Outline

Decision Making: . . .

The Notion of Utility

Subjective Probabilities Quantum Example of Seemingly . . . Relation to Quantum . . .

From Individual to . . .

(A review of a recent book by Alternative (Non- . . .

E. Haven and A. Khrennikov) Why Humans Use . . . Home Page

Presented by Vladik Kreinovich Title Page

Department of Computer Science JJ II University of Texas at El Paso El Paso, TX 79968, USA J I [email protected] Page1 of 29

http://www.cs.utep.edu/vladik Go Back

Full Screen

Close

Quit Outline

Decision Making: . . . 1. Outline The Notion of Utility • Decision theory – how to describe preferences if people Subjective Probabilities behave rationally. Example of Seemingly . . .

Relation to Quantum . . . • In practice, people sometimes deviate from rational be- havior. From Individual to . . . Alternative (Non- . . .

• In some cases, quantum-type models are helpful in ex- Why Humans Use . . .

plaining actual human behavior. Home Page

• The authors use this to explain fluctuations of the stock Title Page market prices. JJ II • They also speculate on why humans use quantum-stype J I reasoning. Page2 of 29

Go Back

Full Screen

Close

Quit Outline

Decision Making: . . . 2. Decision Making: General Need and Traditional Approach The Notion of Utility Subjective Probabilities

• To make a decision, we must: Example of Seemingly . . .

Relation to Quantum . . . – find out the user’s preference, and From Individual to . . . – help the user select an alternative which is the best Alternative (Non- . . . – according to these preferences. Why Humans Use . . . • Traditional approach is based on an assumption that Home Page 0 00 for each two alternatives A and A , a user can tell: Title Page

– whether the first alternative is better for him/her; JJ II we will denote this by A00 < A0; J I – or the second alternative is better; we will denote this by A0 < A00; Page3 of 29 – or the two given alternatives are of equal value to Go Back 0 00 the user; we will denote this by A = A . Full Screen

Close

Quit Outline

Decision Making: . . . 3. The Notion of Utility The Notion of Utility • Under the above assumption, we can form a natural Subjective Probabilities numerical scale for describing preferences. Example of Seemingly . . .

Relation to Quantum . . . • Let us select a very bad alternative A0 and a very good From Individual to . . . alternative A1. Alternative (Non- . . .

• Then, most other alternatives are better than A0 but Why Humans Use . . .

worse than A1. Home Page

• For every prob. p ∈ [0, 1], we can form a lottery L(p) Title Page in which we get A1 w/prob. p and A0 w/prob. 1 − p. JJ II • When p = 0, this lottery simply coincides with the J I alternative A0: L(0) = A0. Page4 of 29 • The larger the probability p of the positive outcome Go Back increases, the better the result: Full Screen p0 < p00 implies L(p0) < L(p00). Close

Quit Outline

Decision Making: . . . 4. The Notion of Utility (cont-d) The Notion of Utility • Finally, for p = 1, the lottery coincides with the alter- Subjective Probabilities native A1: L(1) = A1. Example of Seemingly . . . Relation to Quantum . . . • Thus, we have a continuous scale of alternatives L(p) From Individual to . . . that monotonically goes from L(0) = A0 to L(1) = A1. Alternative (Non- . . .

• Due to monotonicity, when p increases, we first have Why Humans Use . . .

L(p) < A, then we have L(p) > A. Home Page

• The threshold value is called the utility of the alterna- Title Page tive A: JJ II u(A) def= sup{p : L(p) < A} = inf{p : L(p) > A}. J I

• Then, for every ε > 0, we have Page5 of 29

L(u(A) − ε) < A < L(u(A) + ε). Go Back

• We will describe such (almost) equivalence by ≡, i.e., Full Screen

we will write that A ≡ L(u(A)). Close

Quit Outline

Decision Making: . . . 5. Fast Iterative Process for Determining u(A) The Notion of Utility • Initially: we know the values u = 0 and u = 1 such Subjective Probabilities that A ≡ L(u(A)) for some u(A) ∈ [u, u]. Example of Seemingly . . . Relation to Quantum . . . • What we do: we compute the midpoint umid of the From Individual to . . . interval [u, u] and compare A with L(umid). Alternative (Non- . . . • Possibilities: A ≤ L(u ) and L(u ) ≤ A. mid mid Why Humans Use . . .

