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Quaestiones disputatae de observatione

Piotr Mironowicz

Department of Algorithms and System Modeling, Faculty of Electronics, Telecommunications and Informatics, Gdańsk University of Technology National Quantum Information Centre in Gdańsk

18 czerwca 2017, Toruń

Collaborative results with J.K.Korbicz, E.Aguilar, P.Ćwikliński and P.Horodecki.

National Science Centre (NCN) grant 2014/14/E/ST2/00020 and the John Templeton Foundation ID 5603. Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 1 / 24

Ontology and epistemology

Fundamental questions

What does it mean to exist? What does it mean to exist objectively? What does exist objectively? How does existence relate to observations?

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 2 / 24

Ontology and epistemology

Protagoras (refered by Plato’s Socrates)

• Soc. Bravely said, boy; (...) You say that knowledge is perception?
• Theaet. Yes.
• Soc. Well, you have delivered yourself of a very important doctrine about

knowledge; it is indeed the opinion of Protagoras, who has another way of expressing it, Man, he says, is the measure of all things, of the existence of things that are, and of the non-existence of things that are not:-You have read him? (...)

• Soc. Now is the wind, regarded not in relation to us but absolutely, cold or

not; or are we to say, with Protagoras, that the wind is cold to him who is cold, and not to him who is not? (...)

• Soc. Then perception is always of existence, and being the same as

knowledge is unerring?

• Theaet. Clearly.

[Plato, Theaetetus 151e-152c, transl. Benjamin Jowett]

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 3 / 24

Ontology and epistemology

Plato and Aristotle

Plato: only Ideas (Forms) exist, and perceptions are just stimuli for recollection of the knowledge. Forms exist on their own. Aristotle (Hylomorphism): the existing

• bjects (substances) are constituted by both unperceived

matter and perceivable form. Forms exist in real things.

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 4 / 24

Ontology and epistemology

Berkeley and Kant

Berkeley: Esse est percipi, to be is to be perceived (by spirits, i.e. conscious subjects or observers). Since the perception is relative, there exist sensations, but there is no real things. God has to be the source of

• sensations. [Berkeley, Three Dialogues between Hylas and Philonous

(1713)] Kant distinguished the unintelligible but existing things-in-itself, noumenon, and things that appear, phenomenon, perceived by a reason via senses

• f a particular being. [Kant, Critique of Pure Reason (1781)]

Other important statements: 19th century positivism; Husserl and phenomenology.

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 5 / 24

Ontology and epistemology

Epistemic intuition

The philosophers differ greatly in their views on the relation between

• bservations and existence.

Protagoras: observation = ⇒ existence Plato: observation ≪ existence Aristotle: observation existence Berkeley: observation ≡ existence Kant: observation ⊥ existence positivism: OBSERVATION existence Husserl: observation Common epistemic intuition: the objectively existing things should not be perceived in contradictory ways by different observers.

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 6 / 24

Ontology and epistemology

Inter-subjectivity of perceptions

Suitable notion of objectivity of perceptions: A state of the system S is

• bjective if many observers can find out (and agree on) the state of

S independently, and without perturbing it. [W. Żurek, Nat. Phys. 2009; R. Horodecki, J. K. Korbicz, P. Horodecki, Phys. Rev. A 2015.] More precisely: Inter-subjectivity not objectivity [Ajdukiewicz 1949].

• K. Ajdukiewicz, rector of UAM Poznań, 1948-1952.

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 7 / 24

Ontology and epistemology

Quantum inter-subjectivity

Question: What information can being broadcasted (in order to ensure

• bjectivity) to multiple observers and how?

Recall NO-GO theorems: no-cloning theorem, no-broadcast theorem! Related question: How does the classical world emerge out of quantum laws of physics? Żurek: The part of information about the quantum system which is proliferated in many copies in the environment becomes classical.

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 8 / 24

Ontology and epistemology

Inter-subjective perception vs. the measurement problem

Question: What happens when a measurement is performed?, and Question: How (at what circumstances) a measurement is performed? the measurement problem the inter-subjective perception single device multi-party measurement agreement (possibly ontologically significant) (epistemic)

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 9 / 24

Broadcasting of quantum information

Spectrum Broadcast Structures

Intuition: If the results of a measuring device are different, the states of the device should also be different. The state of an observed object and its observers is of the following form called a Spectrum Broadcast Structure (SBS): ̺SBS ≡

• i

pi| ii |

• system

⊗

• k

̺(k)

i

• measuring devices

with ̺(k)

i

⊥ ̺(k)

j

, where k denotes observers. [J. K. Korbicz, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 2014;

• R. Horodecki, J. K. Korbicz, P. Horodecki, Phys. Rev. A 2015]

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 10 / 24

Broadcasting of quantum information

Spectrum Broadcast Structures

Examples of models leading to Spectrum Broadcast Structures (asymptotically in time): Illuminated sphere: J. K. Korbicz, P. Horodecki, R. Horodecki,

• Phys. Rev. Lett. 112, 120402 (2014).

