes as diagrams diagrams symbolically
26 Ch. 1 Processes as diagrams diagrams symbolically
27 Ch. 1 Processes as diagrams diagrams symbolically Thm. Diagrams these symbolic expressions.
28 Ch. 1 Processes as diagrams composing diagrams
29 Ch. 1 Processes as diagrams composing diagrams Two operations: f g := f while g f g := f after g
30 Ch. 1 Processes as diagrams composing diagrams Two operations: f g := f while g f g := f after g
31 Ch. 1 Processes as diagrams Two operations: These are: associative composing diagrams f g := f while g f g := f after g have as respective units: empty -diagram wire -diagram
32 Ch. 1 Processes as diagrams circuits
33 Ch. 1 Processes as diagrams circuits Defn.... := can be build with and.
34 Ch. 1 Processes as diagrams circuits Defn.... := can be build with and. Thm. Circuit no box above itself.
35 Ch. 1 Processes as diagrams circuits Defn.... := can be build with and. Thm. Circuit no box above itself. Corr. Circuit admits causal interpretation. with
36 Ch. 1 Processes as diagrams circuits Defn.... := can be build with and. Thm. Circuit no box above itself. Corr. Circuit admits causal interpretation. Not circuit:
37 Ch. 1 Processes as diagrams why diagrams?
38 Ch. 1 Processes as diagrams why diagrams? Since by definition circuits can be build by means of symbolic connectives, why bother with diagrams?
39 Ch. 1 Processes as diagrams why diagrams? Since by definition circuits can be build by means of symbolic connectives, why bother with diagrams? Since equations come for free!
40 Ch. 1 Processes as diagrams why diagrams? Since by definition circuits can be build by means of symbolic connectives, why bother with diagrams? Since equations come for free! (f g) h = = f (g h)
41 Ch. 1 Processes as diagrams why diagrams? Since by definition circuits can be build by means of symbolic connectives, why bother with diagrams? Since equations come for free! (f g) h = = f (g h) f 1 I = = f
42 Ch. 1 Processes as diagrams why diagrams? Since by definition circuits can be build by means of symbolic connectives, why bother with diagrams? Since equations come for free!?
43 Ch. 1 Processes as diagrams why diagrams? Since by definition circuits can be build by means of symbolic connectives, why bother with diagrams? Since all equations come for free!
44 Ch. 1 Processes as diagrams why diagrams? Since by definition circuits can be build by means of symbolic connectives, why bother with diagrams? Since all equations come for free!
45 Ch. 1 Processes as diagrams why diagrams? Since by definition circuits can be build by means of symbolic connectives, why bother with diagrams? Since all equations come for free!
46 Ch. 1 Processes as diagrams why diagrams? Since by definition circuits can be build by means of symbolic connectives, why bother with diagrams? Since all equations come for free!
47 Ch. 1 Processes as diagrams why diagrams? Since by definition circuits can be build by means of symbolic connectives, why bother with diagrams? Since all equations come for free!
48 Ch. 1 Processes as diagrams why diagrams? Since by definition circuits can be build by means of symbolic connectives, why bother with diagrams? Since all equations come for free!
49 Ch. 1 Processes as diagrams special processes/diagrams
50 State := Ch. 1 Processes as diagrams special processes/diagrams
51 Ch. 1 Processes as diagrams special processes/diagrams State := Effect / Test :=
52 Ch. 1 Processes as diagrams special processes/diagrams State := Effect / Test := Number :=
53 Born rule := Ch. 1 Processes as diagrams special processes/diagrams
54 Ch. 1 Processes as diagrams Dirac notation := special processes/diagrams
55 Ch. 1 Processes as diagrams special processes/diagrams Separable disconnected :=
56 Ch. 1 Processes as diagrams special processes/diagrams Separable disconnected := E.g.:
57 Ch. 1 Processes as diagrams special processes/diagrams Non-separable := way more interesting!
58 Ch. 2 String diagrams When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. Erwin Schrödinger, Here we: introduce a wilder kind of diagram define quantum notions in great generality derive quantum phenomena in great generality
59 Ch. 2 String diagrams process-state duality
60 Ch. 2 String diagrams process-state duality Exists state and effect :
61 Ch. 2 String diagrams process-state duality Exists state and effect : such that:
62 proof of duality: Ch. 2 String diagrams process-state duality
63 proof of duality: Ch. 2 String diagrams process-state duality
64 Change notation: Ch. 2 String diagrams process-state duality
65 Change notation: Ch. 2 String diagrams process-state duality so that now:
66 Ch. 2 String diagrams definition
67 Ch. 2 String diagrams definition Thm. TFAE: circuits with process-state duality and:
68 Ch. 2 String diagrams definition Thm. TFAE: circuits with process-state duality and: diagrams with in-in and out-out connection:
69 Ch. 2 String diagrams definition
70 Ch. 2 String diagrams transpose
71 Ch. 2 String diagrams transpose... :=
72 Ch. 2 String diagrams transpose... :=
73 Ch. 2 String diagrams transpose Prop. The transpose is an involution:
74 Ch. 2 String diagrams transpose Prop. Transpose of cup is cap :
75 Clever new notation: Ch. 2 String diagrams transpose
76 Clever new notation: Ch. 2 String diagrams transpose just what happens when yanking hard!
77 Prop. Sliding: Ch. 2 String diagrams transpose
78 Prop. Sliding: Ch. 2 String diagrams transpose Pf.
79 Prop. Sliding: Ch. 2 String diagrams transpose... so this is a mathematical equation:
80 Ch. 2 String diagrams trace
81 Ch. 2 String diagrams trace... :=
82 Partial... := Ch. 2 String diagrams trace
83 Prop. Cyclicity: Ch. 2 String diagrams trace
84 Prop. Cyclicity: Ch. 2 String diagrams trace Redundant but fun ferris wheel proof:
85 Ch. 2 String diagrams quantum -like features
86 Ch. 2 String diagrams quantum -like features Thm. All states separable rubbish theory.
87 Ch. 2 String diagrams quantum -like features Thm. All states separable rubbish theory. Lem. All states separable wires separable. Pf.
