Quantum Origins of Information and Ignorance Wojciech Hubert Zurek Theory Division, Los Alamos Deriving Probabilities from Symmetries of Entanglement “BEYOND.

Quantum Originsof Information and

Ignorance

Wojciech Hubert Zurek

Theory Division, Los Alamos

Deriving Probabilities from Symmetries of Entanglement

“BEYOND DECOHERENCE”

Robin Blume-Kohout, Fernando Cucchietti, Diego Dalvit, Harold Ollivier, Juan Pablo Paz, David Poulin, Hai-Tao Quan, Michael Zwolak,…

Textbook* Quantum Theory

1. Quantum states of a system are represented by vectors in its Hilbert space. (“Quantum Superposition Principle”)

2. Evolutions are unitary (e.g. generated by Schroedinger equation). (“Unitarity”)

3. Immediate repetition of a measurement yields the same outcome. (“Predictability”)

4. (a) Outcomes are restricted to orthonormal states {|sk>} (eigenstates of the measured observable). (b) Just one outcome is seen each time. (“Collapse Postulate”)

5. Probability of finding an outcome |sk> given a state |ƒ> is pk=|<sk|ƒ>|2. (“Born’s Rule”)

0. State of a composite system is a vector in the tensor product of the constituent Hilbert spaces. (“Complexity”)

Bohr, Dirac, “Copenhagen” -- 4&5 require “classical apparatus”, etc.Everett: Relative states -- no need for

explicit collapse!But need a preferred basis!!!

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Ψ17 ≅ I17 ΨUniverse*Dirac

Decoherence restricts stable states (states that can persist, and, therefore, “exist”) to the exceptional… Pointer states that exist or evolve predictably in spite of the immersion of the system in the environment.Predictability sieve can be used to ‘sift’ through the Hilbert space of any open quantum system in search of these pointer states.EINSELECTION (or Environment INduced superSELECTION) is the process of selection of these preferred pointer states.

DECOHERENCE AND EINSELECTION Thesis: Quantum theory can explain emergence of the classical.Principle of superposition loses its validity in “open” systems, that is,systems interacting with their environments.

For macroscopic systems, decoherence and einselection can be very effective, enforcing ban on Schroedinger cats.Einselection enforces an effective border that divides quantum from classical, making a point of view similar to Bohr’s Copenhagen Interpretation possible, although starting from a rather different standpoint (i. e., no ab initio classical domain of the universe).(Zeh, Joos, Paz, Hu, Caldeira, Leggett, Kiefer, Gell-Mann, Hartle, Omnes, Dalvit, Dziarmaga, Cucchietti …

Haroche, Raimond, Brune, Zeilinger, Arndt, Hasselbach…)

S E

REDUCED DENSITY MATRIX

€

ρS t( ) =TrE ΦSE t( ) ΦSE t( ) = α i

2σ i σ i

i

∑

EINSELECTION*, POINTER BASIS,AND DECOHERENCE

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ΦSE 0( ) =ψS ⊗ ε0

= α i σii

∑⎛

⎝ ⎜ ⎞

⎠ ⎟ ⊗ ε

0

Interaction

Entanglement

€

α i σ ii

∑ ⊗ εi =ΦSE t( )

EINSELECTION* leads to POINTER STATESStable states, appear on the diagonal of after decoherence time; pointer states are effectively classical! Pointer states left unperturbed by the “environmental monitoring”.

*Environment INduced superSELECTION

€

ρS t( )

€

HSE

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HSE , σ i σ i[ ] = 0

Goal

• Why are the measurement outcomes limited to an orthogonal subset of all the possible states in the Hilbert states? (as in “Collapse”)

• Why does “Born’s rule” yield probabilities?• How can “objective classical reality” -- states

we can find out -- arise from the fragile quantum states that are perturbed by measurements? (“Quantum Darwinism”)

Justify axioms 4&5 using the noncontroversial 0-3.

PLAN:

Understand emergence of “objective classical reality” -- how realstates that can be found out by us arise from quantum substrate.

CANNOT USE DECOHERENCE UNTILL BORN’S RULE IS ESTABLISHED!!!

CANNOT USE DECOHERENCE UNTILL BORN’S RULE IS ESTABLISHED!!!

States that can survive “being found out” intact must be orthogonal.

Consider two states that can be “found out” -- that change state of an apparatus:

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u A0 ⇒ u Au

v A0 ⇒ v Av

Consider an initial superposition of these two states:

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α u + β v( ) A0 ⇒ α u Au + β v Av

Norm must be preserved. Hence:

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Re(α *β u v ) = Re(α *β u v Au Av )

Phases of the coefficients can be adjusted at will. So:

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u v = u v Au Av

So either (measurement was not successful)

or QED!!!!

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Au Av =1

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u v = 0

PREDICTABILITY IMPLIES RESTRICTION TO ORTHOGONAL STATES!!!

“collapse”

Consequences and extensions• Derivation of the key to Collapse Postulate from Axioms 1-3:

explains why in general one cannot “find out” preexisting states.• Implies that observables are Hermitean (given an extra

assumption that eigenvalues are real).• Proof similar to “no cloning theorem” -- information about

preexisting states cannot be found out -- passed on. (Cloning means making a “perfect copy”. Here the copy need not be perfect; “information - disturbance”).

• Proof can be extended to the case when apparatus (or environment) is initially in a mixed state.

• Axiom 3 -- predictability -- is the key to the proof!• Information transfer need not be due to a deliberate

measurement: any information transfer that does not perturb outcome states will have to abide by this rule: Pointer states, predictability sieve, and DECOHERENCE.

?