QBM: J. Tuziemski, J. K. Korbicz, EPL 112, 40008 (2015);

• J. Tuziemski, J. K. Korbicz, Photonics 2, 228-240 (2015);
• J. Tuziemski, J. K. Korbicz, J. Phys. A 49, 445301 (2016).

Gravity: J. K. Korbicz, J. Tuziemski, arXiv:1612.08864. Spin-spin: P. Mironowicz, J. K. Korbicz, P. Horodecki,

• Phys. Rev. Lett. 118, 150501 (2017).

Spin-boson: A. Lampo, J. Tuziemski, M. Lewenstein, J. K. Korbicz, arXiv:1702.08427. How Spectrum Broadcast Structures are being formed?

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 11 / 24

Broadcasting of quantum information

A central system with environment

Quantum Setup: An observed system S, M-partite environment, ̺(0) ≡ ̺S(0) ⊗ M

k=1 ̺(k)(0).

Quantum measurement limit interaction: H =

 

dS

• i=1

ai| ii |

 

• A

⊗

k B(k).

Discarding unobserved parts of the environment: ̺Obs(t) =

• i

pi| ii | ⊗

• k∈Obs

̺(k)

i

(t)+

• i=j

σij| ij |

 

k / ∈Obs

γ(k)

ij (t)

  ⊗

• k∈Obs

̺(k)

i,j (t).

̺(k)

i,j (t) ≡ U(k) i

(t)̺(k)(0)U(k)†

j

(t); ̺(k)

i

(t) ≡ ̺(k)

i,i (t); σij ≡ i |̺(0)| j; pi ≡ σi,i ; γ(k) ij

(t) ≡ Tr ̺(k)

i,j (t)

How far is the state from the Spectrum Broadcast Structure form?

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 12 / 24

Broadcasting of quantum information

How far is the state from the Spectrum Broadcast Structure form?

Compare the Spectrum Broadcast Structure: ̺SBS ≡

• i

pi| ii | ⊗

• k

̺(k)

i

with ̺(k)

i

⊥ ̺(k)

j

, and the actual state: ̺Obs(t) =

• i

pi| ii | ⊗

• k∈Obs

how close to ̺(k)

i

(t) ⊥ ̺(k)

j

(t)?

̺(k)

i

(t) +

• i=j

σij| ij |

 

k / ∈Obs

γ(k)

ij (t)

  ⊗

• k∈Obs

̺(k)

i,j (t)

• how close to 0?

.

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 13 / 24

Broadcasting of quantum information

Overlap (fidelity) of states

Overlap of quantum states: B(ρ1, ρ2) ≡ Tr

√ρ1ρ2√ρ1.

B(ρ1, ρ2) = 1 if the states are exactly the same. B(ρ1, ρ2) = 0 if the states are completely different.

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 14 / 24

Broadcasting of quantum information

The cumulative error of discrimination

Discriminate states from ensemble

• pi, ̺(k)

i

• using projectors
• P(k)

i

• .

The cumulative error of discrimination vs. the overlap:

∀

pi,̺(k)

i

∃

P(k)

i

i

pi Tr

• ̺(k)

i

• I − P(k)

i

• probability of error of discrimination

≤

• i=j

√pipjB

• ̺(k)

i

, ̺(k)

j

• .

̺(k)

i

→ P̺(k)

i

≡ P(k)

i

̺(k)

i

P(k)

i

(if successful) The probability of the correct guess: p(k)

i

≡ Tr P̺(k)

i

• .

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 15 / 24

Broadcasting of quantum information

Information broadcasting in time

The decoherence factor: γij(t) ≡

• k /

∈Obs γ(k) ij (t)

• .

The collective decoherence factor: Γ(t) ≡

i=j |σij | γij(t).

Recall the Spectrum Broadcast Structure: ̺SBS(t) ≡

• i

pi| ii | ⊗

• k

̺(k)

i

, with ̺(k)

i

⊥ ̺(k)

j

. Our main result [PM, J. K. Korbicz, P.Horodecki, PRL 2017] For given ̺Obs(t) there exist ̺SBS such that: 1 2

• ̺Obs(t) − ̺SBS
• 1 ≤ Γ(t) +
• k∈Obs
• i=j

√pipjB

• ̺(k)

i

(t), ̺(k)

j

(t)

• verlap (orthogonalization) factor B(t)

.

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 16 / 24

Broadcasting of quantum information

Illustration: Spin-spin system

The upper bound on ǫ(fM) ≡ 1

2

• ̺Obs(t) − ̺SBS(t)
• 1.