88 Perfect correlations: Ch. 2 String diagrams quantum -like features
89 Perfect correlations: Ch. 2 String diagrams quantum -like features
90 Logical reading: Ch. 2 String diagrams quantum -like features
91 Operational reading: Ch. 2 String diagrams quantum -like features
92 Ch. 2 String diagrams quantum -like features Realising time-reversal (and make NY times):
93 Ch. 2 String diagrams quantum -like features Thm. No-cloning from assumptions: ψ, π :
94 Ch. 2 String diagrams quantum -like features Pf.
95 Ch. 2 String diagrams quantum -like features Pf.
96 Ch. 2 String diagrams quantum -like features Pf.
97 Ch. 2 String diagrams quantum -like features Pf.
98 Ch. 2 String diagrams quantum -like features Pf.
99 Ch. 2 String diagrams quantum -like features
100 Ch. 2 String diagrams quantum -like features
101 Ch. 2 String diagrams adjoint & conjugate
102 Ch. 2 String diagrams adjoint & conjugate A ket sometimes wants to be bra :
103 Ch. 2 String diagrams adjoint & conjugate Conjugate := Adjoint :=
104 Unitarity/isometry := Ch. 2 String diagrams adjoint & conjugate
105 Teleportation: Ch. 2 String diagrams adjoint & conjugate
106 Ch. 2 String diagrams Entanglement swapping: adjoint & conjugate
107 Ch. 2 String diagrams designing teleportation
108 Ch. 2 String diagrams designing teleportation
109 Ch. 2 String diagrams designing teleportation
110 Ch. 3 Hilbert space from diagrams I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more. John von Neumann, letter to Garrett Birkhoff, Here we introduce: ONBs, matrices and sums (multi-)linear maps & Hilbert space and relate: string diagrams (multi-)linear maps & Hilbert space
111 Ch. 3 Hilbert space from diagrams ONB
112 A set: Ch. 3 Hilbert space from diagrams ONB is pre-basis if:
113 Ch. 3 Hilbert space from diagrams Orthonormal := ONB
114 Ch. 3 Hilbert space from diagrams Orthonormal := ONB Canonical :=
115 Ch. 3 Hilbert space from diagrams matrix calculus
116 Ch. 3 Hilbert space from diagrams Thm. We have: matrix calculus so there is a matrix: with
117 Ch. 3 Hilbert space from diagrams matrix calculus But one also may want to glue things together:
118 Ch. 3 Hilbert space from diagrams matrix calculus Sums := for {f i } i of the same type there exists: which moves around :
119 Ch. 3 Hilbert space from diagrams matrix calculus In: the intuition is:
120 Ch. 3 Hilbert space from diagrams matrix calculus In: the intuition is: but better (see later):
121 Ch. 3 Hilbert space from diagrams definition
122 Ch. 3 Hilbert space from diagrams definition Defn. Linear maps := String diagrams s.t.: each system has ONB sums numbers are C
123 Ch. 3 Hilbert space from diagrams definition Defn. Linear maps := String diagrams s.t.: each system has ONB sums numbers are C
124 Ch. 3 Hilbert space from diagrams definition Defn. Linear maps := String diagrams s.t.: each system has ONB sums numbers are C
125 Ch. 3 Hilbert space from diagrams definition Defn. Linear maps := String diagrams s.t.: each system has ONB sums numbers are C
126 Ch. 3 Hilbert space from diagrams definition Defn. Linear maps := String diagrams s.t.: each system has ONB sums numbers are C Hilbert space := states for a system with Born-rule.
127 Ch. 3 Hilbert space from diagrams model-theoretic completeness
128 Ch. 3 Hilbert space from diagrams model-theoretic completeness THM. (Selinger, 2008) An equation between string diagrams holds, if and only if it holds for Hilbert spaces and linear maps.
129 Ch. 3 Hilbert space from diagrams model-theoretic completeness THM. (Selinger, 2008) An equation between string diagrams holds, if and only if it holds for Hilbert spaces and linear maps. I.e. defining Hilbert spaces and linear maps in this manner is a conservative extension of string diagrams.
130 Ch. 4 Quantum processes The art of progress is to preserve order amid change, and to preserve change amid order. Alfred North Whitehead, Process and Reality, Here we introduce in terms of diagrams: pure quantum maps mixed/open quantum maps causality & Stinespring dilation general quantum processes done badly
131 Ch. 4 Quantum processes doubling
132 Ch. 4 Quantum processes doubling Goal 1:
133 Ch. 4 Quantum processes doubling Goal 1: Goal 2:
134 Ch. 4 Quantum processes Pure quantum state := doubling
135 Ch. 4 Quantum processes Pure quantum effect := doubling
136 Ch. 4 Quantum processes doubling genuine probabilities
137 Ch. 4 Quantum processes Pure quantum map := doubling
138 Thm. We have: Ch. 4 Quantum processes doubling if and only if there exist λ λ = µ µ:
139 Pf. Setting: Ch. 4 Quantum processes doubling then:
140 Pf. Setting: Ch. 4 Quantum processes doubling then:
141 Ch. 4 Quantum processes open systems
142 Discarding := Ch. 4 Quantum processes open systems
143 Discarding := Ch. 4 Quantum processes open systems Thm. Discarding is not a pure quantum map.
144 Discarding := Ch. 4 Quantum processes open systems Thm. Discarding is not a pure quantum map. Pf. Something connected something disconnected.
145 Ch. 4 Quantum processes open systems Quantum maps := pure quantum maps + discarding
146 Ch. 4 Quantum processes open systems Quantum maps := pure quantum maps + discarding E.g. maximally mixed state :=
147 Ch. 4 Quantum processes open systems Quantum maps := pure quantum maps + discarding Prop. All quantum maps are of the form:
148 Ch. 4 Quantum processes open systems Quantum maps := pure quantum maps + discarding Prop. All quantum maps are of the form:
149 Ch. 4 Quantum processes open systems Quantum maps := pure quantum maps + discarding Prop. All quantum maps are of the form:
150 Ch. 4 Quantum processes causality
151 Ch. 4 Quantum processes... of quantum maps: causality
152 Ch. 4 Quantum processes causality Prop. For pure quantum maps: causality isometry
153 Ch. 4 Quantum processes causality Prop. For pure quantum maps: causality isometry Pf.
154 Ch. 4 Quantum processes causality Prop. For general quantum maps: causality of the form
155 Ch. 4 Quantum processes causality Prop. For general quantum maps: causality of the form Pf.
156 Ch. 4 Quantum processes causality Prop. For general quantum maps: causality of the form Pf. Cor. Stinespring dilation.