• Case 1: if A ≤ L(umid), then u(A) ≤ umid, so Home Page

Title Page u ∈ [u, umid].

JJ II • Case 2: if L(umid) ≤ A, then umid ≤ u(A), so J I u ∈ [umid, u]. Page6 of 29 • After each iteration, we decrease the width of the in- terval [u, u] by half. Go Back • After k iterations, we get an interval of width 2−k which Full Screen −k contains u(A) – i.e., we get u(A) w/accuracy 2 . Close

Quit Outline

Decision Making: . . . 6. How to Make a Decision Based on Utility Val- ues The Notion of Utility Subjective Probabilities

• Suppose that we have found the utilities u(A0), u(A00), Example of Seemingly . . . 0 00 . . . , of the alternatives A , A ,... Relation to Quantum . . . • Which of these alternatives should we choose? From Individual to . . . Alternative (Non- . . .

• By definition of utility, we have: Why Humans Use . . . • A ≡ L(u(A)) for every alternative A, and Home Page • L(p0) < L(p00) if and only if p0 < p00. Title Page • We can thus conclude that A0 is preferable to A00 if and JJ II 0 00 only if u(A ) > u(A ). J I

• In other words, we should always select an alternative Page7 of 29

with the largest possible value of utility. Go Back

Full Screen

Close

Quit Outline

Decision Making: . . . 7. How to Estimate Utility of an Action The Notion of Utility • For each action, we usually know possible outcomes Subjective Probabilities S1,...,Sn. Example of Seemingly . . . Relation to Quantum . . . • We can often estimate the prob. p1, . . . , pn of these out- comes. From Individual to . . . Alternative (Non- . . . • By definition of utility, each situation S is equiv. to a i Why Humans Use . . .

lottery L(u(Si)) in which we get: Home Page

• A1 with probability u(Si) and Title Page

• A0 with the remaining probability 1 − u(Si). JJ II • Thus, the action is equivalent to a complex lottery in J I which: Page8 of 29 • first, we select one of the situations Si with proba- Go Back bility pi: P (Si) = pi; Full Screen • then, depending on Si, we get A1 with probability P (A1 | Si) = u(Si) and A0 w/probability 1 − u(Si). Close

Quit Outline

Decision Making: . . . 8. How to Estimate Utility of an Action (cont-d) The Notion of Utility • Reminder: Subjective Probabilities Example of Seemingly . . . • first, we select one of the situations Si with proba- Relation to Quantum . . . bility pi: P (Si) = pi; From Individual to . . . • then, depending on Si, we get A1 with probability Alternative (Non- . . . P (A1 | Si) = u(Si) and A0 w/probability 1 − u(Si). Why Humans Use . . .

• The prob. of getting A1 in this complex lottery is: Home Page n n X X Title Page P (A1) = P (A1 | Si) · P (Si) = u(Si) · pi. i=1 i=1 JJ II

• In the complex lottery, we get: J I n P • A1 with prob. u = pi · u(Si), and Page9 of 29 i=1 Go Back • A0 w/prob. 1 − u. • So, we should select the action with the largest value Full Screen P of expected utility u = pi · u(Si). Close

Quit Outline

Decision Making: . . . 9. Non-Uniqueness of Utility The Notion of Utility

Subjective Probabilities • The above definition of utility u depends on A0, A1. 0 0 Example of Seemingly . . . • What if we use different alternatives A0 and A1? Relation to Quantum . . .

• Every A is equivalent to a lottery L(u(A)) in which we From Individual to . . . get A w/prob. u(A) and A w/prob. 1 − u(A). 1 0 Alternative (Non- . . . 0 0 • For simplicity, let us assume that A0 < A0 < A1 < A1. Why Humans Use . . . 0 0 0 0 Home Page • Then, A0 ≡ L (u (A0)) and A1 ≡ L (u (A1)). • So, A is equivalent to a complex lottery in which: Title Page JJ II 1) we select A1 w/prob. u(A) and A0 w/prob. 1−u(A); 0 0 0 2) depending on Ai, we get A1 w/prob. u (Ai) and A0 J I 0 w/prob. 1 − u (Ai). Page 10 of 29 0 • In this complex lottery, we get A1 with probability Go Back 0 0 0 0 u (A) = u(A) · (u (A1) − u (A0)) + u (A0). Full Screen • So, in general, utility is defined modulo an (increasing) Close linear transformation u0 = a · u + b, with a > 0. Quit Outline

Decision Making: . . . 10. Subjective Probabilities The Notion of Utility

Subjective Probabilities • In practice, we often do not know the probabilities pi of different outcomes. Example of Seemingly . . . • For each event E, a natural way to estimate its subjec- Relation to Quantum . . . tive probability is to fix a prize (e.g., $1) and compare: From Individual to . . . Alternative (Non- . . .