Equal size of N/2 spins of the observed and unobserved parts of the environment. For small sizes (N/2 = 30, 50) the bound is substantially larger 1 and does not imply an Spectrum Broadcast Structure formation.

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 17 / 24

Typicality of objectivity

Do objective results typically appear in quantum measurements?

How often the dumping of Γ(t) and B(t) occur in nature? In other words: Do objective results typically appear in quantum measurements? Recall: Quantum measurement limit interaction: H = A ⊗

k B(k).

To study typicality, one can use Gaussian Unitary Ensemble (GUE) distribution: i) A is distributed according to its own GUE with a scale factor ηS, ii) B(k) are independently, identically distributed according to a GUE with a scale factor ηE.

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 18 / 24

Typicality of objectivity

The Hilbert-Schmidt and the Bures measures:

• Tr ρ2

HS =

2d d2 + 1,

• Tr ρ2

Bures =

5d2 + 1 2d(d2 + 2). Super-fidelity bound (not tight) [Miszczak et al. 2008]: B

• ̺(k)

i

(t), ̺(k)

j

(t)

2 ≤ G(k)

i,j (t) ≡ Tr

• ̺(k)

i

(t)̺(k)

j

(t)

• + Slin
• ̺(k)(0)
• =Slin
• ̺(k)

i or j(t)

• Super-fidelity is super-multiplicative:

G(A ⊗ B, C ⊗ D) ≥ G(A, C) · G(B, D).

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 19 / 24

Typicality of objectivity

The GUE averages over B(k) for central system states ai and aj and k-th

• bserver:
• γ(k)

ij (t)

• O 1

d

• for large t

= 1 +

Tr ρ2

• d + 1
• O( 1

d )

+ ft(a a a,λ λ λ)

• O
• poly(d)·e− ˜

∆ij t

· 2(d −

Tr ρ2 )

d(d2 − 1)

• O 1

d2

• ,
• G(k)

ij (t)

• 1 − O 1

d

• for large t

= Slin(ρ0) + 1 +

Tr ρ2

• d + 1
• 1−O( 1

d )

+ ft(a a a,λ λ λ)

• O
• poly(d)·e− ˜

∆ij t

· 2(d

Tr ρ2 − 1)

d(d2 − 1)

• O 1

d3

• ,

˜ ∆ij ≡ (ai − aj)/√ηE, ρ0 ≡ ̺(k)(0) ∈ Cd.

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 20 / 24

Typicality of objectivity

Coarse-graining

Recall the decoherence factor: γij(t) ≡

• k /

∈Obs γ(k) ij (t)

• .

For t large enough (depending on ai and aj): γij(t) = O (exp[−Nunobs log d]) , where Nunobs is the number of not observed parts of the environment. Coarse-graining of observed parts into fractions of size of rank Nmac for time large enough using super-multiplicativity of super-fidelity:

• Gmac

ij

(t)

• = O
• e− Nmac

d

• .

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 21 / 24

Typicality of objectivity

Time scales

The time scales for the decoherence and super-fidelity decay: τdec ≡

• 8g2NunobsdS
• d −
• Tr ρ2

− 1

2 d≫1

∼ g−1 √NunobsdSd , τfid ≡

• 8g2NmacdS
• d
• Tr ρ2
• − 1

− 1

2 d≫1

∼ g−1 √NmacdS . This leads to:

∀i,j γij(t) −

− − − →

t≫τdec O (exp[−Nunobs log d]) ,

∀i,j

• Gmac

ij

(t)

• −

− − − →

t≫τfid O

• exp
• −Nmac

d

• .

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 22 / 24

Typicality of objectivity

Combining the above result about formation of Spectrum Broadcast Structures and matrix ensembles: Typicality of objectivity [J. K. Korbicz, E. Aguilar, P. Ćwikliński, P. Horodecki, arXiv:1604.02011] Averaged over interactions and the initial conditions, the distance ǫSBS of the actual state to an Spectrum Broadcast Structure state satisfies: ǫSBS − − − − − − →

t≫τfid,τdec O

• d2

S

• exp
• −Nunobs

2 log d

• + M exp
• −Nmac

2d

• ,

where M is the number of coarse-grained observed fractions.

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 23 / 24

Summary

We have discussed the philosophical tradition of the problem of the relation between reality and perception. We have presented a convenient tool for estimation how far is the state from being “inter-subjective”. We have shown that “inter-subjective” perceptions occur in typical quantum setups. Open (big) question: How are the Spectrum Broadcast Structures relevant as a model of the real inter-subjective perception?

Thank you for attention!

Piotr Mironowicz (Gdańsk Univ. Tech.) Quaestiones disputatae de observatione 18 czerwca 2017, Toruń 24 / 24