157 Ch. 4 Quantum processes non-deterministic quantum processes
158 Ch. 4 Quantum processes non-deterministic quantum processes... := such that E.g. quantum measurements.
159 Ch. 5 Quantum measurement The bureaucratic mentality is the only constant in the universe. Dr. McCoy, Star Trek IV: The Voyage Home, Here we briefly address: Next-best-thing to observing Measurement-induced dynamics Measurement-only quantum computing
160 Ch. 5 Quantum measurement is quantum measurement weird?
161 Ch. 5 Quantum measurement is quantum measurement weird? Thm. Observing is not a quantum process
162 Ch. 5 Quantum measurement is quantum measurement weird? Thm. Observing is not a quantum process i.e. : with = { 1 iff ψ = φ 0 iff ψ φ
163 Ch. 5 Quantum measurement is quantum measurement weird? Thm. Observing is not a quantum process i.e. : with = { 1 iff ψ = φ 0 iff ψ φ Prop. Condition can only hold for orthogonal states.
164 Ch. 5 Quantum measurement is quantum measurement weird? Thm. Observing is not a quantum process i.e. : with = { 1 iff ψ = φ 0 iff ψ φ Prop. Condition can only hold for orthogonal states. measurement is next-best-thing to observing
165 Ch. 5 Quantum measurement is quantum measurement weird? Bohr-Heisenberg: any attempt to observe is bound to disturb
166 Ch. 5 Quantum measurement is quantum measurement weird? Bohr-Heisenberg: any attempt to observe is bound to disturb Newtonian equivalent: locating a baloon by mechanical means
167 Ch. 5 Quantum measurement is quantum measurement weird? Heisenberg-Bohr: any attempt to observe is bound to disturb Newtonian equivalent: locating a baloon by mechanical means Resulting diagnosis: we suffer from quantum-blindness
168 Ch. 5 Quantum measurement is quantum measurement weird? BUT, the stuff that people call quantum measurement turns out to be extremely useful nonetheless!
169 Ch. 5 Quantum measurement what people call measurement ONB-measurement :=
170 Ch. 5 Quantum measurement what people call measurement ONB-measurement := E.g. for {β i } i Pauli-matrices:
171 Ch. 5 Quantum measurement what people call measurement Thm. All quantum maps arise from ONB-measurements.
172 Ch. 5 Quantum measurement what people call measurement Thm. All quantum maps arise from ONB-measurements. Pf. There are enough ONB s such that:
173 Ch. 5 Quantum measurement measurement-induced dynamics
174 Ch. 5 Quantum measurement measurement-induced dynamics
175 Ch. 5 Quantum measurement measurement-induced dynamics
176 Ch. 5 Quantum measurement measurement-induced dynamics
177 Ch. 5 Quantum measurement measurement-only quantum computing
178 Ch. 5 Quantum measurement measurement-only quantum computing
179 Ch. 5 Quantum measurement measurement-only quantum computing
180 Ch. 5 Quantum measurement measurement-only quantum computing
181 Ch. 5 Quantum measurement measurement-only quantum computing
182 Ch. 6 Picturing classical processes Damn it! I knew she was a monster! John! Amy! Listen! Guard your buttholes. David Wong, This Book Is Full of Spiders, Here we fully diagrammatically describe: all quantum processes special ones protocols and introduce the humongously important notion of: spiders
183 Ch. 6 Picturing classical processes classical vs. quantum wires
184 Ch. 6 Picturing classical processes They should meet: classical vs. quantum wires quantum wires classical wires
185 Ch. 6 Picturing classical processes They should meet: classical vs. quantum wires quantum wires but retain their distance: classical wires quantum wires classical wires
186 Ch. 6 Picturing classical processes They should meet: classical vs. quantum wires quantum wires but retain their distance: classical wires quantum wires classical wires which can be realised via un-doubling : classical wire quantum wire = normal (i.e. 1) boldface (i.e. 2)
187 Ch. 6 Picturing classical processes encoding classical data
188 Ch. 6 Picturing classical processes Classical data ONB: encoding classical data
189 Ch. 6 Picturing classical processes Classical data ONB: encoding classical data := providing classical value i
190 Ch. 6 Picturing classical processes Classical data ONB: encoding classical data := providing classical value i := testing for classical value i
191 Ch. 6 Picturing classical processes Classical data ONB: encoding classical data := providing classical value i := testing for classical value i Sanity check:
192 Ch. 6 Picturing classical processes encoding classical data Non-deterministic quantum process:
193 Ch. 6 Picturing classical processes encoding classical data Non-deterministic quantum process: Process controlled by outcome:
194 Ch. 6 Picturing classical processes encoding classical data
195 Ch. 6 Picturing classical processes encoding classical data
196 Ch. 6 Picturing classical processes encoding classical data
197 Ch. 6 Picturing classical processes encoding classical data
198 Ch. 6 Picturing classical processes Prop. Braces sums classical data in diagrams
199 Ch. 6 Picturing classical processes classical data in diagrams Prop. Braces sums Pf.
200 Ch. 6 Picturing classical processes encoding classical data Non-deterministic quantum process:
201 Ch. 6 Picturing classical processes encoding classical data Non-deterministic quantum process: Process controlled by outcome:
202 Ch. 6 Picturing classical processes encoding classical data
203 Ch. 6 Picturing classical processes encoding classical data
204 Ch. 6 Picturing classical processes classical-quantum maps
205 ... := Ch. 6 Picturing classical processes classical-quantum maps
206 Ch. 6 Picturing classical processes Classical map := classical-quantum maps
207 Ch. 6 Picturing classical processes Classical map examples: classical-quantum maps copy delete match compare
208 Ch. 6 Picturing classical processes classical-quantum maps The name explains the action:
209 Ch. 6 Picturing classical processes classical-quantum maps The name explains the action:
210 Ch. 6 Picturing classical processes classical-quantum maps Classical-quantum map examples: encode measure
211 Ch. 6 Picturing classical processes classical-quantum maps Thm.... are always of the form:
212 Ch. 6 Picturing classical processes classical-quantum maps Thm.... are always of the form: where f is a quantum map.
213 Ch. 6 Picturing classical processes classical-quantum maps Thm.... are always of the form:
214 Ch. 6 Picturing classical processes classical-quantum maps Thm.... are always of the form:
215 Ch. 6 Picturing classical processes classical maps
216 Ch. 6 Picturing classical processes classical maps Thm.... are always of the form:
217 Ch. 6 Picturing classical processes classical-quantum processes
218 Ch. 6 Picturing classical processes Thm. Causality: classical-quantum processes
219 Lem. Ch. 6 Picturing classical processes classical-quantum processes
220 Ch. 6 Picturing classical processes Thm. Causality: classical-quantum processes
221 Ch. 6 Picturing classical processes Thm. Causality: classical-quantum processes
222 ... := Ch. 6 Picturing classical processes classical-quantum processes s.t.:
223 Ch. 6 Picturing classical processes teleportation diagrammatically
224 Ch. 6 Picturing classical processes teleportation diagrammatically Thm. Controlled isometry:
225 Ch. 6 Picturing classical processes teleportation diagrammatically Thm. Controlled isometry:
226 Ch. 6 Picturing classical processes teleportation diagrammatically Pf.
227 Ch. 6 Picturing classical processes teleportation diagrammatically Pf.
228 Ch. 6 Picturing classical processes teleportation diagrammatically
229 Ch. 6 Picturing classical processes teleportation diagrammatically
230 Ch. 6 Picturing classical processes teleportation diagrammatically
231 Ch. 6 Picturing classical processes dense coding
232 Ch. 6 Picturing classical processes dense coding
233 Ch. 6 Picturing classical processes dense coding
234 Ch. 6 Picturing classical processes dense coding
235 Ch. 6 Picturing classical processes Naimark dilation
236 Ch. 6 Picturing classical processes Naimark dilation
237 Ch. 6 Picturing classical processes Naimark dilation
238 Ch. 6 Picturing classical processes Naimark dilation
239 Ch. 6 Picturing classical processes spiders
240 ... := Ch. 6 Picturing classical processes spiders
241 Ch. 6 Picturing classical processes spiders Cf. copy delete match compare
242 Ch. 6 Picturing classical processes spiders
243 Ch. 6 Picturing classical processes spiders
244 Ch. 6 Picturing classical processes spiders
245 Ch. 6 Picturing classical processes spiders
246 Ch. 6 Picturing classical processes Prop. Spiders obey: spiders
247 Ch. 6 Picturing classical processes Prop. Spiders obey: spiders
248 Ch. 6 Picturing classical processes For example: spiders
249 Ch. 6 Picturing classical processes... and in particular: spiders
250 Ch. 6 Picturing classical processes Thm. Spiders ONBs spiders
251 Ch. 6 Picturing classical processes spiders Thm. Spiders ONBs Pf. Consider copy spider: so claim follows by only-orthogonals-are-clonable.
252 Ch. 6 Picturing classical processes spiders THM. (CPV) All families of linear maps: which behave like spiders, are spiders.
253 Ch. 6 Picturing classical processes Classical spider := spiders
254 Ch. 6 Picturing classical processes Quantum spider := spiders
255 Ch. 6 Picturing classical processes Bastard spider := spiders
256 Ch. 6 Picturing classical processes Bastard spider := spiders =:
257 References (for Ch. 1 till Ch. 6) Penrose, R. (1971) Applications of negative dimensional tensors. Pages of: Combinatorial Mathematics and its Applications. Academic Press. Coecke, B. (2003) The Logic of Entanglement. Tech. Rept. RR Department of Computer Science, Oxford University. arxiv:quant-ph/ Abramsky, S., and Coecke, B. (2004) A categorical semantics of quantum protocols. LICS. arxiv:quant-ph/ Coecke, B. (2005) Kindergarten quantum mechanics lecture notes. In: Quantum Theory: Reconsiderations of the Foundations III. AIP Press. arxiv:quantph/ Coecke, B. (2007) De-linearizing linearity: Projective Quantum Axiomatics From Strong Compact Closure. ENTCS. arxiv:quant-ph/ Selinger, P. (2007) Dagger compact closed categories and completely positive maps. ENTCS. Coecke, B., and Pavlovic, D. (2007) Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology. Taylor and Francis. arxiv:quant-ph/
258 Coecke, B., Pavlovic, D., and Vicary, J. (2011) A new description of orthogonal bases. MSCS. arxiv:quant-ph/ Abramsky, S. (2010) No-cloning in categorical quantum mechanics. In: Semantic Techniques in Quantum Computation. CUP. arxiv: Coecke, B., Paquette, E. O., and Pavlovic, D. (2010) Classical and quantum structuralism. In: Semantic Techniques in Quantum Computation. CUP. arxiv: Selinger, P. (2011) Finite dimensional Hilbert spaces are complete for dagger compact closed categories. ENTCS. Coecke, B., and Perdrix, S. (2010) Environment and classical channels in categorical quantum mechanics. CSL. arxiv: Chiribella, G., D Ariano, G.M., and Perinotti, P. (2010) Probabilistic theories with purification. Physical Review. arxiv: Hardy, L. (2013) On the theory of composition in physics. In: Computation, Logic, Games, and Quantum Foundations. Springer. arxiv:
Physics, Language, Maths & Music
Physics, Language, Maths & Music (partly in arxiv:1204.3458) Bob Coecke, Oxford, CS-Quantum SyFest, Vienna, July 2013 ALICE f f = f f = f f = ALICE BOB BOB meaning vectors of words does not Alice not like
Selling Category Theory to the Masses
Selling Category Theory to the Masses Bob Coecke Quantum Group - Computer Science - Oxford University ALICE f f = f f = f f = ALICE BOB BOB does not Alice not like Bob = Alice like not Bob Samson Abramsky
In the beginning God created tensor... as a picture
In the beginning God created tensor... as a picture Bob Coecke coecke@comlab.ox.ac.uk EPSRC Advanced Research Fellow Oxford University Computing Laboratory se10.comlab.ox.ac.uk:8080/bobcoecke/home en.html
Baby's First Diagrammatic Calculus for Quantum Information Processing
Baby's First Diagrammatic Calculus for Quantum Information Processing Vladimir Zamdzhiev Department of Computer Science Tulane University 30 May 2018 1 / 38 Quantum computing ˆ Quantum computing is usually
Causal categories: a backbone for a quantumrelativistic universe of interacting processes
Causal categories: a backbone for a quantumrelativistic universe of interacting processes ob Coecke and Ray Lal Oxford University Computing Laboratory bstract We encode causal space-time structure within
The Logic of Quantum Mechanics - take II Bob Coecke Oxford Univ. Computing Lab. Quantum Group
The Logic of Quantum Mechanics - take II Bob Coecke Oxford Univ. Computing Lab. Quantum Group ALICE f f BOB = f f = f f = ALICE BOB does not Alice not like BC (2010) Quantum picturalism. Contemporary physics
Qubits vs. bits: a naive account A bit: admits two values 0 and 1, admits arbitrary transformations. is freely readable,
Qubits vs. bits: a naive account A bit: admits two values 0 and 1, admits arbitrary transformations. is freely readable, A qubit: a sphere of values, which is spanned in projective sense by two quantum
Categorical quantum mechanics
Categorical quantum mechanics Chris Heunen 1 / 76 Categorical Quantum Mechanics? Study of compositional nature of (physical) systems Primitive notion: forming compound systems 2 / 76 Categorical Quantum
Universal Properties in Quantum Theory
Universal Properties in Quantum Theory Mathieu Huot ENS Paris-Saclay Sam Staton University of Oxford We argue that notions in quantum theory should have universal properties in the sense of category theory.