– the lottery `E in which we get the fixed prize if the Why Humans Use . . .

event E occurs and 0 is it does not occur, with Home Page – a lottery `(p) in which we get the same amount Title Page with probability p. JJ II • Here, similarly to the utility case, we get a value ps(E) for which, for every ε > 0: J I

Page 11 of 29 `(ps(E) − ε) < `E < `(ps(E) + ε). Go Back • Then, the utility of an action with possible outcomes n Full Screen P S1,...,Sn is equal to u = ps(Ei) · u(Si). i=1 Close

Quit Outline

Decision Making: . . . 11. Example of Seemingly Irrational Behavior The Notion of Utility • An urn contains red (R), green (G), and blue (B) balls. Subjective Probabilities Example of Seemingly . . . • We know that exactly 1/3 of balls are red: p(R) = 1/3. Relation to Quantum . . .

• We do not know p(G). From Individual to . . . • 1st experiment: choose between two alternatives: Alternative (Non- . . . Why Humans Use . . .

R: $1 if a randomly chosen ball is red, $0 otherwise; Home Page G: $1 if a random ball is green, $0 otherwise. Title Page • Result: most people select alternative R: G < R. JJ II

• In terms of utility: we can take u($0) = 0, u($1) = 1. J I

• In this case, u(R) = (1/3) · u($1) + (2/3) · u($0) = 1/3 Page 12 of 29 and u(G) = ps(G)·u($1)+(1−ps(G))·u($0) = ps(G). Go Back

• Since G < R, this means that ps(G) < 1/3. Full Screen

Close

Quit Outline

Decision Making: . . . 12. Seemingly Irrational Behavior (cont-d) The Notion of Utility • Second experiment: a person is asked to choose be- Subjective Probabilities tween two alternatives: Example of Seemingly . . . GB: you get $1 if a randomly picked ball is either green Relation to Quantum . . . or blue and $0 otherwise; From Individual to . . . RB: you get $1 if a randomly picked ball is either red or Alternative (Non- . . . blue and $0 otherwise. Why Humans Use . . . Home Page • Result: most people select alternative GB: RB < GB. Title Page • In terms of utility: JJ II u(GB) = (2/3) · u($1) + (1/3) · u($0) = 2/3; J I u(RB) = (1−ps(G))·u($1)+ps(G)·u($0) = 1−ps(G). Page 13 of 29 • Since RB < GB, this means that for most people, 1 − ps(G) < 2/3 and thus, ps(G) > 1/3. Go Back • Contradiction: this conclusion is inconsistent with the Full Screen previous conclusion ps(G) < 1/3. Close

Quit Outline

Decision Making: . . . 13. Relation to Quantum Physics The Notion of Utility • Here, ps(G) < 1/3, ps(B) < 1/3, but ps(G∨B) = 2/3. Subjective Probabilities Example of Seemingly . . . • In other words, here, ps(G ∨ B) 6= ps(G) + ps(B). Relation to Quantum . . .

• Such situations are typical for quantum processes. From Individual to . . . • Example: a photon source is separated from the sensors Alternative (Non- . . . by a wall with two doors. Why Humans Use . . . Home Page • If we open one door or both doors, some photons pass to the sensors. Title Page • We can open the first door and count how many photos JJ II passed to the sensors. J I

• Let p1 be the resulting probability. Page 14 of 29 • We can open the second door and count how many Go Back

photos passed to the sensors. Full Screen

• Let p2 be the resulting probability. Close

Quit Outline

Decision Making: . . . 14. Relation to Quantum Physics (cont-d) The Notion of Utility • If both doors are open, then a photon can pass: Subjective Probabilities Example of Seemingly . . . – either through the 1st door Relation to Quantum . . .

– or through the 2nd door From Individual to . . .

– but not through both doors. Alternative (Non- . . .

