Undecidability as a genuine quantum property
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1 Undecidability as a genuine quantum property 1 / 20 Undecidability as a genuine quantum property Jens Eisert 1, Markus P. Müller 2, and Christian Gogolin 1 1 Dahlem Center for Complex Quantum Systems, Freie Universität Berlin 2 Perimeter Institute for Theoretical Physics, Waterloo, Canada UNAM Mexico
2 Undecidability as a genuine quantum property Motivation 2 / 20 What I hope to get across An operationally defined problem about measurements
3 Undecidability as a genuine quantum property Motivation 2 / 20 What I hope to get across An operationally defined problem about measurements Question: Are certain outcome sequences impossible?
4 Undecidability as a genuine quantum property Motivation 2 / 20 What I hope to get across An operationally defined problem about measurements Question: Are certain outcome sequences impossible? Two versions: Quantum Classical algorithmically undecidable algorithmically decidable
5 Undecidability as a genuine quantum property Motivation 2 / 20 What I hope to get across An operationally defined problem about measurements Question: Are certain outcome sequences impossible? Two versions: Quantum Classical What it means: algorithmically undecidable algorithmically decidable
6 Undecidability as a genuine quantum property Motivation 2 / 20 What I hope to get across An operationally defined problem about measurements Question: Are certain outcome sequences impossible? Two versions: Quantum Classical algorithmically undecidable algorithmically decidable What it means: Statement about the complexity of the mathematical theory of quantum mechanics
7 Undecidability as a genuine quantum property Motivation 2 / 20 What I hope to get across An operationally defined problem about measurements Question: Are certain outcome sequences impossible? Two versions: Quantum Classical algorithmically undecidable algorithmically decidable What it means: Statement about the complexity of the mathematical theory of quantum mechanics Arguably the strongest complexity theoretic quantum/classical separation imaginable (almost)
8 Undecidability as a genuine quantum property Outline 3 / 20 Outline 1 Undecidability Turing Machines and Computability Undecidability and the halting problem 2 Measurement occurrence problem Operational definition Quantum and classical version 3 Results Decidability of the classical problem Undecidability of the quantum problem
9 Undecidability as a genuine quantum property Undecidability 4 / 20 Undecidability
10 Undecidability as a genuine quantum property Undecidability Turing Machines and Computability 5 / 20 Turing machines Turing Machine t 0, 1 0, 2, 1, 2 0, h, 5 1, 5 0, 1,. input x
11 Undecidability as a genuine quantum property Undecidability Turing Machines and Computability 5 / 20 Turing machines Turing Machine t 0, 1 0, 2, 1, 2 0, h, 1 1, 5 0, 1,.
12 Undecidability as a genuine quantum property Undecidability Turing Machines and Computability 5 / 20 Turing machines Turing Machine t 0, 1 0, 2, 1, 2 0, h, 2 1, 5 0, 1,.
13 Undecidability as a genuine quantum property Undecidability Turing Machines and Computability 5 / 20 Turing machines Turing Machine t 0, 1 0, 2, 1, 2 0, h, h 1, 5 0, 1,. output y
14 Undecidability as a genuine quantum property Undecidability Turing Machines and Computability 5 / 20 Turing machines Turing Machine t 0, 1 0, 2, 1, 2 0, h, h 1, 5 0, 1,. Input x and output y are (bit) strings (countable). output y
15 Undecidability as a genuine quantum property Undecidability Turing Machines and Computability 5 / 20 Turing machines Turing Machine t 0, 1 0, 2, 1, 2 0, h, h 1, 5 0, 1,. Input x and output y are (bit) strings (countable). The set of Turing machines is countable (Turing Number t N). output y
16 Undecidability as a genuine quantum property Undecidability Turing Machines and Computability 5 / 20 Turing machines Turing Machine t 0, 1 0, 2, 1, 2 0, h, h 1, 5 0, 1,. Input x and output y are (bit) strings (countable). The set of Turing machines is countable (Turing Number t N). Turing Machine computes a function f : X Y iff it halts and y = f(x) for each x X. output y
17 Undecidability as a genuine quantum property Undecidability Turing Machines and Computability 5 / 20 Turing machines Turing Machine t 0, 1 0, 2, 1, 2 0, h, h 1, 5 0, 1,. output y Input x and output y are (bit) strings (countable). The set of Turing machines is countable (Turing Number t N). Turing Machine computes a function f : X Y iff it halts and y = f(x) for each x X. Universal model of computation
18 Undecidability as a genuine quantum property Undecidability Turing Machines and Computability 5 / 20 Turing machines Turing Machine t 0, 1 0, 2, 1, 2 0, h, h 1, 5 0, 1,. output y Input x and output y are (bit) strings (countable). The set of Turing machines is countable (Turing Number t N). Turing Machine computes a function f : X Y iff it halts and y = f(x) for each x X. Universal model of computation Question: Are all functions f : N {0, 1} computable?