Hilbert Space, Entanglement, Quantum Gates, Bell States, Superdense Coding.
CS 94- Bell States Bell Inequalities 9//04 Fall 004 Lecture Hilbert Space Entanglement Quantum Gates Bell States Superdense Coding 1 One qubit: Recall that the state of a single qubit can be written as
Categorical Models for Quantum Computing
Categorical odels for Quantum Computing Linde Wester Worcester College University of Oxford thesis submitted for the degree of Sc in athematics and the Foundations of Computer Science September 2013 cknowledgements
Quantum information processing: a new light on the Q-formalism and Q-foundations
Bob Coecke - Oxford University Computing Laboratory Quantum information processing: a new light on the Q-formalism and Q-foundations ASL 2010 North American Annual Meeting The George Washington University
1. Basic rules of quantum mechanics
1. Basic rules of quantum mechanics How to describe the states of an ideally controlled system? How to describe changes in an ideally controlled system? How to describe measurements on an ideally controlled
Design and Analysis of Quantum Protocols and Algorithms
Design and Analysis of Quantum Protocols and Algorithms Krzysztof Bar University College MMathCompSci May 20, 2012 Acknowledgements I would like to thank my supervisor prof. Bob Coecke for all the help
Infinite-dimensional Categorical Quantum Mechanics
Infinite-dimensional Categorical Quantum Mechanics A talk for CLAP Scotland Stefano Gogioso and Fabrizio Genovese Quantum Group, University of Oxford 5 Apr 2017 University of Strathclyde, Glasgow S Gogioso,
Quantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras
Quantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 6 Postulates of Quantum Mechanics II (Refer Slide Time: 00:07) In my last lecture,
arxiv: v2 [quant-ph] 26 Mar 2012
![arxiv: v2 [quant-ph] 26 Mar 2012](/thumbs/84/91205204.jpg "arxiv: v2 [quant-ph] 26 Mar 2012")
Optimal Probabilistic Simulation of Quantum Channels from the Future to the Past Dina Genkina, Giulio Chiribella, and Lucien Hardy Perimeter Institute for Theoretical Physics, 31 Caroline Street North,
Lecture 11 September 30, 2015
PHYS 7895: Quantum Information Theory Fall 015 Lecture 11 September 30, 015 Prof. Mark M. Wilde Scribe: Mark M. Wilde This document is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike
Quantum Quandaries: A Category Theoretic Perspective
Quantum Quandaries: A Category Theoretic Perspective John C. Baez Les Treilles April 24, 2007 figures by Aaron Lauda for more, see http://math.ucr.edu/home/baez/quantum/ The Big Idea Once upon a time,
Lecture: Quantum Information
Lecture: Quantum Information Transcribed by: Crystal Noel and Da An (Chi Chi) November 10, 016 1 Final Proect Information Find an issue related to class you are interested in and either: read some papers
Quantum Gates, Circuits & Teleportation
Chapter 3 Quantum Gates, Circuits & Teleportation Unitary Operators The third postulate of quantum physics states that the evolution of a quantum system is necessarily unitary. Geometrically, a unitary
Quantum Computing: Foundations to Frontier Fall Lecture 3
Quantum Computing: Foundations to Frontier Fall 018 Lecturer: Henry Yuen Lecture 3 Scribes: Seyed Sajjad Nezhadi, Angad Kalra Nora Hahn, David Wandler 1 Overview In Lecture 3, we started off talking about
2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
Introduction to Quantum Computing for Folks
Introduction to Quantum Computing for Folks Joint Advanced Student School 2009 Ing. Javier Enciso encisomo@in.tum.de Technische Universität München April 2, 2009 Table of Contents 1 Introduction 2 Quantum
Quantum Error Correcting Codes and Quantum Cryptography. Peter Shor M.I.T. Cambridge, MA 02139
Quantum Error Correcting Codes and Quantum Cryptography Peter Shor M.I.T. Cambridge, MA 02139 1 We start out with two processes which are fundamentally quantum: superdense coding and teleportation. Superdense
Categorical quantum channels
Attacking the quantum version of with category theory Ian T. Durham Department of Physics Saint Anselm College 19 March 2010 Acknowledgements Some of this work has been the result of some preliminary collaboration
Diagrammatic Methods for the Specification and Verification of Quantum Algorithms
Diagrammatic Methods or the Speciication and Veriication o Quantum lgorithms William Zeng Quantum Group Department o Computer Science University o Oxord Quantum Programming and Circuits Workshop IQC, University
Private quantum subsystems and error correction
Private quantum subsystems and error correction Sarah Plosker Department of Mathematics and Computer Science Brandon University September 26, 2014 Outline 1 Classical Versus Quantum Setting Classical Setting
A Graph Theoretic Perspective on CPM(Rel)
A Graph Theoretic Perspective on CPM(Rel) Daniel Marsden Mixed states are of interest in quantum mechanics for modelling partial information. More recently categorical approaches to linguistics have also
Finite Dimensional Hilbert Spaces are Complete for Dagger Compact Closed Categories (Extended Abstract)
Electronic Notes in Theoretical Computer Science 270 (1) (2011) 113 119 www.elsevier.com/locate/entcs Finite Dimensional Hilbert Spaces are Complete or Dagger Compact Closed Categories (Extended bstract)
Introduction to Quantum Information Processing QIC 710 / CS 768 / PH 767 / CO 681 / AM 871