Why Humans Use . . . • So, we expect p12 = p1 + p2. Home Page • In practice, p12 6= p1 + p2. Title Page

• Quantum description: we have complex amplitudes Ψ1 2 2 JJ II and Ψ2 such that p1 = |Ψ1| and p2 = |Ψ2| . 2 J I • Here, p12 = |Ψ12| , where Ψ12 = Ψ1 + Ψ2. Page 15 of 29 2 2 2 • In general, |Ψ1 + Ψ2| 6= |Ψ1| + |Ψ| , so p12 6= p1 + p2. Go Back • Interesting: kids going to candy boxes behave similarly Full Screen to photons. Close

Quit Outline

Decision Making: . . . 15. From Individual to Collective Behavior The Notion of Utility • Psychologists have performed many experiments show- Subjective Probabilities ing such irrational behavior. Example of Seemingly . . . • Several models have been proposed explaining such be- Relation to Quantum . . . havior. From Individual to . . . Alternative (Non- . . . • These models, however, are far from perfect when de- Why Humans Use . . .

scribing individual decision making. Home Page

• At first glance, the problem of describing such irra- Title Page tional behavior, JJ II – while important in , J I – is not of large practical interest. Page 16 of 29 • The authors notice, however, that: Go Back – a combination of such irrational behaviors Full Screen – leads, e.g., to seemingly irrational fluctuations of the stock market. Close

Quit Outline

Decision Making: . . . 16. Collective Behavior (cont-d) The Notion of Utility • In contrast to highly irregular individual behavior, such Subjective Probabilities group behavior is much more regular. Example of Seemingly . . . Relation to Quantum . . . • It can be reasonably well described by known mathe- From Individual to . . . matical models. Alternative (Non- . . .

• For example, we can use quantum-type stochastic dif- Why Humans Use . . .

ferential equations. Home Page

• In the first approximation, the resulting equations are Title Page similar to Schroedinger’s equations of quantum physics. JJ II • A detailed analysis shows that the stock market dy- J I namics is somewhat different from . Page 17 of 29

Go Back

Full Screen

Close

Quit Outline

Decision Making: . . . 17. Alternative (Non-Quantum) Explanation The Notion of Utility • Previously, we assumed that a user can always decide Subjective Probabilities which of the two alternatives A0 and A00 is better: Example of Seemingly . . . Relation to Quantum . . . – either A0 < A00, From Individual to . . . – or A00 < A0, Alternative (Non- . . . 0 00 – or A ≡ A . Why Humans Use . . . • In practice, a user is sometimes unable to meaningfully Home Page 0 00 decide between the two alternatives; denoted A k A . Title Page

• In mathematical terms, this means that the preference JJ II relation: J I

– is no longer a total (linear) order, Page 18 of 29

– it can be a partial order. Go Back

Full Screen

Close

Quit Outline

Decision Making: . . . 18. From Utility to Interval-Valued Utility The Notion of Utility • Similarly to the traditional decision making approach: Subjective Probabilities Example of Seemingly . . . – we select two alternatives A0 < A1 and Relation to Quantum . . . – we compare each alternative A which is better than From Individual to . . . A and worse than A with lotteries L(p). 0 1 Alternative (Non- . . .

• Since preference is a partial order, in general: Why Humans Use . . . Home Page u(A) def= sup{p : L(p) < A} < u(A) def= inf{p : L(p) > A}. Title Page

• For each alternative A, instead of a single value u(A) JJ II of the utility, we now have an interval [u(A), u(A)] s.t.: J I – if p < u(A), then L(p) < A; Page 19 of 29 – if p > u(A), then A < L(p); and Go Back – if u(A) < p < u(A), then A k L(p). Full Screen • We will call this interval the utility of the alternative A. Close

Quit Outline

Decision Making: . . . 19. Interval-Valued Utilities and Interval-Valued Subjective Probabilities The Notion of Utility Subjective Probabilities

• To feasibly elicit the values u(A) and u(A), we: Example of Seemingly . . . 1) starting w/[u, u] = [0, 1], bisect an interval s.t. Relation to Quantum . . . From Individual to . . . L(u) < A < L(u) until we find u0 s.t. A k L(u0); Alternative (Non- . . . 2) by bisecting an interval [u, u0] for which Why Humans Use . . . L(u) < A k L(u0), we find u(A); Home Page 3) by bisecting an interval [u0, u] for which Title Page L(u0) k A < L(u), we find u(A). • Similarly, when we estimate the probability of an event E: JJ II – we no longer get a single value ps(E); J I – we get an interval ps(E), ps(E) of possible values Page 20 of 29 of probability. Go Back