19 Undecidability as a genuine quantum property Undecidability Turing Machines and Computability 5 / 20 Turing machines Turing Machine t 0, 1 0, 2, 1, 2 0, h, h 1, 5 0, 1,. output y Input x and output y are (bit) strings (countable). The set of Turing machines is countable (Turing Number t N). Turing Machine computes a function f : X Y iff it halts and y = f(x) for each x X. Universal model of computation Question: Are all functions f : N {0, 1} computable? Answer: No!
20 Undecidability as a genuine quantum property Undecidability Undecidability and the halting problem 6 / 20 The halting problem Halt(x) Halt(x) = { true false if TM x halts on input x if TM x does not halt on input x
21 Undecidability as a genuine quantum property Undecidability Undecidability and the halting problem 6 / 20 The halting problem Halt(x) Halt(x) = { true false if TM x halts on input x if TM x does not halt on input x Assume Halt(x) is computable, then there must be a Turing machine (with Turing number f) which behaves according to the following program: f(x) if Halt(x) then loop forever else halt endif
22 Undecidability as a genuine quantum property Undecidability Undecidability and the halting problem 6 / 20 The halting problem Halt(x) Halt(x) = { true false if TM x halts on input x if TM x does not halt on input x Assume Halt(x) is computable, then there must be a Turing machine (with Turing number f) which behaves according to the following program: f(x) if Halt(x) then loop forever else halt endif What is f( f)?
23 Undecidability as a genuine quantum property Undecidability Undecidability and the halting problem 7 / 20 Turing undecidability A problem is undecidable iff there is no algorithm solving each instance of the problem.
24 Undecidability as a genuine quantum property Measurement occurrence problem 8 / 20 Measurement occurrence problem
25 Undecidability as a genuine quantum property Measurement occurrence problem Operational definition 9 / 20 A problem from measurement theory..
26 Undecidability as a genuine quantum property Measurement occurrence problem Operational definition 9 / 20 A problem from measurement theory outcomes input K
27 Undecidability as a genuine quantum property Measurement occurrence problem Operational definition 9 / 20 A problem from measurement theory outcomes input K
28 Undecidability as a genuine quantum property Measurement occurrence problem Operational definition 9 / 20 A problem from measurement theory input K
29 Undecidability as a genuine quantum property Measurement occurrence problem Operational definition 9 / 20 A problem from measurement theory input K w 1 = 2
30 Undecidability as a genuine quantum property Measurement occurrence problem Operational definition 9 / 20 A problem from measurement theory.. w 1 = 2
31 Undecidability as a genuine quantum property Measurement occurrence problem Operational definition 9 / 20 A problem from measurement theory.. w 2 = 1
32 Undecidability as a genuine quantum property Measurement occurrence problem Operational definition 10 / 20 A problem from measurement theory... w 1 w 2 w 3...