Introduction to Quantum Information Processing QIC 710 / CS 768 / PH 767 / CO 681 / AM 871 Lecture 1 (2017) Jon Yard QNC 3126 jyard@uwaterloo.ca TAs Nitica Sakharwade nsakharwade@perimeterinstitute.ca
An Introduction to Quantum Information. By Aditya Jain. Under the Guidance of Dr. Guruprasad Kar PAMU, ISI Kolkata
An Introduction to Quantum Information By Aditya Jain Under the Guidance of Dr. Guruprasad Kar PAMU, ISI Kolkata 1. Introduction Quantum information is physical information that is held in the state of
Chapter 5. Density matrix formalism
Chapter 5 Density matrix formalism In chap we formulated quantum mechanics for isolated systems. In practice systems interect with their environnement and we need a description that takes this feature
Computational Algebraic Topology Topic B Lecture III: Quantum Realizability
Computational Algebraic Topology Topic B Lecture III: Quantum Realizability Samson Abramsky Department of Computer Science The University of Oxford Samson Abramsky (Department of Computer ScienceThe Computational
226 My God, He Plays Dice! Entanglement. Chapter This chapter on the web informationphilosopher.com/problems/entanglement
226 My God, He Plays Dice! Entanglement Chapter 29 20 This chapter on the web informationphilosopher.com/problems/entanglement Entanglement 227 Entanglement Entanglement is a mysterious quantum phenomenon
COMPOSITIONAL THERMODYNAMICS
COMPOSITIONL THERMODYNMICS Giulio Chiribella Department of Computer Science, The University of Hong Kong, CIFR-zrieli Global Scholars Program Carlo Maria Scandolo, Department of Computer Science, The University
Manipulating Causal Order of Unitary Operations using Adiabatic Quantum Computation
Manipulating Causal Order of Unitary Operations using Adiabatic Quantum Computation Kosuke Nakago, Quantum Information group, The University of Tokyo, Tokyo,Japan. Joint work with, Mio murao, Michal Hajdusek,
Towards a quantum calculus
QPL 2006 Preliminary Version Towards a quantum calculus (work in progress, extended abstract) Philippe Jorrand 1 Simon Perdrix 2 Leibniz Laboratory IMAG-INPG Grenoble, France Abstract The aim of this paper
On Paradoxes in Proof-Theoretic Semantics
Quantum Group Dept. of Computer Science Oxford University 2nd PTS Conference, Tübingen, March 8, 2013 Outline 1 Categorical Harmony Logical Constants as Adjoint Functors Comparison with Other Concepts
Quantum information-flow, concretely, abstractly
Proc. QPL 2004, pp. 57 73 Quantum information-flow, concretely, abstractly Bob Coecke Oxford University Computing Laboratory Abstract These lecture notes are based on joint work with Samson Abramsky. I
Basics on quantum information
Basics on quantum information Mika Hirvensalo Department of Mathematics and Statistics University of Turku mikhirve@utu.fi Thessaloniki, May 2016 Mika Hirvensalo Basics on quantum information 1 of 52 Brief
QLang: Qubit Language
QLang: Qubit Language Christopher Campbell Clément Canonne Sankalpa Khadka Winnie Narang Jonathan Wong September 24, 24 Introduction In 965, Gordon Moore predicted that the number of transistors in integrated
Categories and Quantum Informatics: Hilbert spaces
Categories and Quantum Informatics: Hilbert spaces Chris Heunen Spring 2018 We introduce our main example category Hilb by recalling in some detail the mathematical formalism that underlies quantum theory:
A review on quantum teleportation based on: Teleporting an unknown quantum state via dual classical and Einstein- Podolsky-Rosen channels
JOURNAL OF CHEMISTRY 57 VOLUME NUMBER DECEMBER 8 005 A review on quantum teleportation based on: Teleporting an unknown quantum state via dual classical and Einstein- Podolsky-Rosen channels Miri Shlomi
1 Measurements, Tensor Products, and Entanglement
Stanford University CS59Q: Quantum Computing Handout Luca Trevisan September 7, 0 Lecture In which we describe the quantum analogs of product distributions, independence, and conditional probability, and
A short and personal introduction to the formalism of Quantum Mechanics
A short and personal introduction to the formalism of Quantum Mechanics Roy Freeman version: August 17, 2009 1 QM Intro 2 1 Quantum Mechanics The word quantum is Latin for how great or how much. In quantum
Quantum theory without predefined causal structure
Quantum theory without predefined causal structure Ognyan Oreshkov Centre for Quantum Information and Communication, niversité Libre de Bruxelles Based on work with Caslav Brukner, Nicolas Cerf, Fabio
Seminar 1. Introduction to Quantum Computing
Seminar 1 Introduction to Quantum Computing Before going in I am also a beginner in this field If you are interested, you can search more using: Quantum Computing since Democritus (Scott Aaronson) Quantum
Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics Indian Institute of Technology, Delhi
Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics Indian Institute of Technology, Delhi Module No. # 07 Bra-Ket Algebra and Linear Harmonic Oscillator - II Lecture No. # 01 Dirac s Bra and
Lecture 1: Overview of quantum information
CPSC 59/69: Quantum Computation John Watrous, University of Calgary References Lecture : Overview of quantum information January 0, 006 Most of the material in these lecture notes is discussed in greater
Lecture 4: Postulates of quantum mechanics