• By using bisection, we can feasibly elicit the values Full Screen

ps(E) and ps(E). Close

Quit Outline

Decision Making: . . . 20. Decision Making Under Interval Uncertainty The Notion of Utility • Situation: for each possible decision d, we know the Subjective Probabilities interval [u(d), u(d)] of possible values of utility. Example of Seemingly . . . • Questions: which decision shall we select? Relation to Quantum . . . From Individual to . . . • Natural idea: select all decisions d that may be opti- 0 Alternative (Non- . . . mal, i.e., which are optimal for some function Why Humans Use . . . u(d) ∈ [u(d), u(d)]. Home Page • Problem: checking all possible functions is not feasible. Title Page • Solution: the above condition is equivalent to an easier- JJ II

to-check one: J I u(d0) ≥ max u(d). d Page 21 of 29

• Interval computations can help in describing the range Go Back of all such d0. Full Screen • Remaining problem: in practice, we would like to select Close one decision; which one should be select? Quit Outline

Decision Making: . . . 21. Decisions under Interval Uncertainty: Hur- wicz Optimism-Pessimism Criterion The Notion of Utility Subjective Probabilities

• Reminder: we need to assign, to each interval [u, u], a Example of Seemingly . . . utility value u(u, u) ∈ [u, u]. Relation to Quantum . . . • : this problem was first handled in 1951, by the From Individual to . . . future Nobelist Leonid Hurwicz. Alternative (Non- . . . def Why Humans Use . . . • Notation: let us denote αH = u(0, 1). Home Page

• Reminder: utility is determined modulo a linear trans- Title Page formation u0 = a · u + b. JJ II • Reasonable to require: the equivalent utility does not change with re-scaling: for a > 0 and b, J I Page 22 of 29 u(a · u− + b, a · u+ + b) = a · u(u−, u+) + b. Go Back • For u− = 0, u+ = 1, a = u − u, and b = u, we get Full Screen

u(u, u) = αH · (u − u) + u = αH · u + (1 − αH ) · u. Close

Quit Outline

Decision Making: . . . 22. Hurwicz Optimism-Pessimism Criterion (cont) The Notion of Utility

Subjective Probabilities • The expression αH · u + (1 − αH ) · u is called optimism- pessimism criterion, because: Example of Seemingly . . . Relation to Quantum . . . – when αH = 1, we make a decision based on the From Individual to . . . most optimistic possible values u = u; Alternative (Non- . . . – when αH = 0, we make a decision based on the Why Humans Use . . .

most pessimistic possible values u = u; Home Page

– for intermediate values αH ∈ (0, 1), we take a weighted Title Page average of the optimistic and pessimistic values. JJ II • According to this criterion: J I – if we have several alternatives A0, . . . , with interval- valued utilities [u(A0), u(A0)], . . . , Page 23 of 29 – we recommend an alternative A that maximizes Go Back

Full Screen αH · u(A) + (1 − αH ) · u(A). Close

Quit Outline

Decision Making: . . . 23. Alternative Explanation of the Seemingly Ir- rational Human Behavior The Notion of Utility Subjective Probabilities

• Situation: an urn contains red (R), green (G), and blue Example of Seemingly . . . (B) balls; p(R) = 1/3. Relation to Quantum . . . • We do not know: the proportion of green and blue From Individual to . . . balls, so we only know that p(G) ∈ [0, 2/3]. Alternative (Non- . . . Why Humans Use . . .

• 1st experiment: choose between two alternatives: Home Page

R: $1 if a random ball is red, $0 otherwise; Title Page G: $1 if a random ball is green, $0 otherwise. JJ II • In this case, u(R) = (1/3) · u($1) + (2/3) · u($0) = 1/3. J I

• Here, u(G) = p(G) · u($1) + (1 − p(G)) · u($0) = p(G). Page 24 of 29

• So, possible values of u(G) form an interval [0, 2/3]. Go Back

• This is equivalent to αH ·(2/3)+(1−αH )·0 = (2/3)·αH . Full Screen

• Since G < R, we have (2/3) · αH < 1/3, so αH < 1/2. Close

Quit Outline

Decision Making: . . . 24. Alternative Explanation (cont-d) The Notion of Utility • 2nd experiment: choose between two alternatives: Subjective Probabilities Example of Seemingly . . . GB: $1 if a random ball is green or blue, $0 otherwise; Relation to Quantum . . .