33 Undecidability as a genuine quantum property Measurement occurrence problem Operational definition 10 / 20 A problem from measurement theory... w n never occurs w 1 w 2 w 3...
34 Undecidability as a genuine quantum property Measurement occurrence problem Operational definition 10 / 20 A problem from measurement theory Measurement occurrence problem (MOP) Given a description of a measurement device decide whether there exists a sequence of outcomes... w w 1,..., w n that can never occur, regardless of the input. n never occurs w 1 w 2 w 3...
35 Undecidability as a genuine quantum property Measurement occurrence problem Quantum and classical version 11 / 20 QMOP vs. CMOP QMOP state: ρ = ρ 0, Tr ρ = 1
36 Undecidability as a genuine quantum property Measurement occurrence problem Quantum and classical version 11 / 20 QMOP vs. CMOP QMOP state: ρ = ρ 0, Tr ρ = 1 device: {A j } K j=1 Q d d K A j A j = 1 j=1
37 Undecidability as a genuine quantum property Measurement occurrence problem Quantum and classical version 11 / 20 QMOP vs. CMOP QMOP state: ρ = ρ 0, Tr ρ = 1 device: {A j } K j=1 Q d d K A j A j = 1 j=1 on outcome j: ρ A jρa j Tr[A j ρa j ]
38 Undecidability as a genuine quantum property Measurement occurrence problem Quantum and classical version 11 / 20 QMOP vs. CMOP QMOP state: ρ = ρ 0, Tr ρ = 1 device: {A j } K j=1 Q d d K A j A j = 1 j=1 on outcome j: ρ A jρa j Tr[A j ρa j ] Prob(w 1,..., w n ) = Tr[A w 1... A w n A wn... A w1 ρ]
39 Undecidability as a genuine quantum property Measurement occurrence problem Quantum and classical version 11 / 20 QMOP vs. CMOP QMOP CMOP d state: ρ = ρ 0, Tr ρ = 1 p 0, p i = 1 device: {A j } K j=1 Q d d {Q j } K j=1 (Q + 0 ) d d i=1 K A j A j = 1 j=1 K Q j is stochastic j=1 on outcome j: ρ A jρa j Tr[A j ρa j ] Prob(w 1,..., w n ) = Tr[A w 1... A w n A wn... A w1 ρ] p Q j p d i=1 (Q j p) i d (Q wn... Q w1 p) i i=1
40 Undecidability as a genuine quantum property Measurement occurrence problem Quantum and classical version 12 / 20 When is w 1,... w n an impossible outcome sequence? QMOP: Prob(w 1,..., w n ) = Tr[A w 1... A w n A wn... A w1 ρ] = 0 ρ A w 1... A w n A wn... A w1 = 0 A wn... A w1 = 0
41 Undecidability as a genuine quantum property Measurement occurrence problem Quantum and classical version 12 / 20 When is w 1,... w n an impossible outcome sequence? QMOP: Prob(w 1,..., w n ) = Tr[A w 1... A w n A wn... A w1 ρ] = 0 ρ A w 1... A w n A wn... A w1 = 0 A wn... A w1 = 0 CMOP: Prob(w 1,..., w n ) = d (Q wn... Q w1 p) i = 0 p i=1 Q wn... Q w1 = 0
42 Undecidability as a genuine quantum property Measurement occurrence problem Quantum and classical version 12 / 20 When is w 1,... w n an impossible outcome sequence? QMOP: Prob(w 1,..., w n ) = Tr[A w 1... A w n A wn... A w1 ρ] = 0 ρ A w 1... A w n A wn... A w1 = 0 A wn... A w1 = 0 CMOP: Prob(w 1,..., w n ) = d (Q wn... Q w1 p) i = 0 p i=1 Q wn... Q w1 = 0 Remember: different restrictions on A j and Q j!