Lecture 4: Postulates of quantum mechanics Rajat Mittal IIT Kanpur The postulates of quantum mechanics provide us the mathematical formalism over which the physical theory is developed. For people studying
Stochastic Histories. Chapter Introduction
Chapter 8 Stochastic Histories 8.1 Introduction Despite the fact that classical mechanics employs deterministic dynamical laws, random dynamical processes often arise in classical physics, as well as in
Mix Unitary Categories
1/31 Mix Unitary Categories Robin Cockett, Cole Comfort, and Priyaa Srinivasan CT2018, Ponta Delgada, Azores Dagger compact closed categories Dagger compact closed categories ( -KCC) provide a categorical
Chapter 10. Quantum algorithms
Chapter 10. Quantum algorithms Complex numbers: a quick review Definition: C = { a + b i : a, b R } where i = 1. Polar form of z = a + b i is z = re iθ, where r = z = a 2 + b 2 and θ = tan 1 y x Alternatively,
Quantum Entanglement- Fundamental Aspects
Quantum Entanglement- Fundamental Aspects Debasis Sarkar Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata- 700009, India Abstract Entanglement is one of the most useful
Basics on quantum information
Basics on quantum information Mika Hirvensalo Department of Mathematics and Statistics University of Turku mikhirve@utu.fi Thessaloniki, May 2014 Mika Hirvensalo Basics on quantum information 1 of 49 Brief
Introduction to Quantum Computing
Introduction to Quantum Computing Part I Emma Strubell http://cs.umaine.edu/~ema/quantum_tutorial.pdf April 12, 2011 Overview Outline What is quantum computing? Background Caveats Fundamental differences
Connecting the categorical and the modal logic approaches to Quantum Mech
Connecting the categorical and the modal logic approaches to Quantum Mechanics based on MSc thesis supervised by A. Baltag Institute for Logic, Language and Computation University of Amsterdam 30.11.2013
Copy-Cat Strategies and Information Flow in Physics, Geometry, Logic and Computation
Copy-Cat Strategies and Information Flow in Physics, Geometry, and Computation Samson Abramsky Oxford University Computing Laboratory Copy-Cat Strategies and Information Flow CQL Workshop August 2007 1
Informational derivation of quantum theory
Selected for a Viewpoint in Physics PHYSICAL REVIEW A 84, 01311 (011) Informational derivation of quantum theory Giulio Chiribella Perimeter Institute for Theoretical Physics, 31 Caroline Street North,
Quantum Cryptography. Areas for Discussion. Quantum Cryptography. Photons. Photons. Photons. MSc Distributed Systems and Security
Areas for Discussion Joseph Spring Department of Computer Science MSc Distributed Systems and Security Introduction Photons Quantum Key Distribution Protocols BB84 A 4 state QKD Protocol B9 A state QKD
5. Communication resources
5. Communication resources Classical channel Quantum channel Entanglement How does the state evolve under LOCC? Properties of maximally entangled states Bell basis Quantum dense coding Quantum teleportation
Complex numbers: a quick review. Chapter 10. Quantum algorithms. Definition: where i = 1. Polar form of z = a + b i is z = re iθ, where
Chapter 0 Quantum algorithms Complex numbers: a quick review / 4 / 4 Definition: C = { a + b i : a, b R } where i = Polar form of z = a + b i is z = re iθ, where r = z = a + b and θ = tan y x Alternatively,
Short Course in Quantum Information Lecture 2
Short Course in Quantum Information Lecture Formal Structure of Quantum Mechanics Course Info All materials downloadable @ website http://info.phys.unm.edu/~deutschgroup/deutschclasses.html Syllabus Lecture
Entanglement Manipulation
Entanglement Manipulation Steven T. Flammia 1 1 Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5 Canada (Dated: 22 March 2010) These are notes for my RIT tutorial lecture at the
Lecture 3: Superdense coding, quantum circuits, and partial measurements
CPSC 59/69: Quantum Computation John Watrous, University of Calgary Lecture 3: Superdense coding, quantum circuits, and partial measurements Superdense Coding January 4, 006 Imagine a situation where two
Ensembles and incomplete information
p. 1/32 Ensembles and incomplete information So far in this course, we have described quantum systems by states that are normalized vectors in a complex Hilbert space. This works so long as (a) the system
Quantum decoherence. Éric Oliver Paquette (U. Montréal) -Traces Worshop [Ottawa]- April 29 th, Quantum decoherence p. 1/2
![Quantum decoherence. Éric Oliver Paquette (U. Montréal) -Traces Worshop [Ottawa]- April 29 th, Quantum decoherence p. 1/2](/thumbs/78/77811482.jpg "Quantum decoherence. Éric Oliver Paquette (U. Montréal) -Traces Worshop [Ottawa]- April 29 th, Quantum decoherence p. 1/2")
Quantum decoherence p. 1/2 Quantum decoherence Éric Oliver Paquette (U. Montréal) -Traces Worshop [Ottawa]- April 29 th, 2007 Quantum decoherence p. 2/2 Outline Quantum decoherence: 1. Basics of quantum
Lecture 3: Hilbert spaces, tensor products
CS903: Quantum computation and Information theory (Special Topics In TCS) Lecture 3: Hilbert spaces, tensor products This lecture will formalize many of the notions introduced informally in the second
II. The Machinery of Quantum Mechanics
II. The Machinery of Quantum Mechanics Based on the results of the experiments described in the previous section, we recognize that real experiments do not behave quite as we expect. This section presents
CHAPTER 0: A (VERY) BRIEF TOUR OF QUANTUM MECHANICS, COMPUTATION, AND CATEGORY THEORY.