RB: $1 if a random ball is red or blue, $0 otherwise. From Individual to . . .

• Result: most people select alternative GB: RB < RB. Alternative (Non- . . . • Here, u(GB) = (2/3) · u($1) + (2/3) · u($0) = 2/3. Why Humans Use . . . Home Page

• In this case, p(RB) = 1 − p(G), so Title Page

u(RB) = (1 − p(G)) · u($1) + p(G) · u($0) = 1 − p(G). JJ II

• Since ps(G) ∈ [0, 2/3], we have u(RB) ∈ [1/3, 1], which J I is equiv. to αH · 1 + (1 − αH ) · (1/3) = (2/3) · αH + 1/3. Page 25 of 29

• RB < GB means (2/3)·αH +1/3 < 2/3, i.e., αH < 1/2. Go Back

Full Screen • This is the same restriction on αH that we obtained

from the 1st experiment. Close

Quit Outline

Decision Making: . . . 25. Why Humans Use Quantum-Type Reasoning? The Notion of Utility • The authors also speculate on why humans use quantum- Subjective Probabilities type reasoning. Example of Seemingly . . .

Relation to Quantum . . . • One possible reason: From Individual to . . .

– many real-life processes are quantum, and Alternative (Non- . . .

– we want to simulate them. Why Humans Use . . . Home Page • Another possible reason: Title Page – our brain, as a product of billion years of evolution, implements the best possible algorithms, and JJ II – quantum algorithms are known to be very efficient. J I

Page 26 of 29

Go Back

Full Screen

Close

Quit Outline

Decision Making: . . . 26. Why The Notion of Utility • The speed of all processes is limited by the speed of Subjective Probabilities light c. Example of Seemingly . . .

Relation to Quantum . . . • To send a signal across a 30 cm laptop, we need at least 1 ns; this corresponds to only 1 Gflop. From Individual to . . . Alternative (Non- . . .

• If we want to make computers faster, we need to make Why Humans Use . . .

processing elements smaller. Home Page

• Already, each processing cell consists of a few dozen Title Page molecules. JJ II • If we decrease the size further, we get to the level of J I individual atoms and molecules. Page 27 of 29 • On this level, physics is different, it is quantum physics. Go Back • One of the properties of quantum physics is its proba- Full Screen bilistic nature (example: radioactive decay). Close

Quit Outline

Decision Making: . . . 27. Why Quantum Computing (cont-d) The Notion of Utility • At first glance, this interferes with our desire to make Subjective Probabilities reproducible computations. Example of Seemingly . . . • However, scientists learned how to make lemonade out Relation to Quantum . . . of this lemon. From Individual to . . . Alternative (Non- . . . • First main discovery: Grover’s quantum search algo- Why Humans Use . . .

rithm. Home Page

• To search for an object in an unsorted array of n ele- Title Page ments, we need, in the worst case, at least n steps. JJ II • Reason: if we use fewer steps, we do not cover all the elements, and thus, we may miss the desired object. J I √ • In quantum physics, we can find an element in n Page 28 of 29 steps. Go Back

• For a Terabyte database, we get a million times speedup. Full Screen

• Main idea: we can use superposition of different searches. Close

Quit 28. Why Quantum Computing (cont-d) Outline Decision Making: . . . The Notion of Utility • Another discovery: Shor’s cracking RSA coding. Subjective Probabilities Example of Seemingly . . . • The RSA algorithm is behind most secure transactions. Relation to Quantum . . . • A person selects two large prime numbers P and P , From Individual to . . . 1 2 Alternative (Non- . . . and advertises their product n = P1 · P2. Why Humans Use . . .

• By using this open code n, anyone can encode their Home Page message. Title Page • To decode this message, one needs to know the factors JJ II P1 and P2. • Factoring a large integer is known to be a computa- J I tionally difficult problem. Page 29 of 29 • It turns out that with quantum computers, we can fac- Go Back tor fast and thus, read all encrypted messages. Full Screen

• The situation is not so bad: there is also a quantum Close

encryption which cannot be easily cracked. Quit