43 Undecidability as a genuine quantum property Measurement occurrence problem Quantum and classical version 13 / 20 QMOP vs. CMOP QMOP CMOP d state: ρ = ρ 0, Tr ρ = 1 p 0, p i = 1 device: {A j } K j=1 Q d d {Q j } K j=1 (Q + 0 ) d d i=1 K A j A j = 1 j=1 K Q j is stochastic j=1 on outcome j: ρ A jρa j Tr A j ρa j Prob(w 1,..., w n ) = Tr[A w 1... A w n A wn... A w1 ρ] p Q j p d i=1 (Q j p) i d (Q wn... Q w1 p) i i=1
44 Undecidability as a genuine quantum property Results 14 / 20 Results
45 Undecidability as a genuine quantum property Results 15 / 20 Quantum vs. classical separation Theorem 1 (Undecidability) The quantum measurement occurrence problem (QMOP) for K 9 and d 15 is undecidable. Theorem 2 (Decidability) For any K and d, both QMOP with Kraus operators A j with non-negative entries and CMOP are decidable. [1] J. Eisert, M. P. Mueller, and C. Gogolin, Phys. Rev. Lett. 108, (2012)
46 Undecidability as a genuine quantum property Results Decidability of the classical problem 16 / 20 Proof idea for decidability Proof of Theorem 2:
47 Undecidability as a genuine quantum property Results Decidability of the classical problem 16 / 20 Proof idea for decidability Proof of Theorem 2: For any entry wise non-negative matrix M define its indicator matrix M a,b := { 0 if M a,b = 0 1 if M a,b > 0.
48 Undecidability as a genuine quantum property Results Decidability of the classical problem 16 / 20 Proof idea for decidability Proof of Theorem 2: For any entry wise non-negative matrix M define its indicator matrix { M a,b := 0 if M a,b = 0 1 if M a,b > 0. and let M N := (M N ), then M wn... M w1 = 0 M w n... M w 1 = 0.
49 Undecidability as a genuine quantum property Results Decidability of the classical problem 16 / 20 Proof idea for decidability Proof of Theorem 2: For any entry wise non-negative matrix M define its indicator matrix { M a,b := 0 if M a,b = 0 1 if M a,b > 0. and let M N := (M N ), then M wn... M w1 = 0 M w n... M w 1 = 0. Moreover: M wn... M w1 = 0 {i j } J j=1 with J 2 (d2) : M i J... M i 1 = 0
50 Undecidability as a genuine quantum property Results Decidability of the classical problem 16 / 20 Proof idea for decidability Proof of Theorem 2: For any entry wise non-negative matrix M define its indicator matrix { M a,b := 0 if M a,b = 0 1 if M a,b > 0. and let M N := (M N ), then M wn... M w1 = 0 M w n... M w 1 = 0. Moreover: M wn... M w1 = 0 {i j } J j=1 with J 2 (d2) : M i J... M i 1 = 0 Why? Because all partial products in the shortest sequence must be different and the number of d d binary matrices is 2 (d2).
51 Undecidability as a genuine quantum property Results Decidability of the classical problem 16 / 20 Proof idea for decidability Proof of Theorem 2: For any entry wise non-negative matrix M define its indicator matrix { M a,b := 0 if M a,b = 0 1 if M a,b > 0. and let M N := (M N ), then M wn... M w1 = 0 M w n... M w 1 = 0. Moreover: M wn... M w1 = 0 {i j } J j=1 with J 2 (d2) : M i J... M i 1 = 0 Why? Because all partial products in the shortest sequence must be different and the number of d d binary matrices is 2 (d2). Ergo: Check all words M w n... M w 1 of length 2 (d2).
52 Undecidability as a genuine quantum property Results Undecidability of the quantum problem 17 / 20 Undecidability due to destructive interference Kraus operators A j in the QMOP can have negative (complex) entries!