CHPTER 0: (VERY) BRIEF TOUR OF QUNTUM MECHNICS, COMPUTTION, ND CTEGORY THEORY. This chapter is intended to be a brief treatment of the basic mechanics, framework, and concepts relevant to the study of
Practice Problems for Test II
Math 117 Practice Problems for Test II 1. Let f() = 1/( + 1) 2, and let g() = 1 + 4 3. (a) Calculate (b) Calculate f ( h) f ( ) h g ( z k) g( z) k. Simplify your answer as much as possible. Simplify your
University of Bristol - Explore Bristol Research. Publisher's PDF, also known as Version of record
Backens, M., Perdrix, S., & Wang, Q. (017). A simplified stabilizer ZXcalculus. In EPTCS: Proceedings 13th International Conference on Quantum Physics and Logic (Vol. 36, pp. 1-0). (Electronic Proceedings
Introduction to Quantum Cryptography
Università degli Studi di Perugia September, 12th, 2011 BunnyTN 2011, Trento, Italy This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Quantum Mechanics
Categories and Quantum Informatics
Categories and Quantum Informatics Week 6: Frobenius structures Chris Heunen 1 / 41 Overview Frobenius structure: interacting co/monoid, self-duality Normal forms: coherence theorem Frobenius law: coherence
Endomorphism Semialgebras in Categorical Quantum Mechanics
Endomorphism Semialgebras in Categorical Quantum Mechanics Kevin Dunne University of Strathclyde November 2016 Symmetric Monoidal Categories Definition A strict symmetric monoidal category (A,, I ) consists
Bell s inequalities and their uses
The Quantum Theory of Information and Computation http://www.comlab.ox.ac.uk/activities/quantum/course/ Bell s inequalities and their uses Mark Williamson mark.williamson@wofson.ox.ac.uk 10.06.10 Aims
Linear Algebra and Dirac Notation, Pt. 1
Linear Algebra and Dirac Notation, Pt. 1 PHYS 500 - Southern Illinois University February 1, 2017 PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 1 / 13
Types in categorical linguistics (& elswhere)
Types in categorical linguistics (& elswhere) Peter Hines Oxford Oct. 2010 University of York N. V. M. S. Research Topic of the talk: This talk will be about: Pure Category Theory.... although it might
Structure of Unital Maps and the Asymptotic Quantum Birkhoff Conjecture. Peter Shor MIT Cambridge, MA Joint work with Anand Oza and Dimiter Ostrev
Structure of Unital Maps and the Asymptotic Quantum Birkhoff Conjecture Peter Shor MIT Cambridge, MA Joint work with Anand Oza and Dimiter Ostrev The Superposition Principle (Physicists): If a quantum
What is a quantum computer? Quantum Architecture. Quantum Mechanics. Quantum Superposition. Quantum Entanglement. What is a Quantum Computer (contd.
What is a quantum computer? Quantum Architecture by Murat Birben A quantum computer is a device designed to take advantage of distincly quantum phenomena in carrying out a computational task. A quantum
The Elements for Logic In Compositional Distributional Models of Meaning
The Elements for Logic In Compositional Distributional Models of Meaning Joshua Steves: 1005061 St Peter s College University of Oxford A thesis submitted for the degree of MSc Mathematics and Foundations
DECAY OF SINGLET CONVERSION PROBABILITY IN ONE DIMENSIONAL QUANTUM NETWORKS
DECAY OF SINGLET CONVERSION PROBABILITY IN ONE DIMENSIONAL QUANTUM NETWORKS SCOTT HOTTOVY Abstract. Quantum networks are used to transmit and process information by using the phenomena of quantum mechanics.
Richard Cleve David R. Cheriton School of Computer Science Institute for Quantum Computing University of Waterloo
CS 497 Frontiers of Computer Science Introduction to Quantum Computing Lecture of http://www.cs.uwaterloo.ca/~cleve/cs497-f7 Richard Cleve David R. Cheriton School of Computer Science Institute for Quantum
Clifford Algebras and Spin Groups
Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of
Stochastic Processes
qmc082.tex. Version of 30 September 2010. Lecture Notes on Quantum Mechanics No. 8 R. B. Griffiths References: Stochastic Processes CQT = R. B. Griffiths, Consistent Quantum Theory (Cambridge, 2002) DeGroot
The dagger lambda calculus
The dagger lambda calculus Philip Atzemoglou University of Oxford Quantum Physics and Logic 2014 Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 1 / 20 Why higher-order? Teleportation
Compression and entanglement, entanglement transformations
PHYSICS 491: Symmetry and Quantum Information April 27, 2017 Compression and entanglement, entanglement transformations Lecture 8 Michael Walter, Stanford University These lecture notes are not proof-read
Quantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras
Quantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 4 Postulates of Quantum Mechanics I In today s lecture I will essentially be talking
PLEASE LET ME KNOW IF YOU FIND TYPOS (send to
Teoretisk Fysik KTH Advanced QM (SI2380), Lecture 2 (Summary of concepts) 1 PLEASE LET ME KNOW IF YOU FIND TYPOS (send email to langmann@kth.se) The laws of QM 1. I now discuss the laws of QM and their
1 Mathematical preliminaries
1 Mathematical preliminaries The mathematical language of quantum mechanics is that of vector spaces and linear algebra. In this preliminary section, we will collect the various definitions and mathematical
A history of entanglement
A history of entanglement Jos Uffink Philosophy Department, University of Minnesota, jbuffink@umn.edu May 17, 2013 Basic mathematics for entanglement of pure states Let a compound system consists of two
Rewriting measurement-based quantum computations with generalised flow
Rewriting measurement-based quantum computations with generalised flow Ross Duncan 1 and Simon Perdrix 2 1 Oxford University Computing Laboratory ross.duncan@comlab.ox.ac.uk 2 CNRS, LIG, Université de
Closing the Debates on Quantum Locality and Reality: EPR Theorem, Bell's Theorem, and Quantum Information from the Brown-Twiss Vantage
Closing the Debates on Quantum Locality and Reality: EPR Theorem, Bell's Theorem, and Quantum Information from the Brown-Twiss Vantage C. S. Unnikrishnan Fundamental Interactions Laboratory Tata Institute
From Quantum Cellular Automata to Quantum Field Theory
From Quantum Cellular Automata to Quantum Field Theory Alessandro Bisio Frontiers of Fundamental Physics Marseille, July 15-18th 2014 in collaboration with Giacomo Mauro D Ariano Paolo Perinotti Alessandro
CSCI 2570 Introduction to Nanocomputing. Discrete Quantum Computation
CSCI 2570 Introduction to Nanocomputing Discrete Quantum Computation John E Savage November 27, 2007 Lect 22 Quantum Computing c John E Savage What is Quantum Computation It is very different kind of computation
Lecture 20: Bell inequalities and nonlocality
CPSC 59/69: Quantum Computation John Watrous, University of Calgary Lecture 0: Bell inequalities and nonlocality April 4, 006 So far in the course we have considered uses for quantum information in the
1 Fundamental physical postulates. C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12
C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12 In this and the next lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to
Ph 219/CS 219. Exercises Due: Friday 3 November 2006
Ph 9/CS 9 Exercises Due: Friday 3 November 006. Fidelity We saw in Exercise. that the trace norm ρ ρ tr provides a useful measure of the distinguishability of the states ρ and ρ. Another useful measure
Chapter 3 Vectors 3-1
Chapter 3 Vectors Chapter 3 Vectors... 2 3.1 Vector Analysis... 2 3.1.1 Introduction to Vectors... 2 3.1.2 Properties of Vectors... 2 3.2 Cartesian Coordinate System... 6 3.2.1 Cartesian Coordinates...