53 Undecidability as a genuine quantum property Results Undecidability of the quantum problem 17 / 20 Undecidability due to destructive interference Kraus operators A j in the QMOP can have negative (complex) entries! Proof of Theorem 1:
54 Undecidability as a genuine quantum property Results Undecidability of the quantum problem 17 / 20 Undecidability due to destructive interference Kraus operators A j in the QMOP can have negative (complex) entries! Proof of Theorem 1: Reduction to the MMP:
55 Undecidability as a genuine quantum property Results Undecidability of the quantum problem 17 / 20 Undecidability due to destructive interference Kraus operators A j in the QMOP can have negative (complex) entries! Proof of Theorem 1: Reduction to the MMP: Matrix mortality problem (MMP) is undecidable [2, 3] It is undecidable whether the semi-group generated by {M i } K i=1 Z d d (with K 8 and d 3) contains the zero matrix.
56 Undecidability as a genuine quantum property Results Undecidability of the quantum problem 17 / 20 Undecidability due to destructive interference Kraus operators A j in the QMOP can have negative (complex) entries! Proof of Theorem 1: Reduction to the MMP: Matrix mortality problem (MMP) is undecidable [2, 3] It is undecidable whether the semi-group generated by {M i } K i=1 Z d d (with K 8 and d 3) contains the zero matrix. Find clever encoding of the MMP into the QMOP such that w n,..., w 1 : A wn... A w1 = 0 i n,..., i 1 : M in... M in = 0. and which takes the restrictions on the A j into account.
57 Undecidability as a genuine quantum property Conclusions 18 / 20 Conclusions An operationally defined problem about measurements
58 Undecidability as a genuine quantum property Conclusions 18 / 20 Conclusions An operationally defined problem about measurements Question: Are certain outcome sequences in repeated measurements impossible? QMOP Turing undecidable CMOP Turing decidable
59 Undecidability as a genuine quantum property Conclusions 18 / 20 Conclusions An operationally defined problem about measurements Question: Are certain outcome sequences in repeated measurements impossible? QMOP Turing undecidable CMOP Turing decidable What it means:
60 Undecidability as a genuine quantum property Conclusions 18 / 20 Conclusions An operationally defined problem about measurements Question: Are certain outcome sequences in repeated measurements impossible? QMOP Turing undecidable CMOP Turing decidable What it means: Destructive interference makes the quantum problem undecidable.
61 Undecidability as a genuine quantum property Conclusions 18 / 20 Conclusions An operationally defined problem about measurements Question: Are certain outcome sequences in repeated measurements impossible? QMOP Turing undecidable CMOP Turing decidable What it means: Destructive interference makes the quantum problem undecidable. Statement about the complexity of the mathematical theory of quantum mechanics
62 Undecidability as a genuine quantum property Conclusions 18 / 20 Conclusions An operationally defined problem about measurements Question: Are certain outcome sequences in repeated measurements impossible? QMOP Turing undecidable CMOP Turing decidable What it means: Destructive interference makes the quantum problem undecidable. Statement about the complexity of the mathematical theory of quantum mechanics Arguably the strongest complexity theoretic quantum/classical separation imaginable (almost: CMOP is NP-complete)
63 Undecidability as a genuine quantum property References 19 / 20 References Thank you for your attention! slides: [1] J. Eisert, M. Müller, and C Gogolin. Quantum measurement occurrence is undecidable. Physical Review Letters, 108(26):1 5, June [2] Vesa Halava, Tero Harju, and Mika Hirvensalo. Undecidability Bounds for Integer Matrices using Claus Instances. Technical Report 766, TUCS Turku Center for Computer Science, April [3] M S Paterson. Unsolvability in 3 {\times} 3 matrices. Stud. Appl. Math., (49): , 1970.
64 Undecidability as a genuine quantum property References 20 / 20 Making the result more physical The QMOP asks w 1,..., w n : Prob(w 1,..., w n ) = 0. The undecidability of the QMOP is stable in the sens that the question w 1,..., w n : Prob(w 1,..., w n ) < δ n is still undecidable, where δ > 1 is a simple function of ρ and the Kraus Operators {A j } K j=1.
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