Quantum Origin of Quantum Jumps…, Phys. Rev. A Nov ‘07; arXiv:quant-ph/0703160

Summary: Observables are HermiteanTheorem: Outcomes of a measurement that satisfy

postulates1-3 must be orthogonal.Proof (another version): measurement is an information transfer from a quantum system S to a quantum apparatus A. So, for any two possible repeatable (predictable) (Axiom 3) outcome states of the same measurement it must be true that:

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u A0 ⇒ u Au

v A0 ⇒ v Av

By unitarity (Axiom 2) scalar product of the total (S+A) state before and after must be the same. So:

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u v A0 A0 = u v Au Av

But . So either (measurement was not

successful) or . QED!!!!

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A0 A0 =1

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Au Av =1

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u v = 0

NOTE: IN CONTRAST WITH DECOHERENCE, WE DO NOT INVOKE BORN’S RULE!!!

DISTINGUISHABLE

“EVENTS”

Plan

• Why the measurement outcomes are limited to an orthogonal subset of all the possible states in the Hilbert states?

• Why does “Born’s rule” yield probabilities?

• How can “objective classical reality” -- states we can find out -- arise from the fragile quantum states that are perturbed by measurements? (“Quantum Darwinism”)

Derive controversial axioms 4&5 from the noncontroversial 0-3.Understand emergence of “objective classical reality” -- how realstates that can be found out by us arise from quantum substrate.

WE HAVE “EVENTS”!

€

pk = ψ k

2

( )

{{

Copy from a translation, p. 52-57 in “Quantum Theory and Measurement”, John Archibald Wheeler & WHZ, eds. (Princeton U. Press, 1983)

{ {

!{

!{

S E

EINSELECTION, POINTER BASIS,AND DECOHERENCE

€

ΦSE 0( ) =ψS ⊗ ε0

= α i σii

∑⎛

⎝ ⎜ ⎞

⎠ ⎟ ⊗ ε

0

Interaction

Entanglement

€

α i σ ii

∑ ⊗ εi =ΦSE t( )

……..Depends on Born’s Rule!!!

€

ρS t( ) =TrE ΦSE t( ) ΦSE t( ) = α i

2σ i σ i

i

∑EINSELECTION leads to POINTER STATES(same states appear on the diagonal of for times long compared to decoherence time)

€

ρS t( )

€

????????

REDUCED DENSITY MATRIX

€

HSE , σ i σ i[ ] = 0

€

!!!!!WE HAVE “EVENTS”!

ENVARIANCE (Entanglement-Assisted Invariance)

Consider a composite quantum object consisting of system S and environment E. When the combined state is transformed by:

€

ψSE

€

US = uS ⊗ 1E

but can be “untransformed” by acting solely on E, that is, if there exists:

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UE =1S ⊗ uE

then is ENVARIANT with respect to .

€

ψSE

€

uS

€

UE(US ψSE ) =UE ϕSE =ψSE

Envariance is a property of and the joint state of two systems, S & E .

€

ψSE

DEFINITION:

€

US

ENTANGLED STATE AS AN EXAMPLE OF ENVARIANCE:

€

ψSE = αkk=1

N

∑ sk εk

Schmidt decomposition:

Above Schmidt states are orthonormal and complex.

€

€

αk

€

sk , εk

Lemma 1: Unitary transformations with Schmidt eigenstates:

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uS (sk)= exp(iφkk=1

∑ ) sk sk

leave envariant.

€

ψSE

Proof:

€

€

uE (εk ){uS (sk )ψ SE } = α k exp{i(φk −k=1

∑ φk + 2π lk ) sk εk = α k

k=1

∑ sk εk = ψ SE

€

uS (sk)ψSE = αk exp(iφkk=1

∑ ) sk εk , uE (εk) = exp{i(−φkk=1

∑ +2π lk)}εk εk

LOCALLY, SCHMIDT PHASES DO NOT MATTER: DECOHERENCE!!!

PHASE ENVARIANCE THEOREMFact 1: Unitary transformations must act on the

system to alter its state (if they act only somewhere else, system is not effected).

Fact 2: The state of the system is all that is necessary/available to predict measurement outcomes (including their probabilities).

Fact 3: A state of the composite system is all that is needed/available to determine the state of the system.

Moreover, “entanglement happens”:

€

ψSE ∝ αkk=1

N

∑ sk εk

THEOREM 1: State (and probabilities) of S alone can depend only on the absolute values of Schmidt coefficients , and not on their phases.

€

α k

Proof: Phases of can be changed by acting on S alone. But thestate of the whole can be restored by acting only on E. So changeof phases of Schmidt coefficients could not have affected S! QED.

€

αk

€

∴By phase envariance, { } must provide a complete local description of the system alone. Same info as reduced density matrix!!!

€

αk , sk

Envariance of entangled states:the case of equal coefficients

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ψSE ∝ exp(iφk)k=1

N

∑ sk εk

In this case ANY orthonormal basis is Schmidt. In particular, in the

Hilbert subspace spanned by any two { } one can define aHadamard basis;

€

± =( sk ± sl )/ 2

€

sk , sl

This can be used to generate ‘new kind’of envariant transformations:

€

A SWAP:

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uS (k↔ l) =exp(iϕkl) sk sl +h.c.

Can be ‘undone’ by the COUNTERSWAP:

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uE (k↔ l)=exp{i(−ϕkl −ϕk +ϕl)}εl εk +h.c.

LEMMA 3: Swaps of states are envariant when Schmidt coefficients have same absolute value.

“Probability from certainty”Probabilities of Schmidt partners are the same

(detecting 0 in S implies 0 in E, etc.).

|0>|0> + |1>|1> (initial state -- equal abs. values of coeff’s)

SWAP on S

|1>|0> + |0>|1> (prob’s in S must have swapped, because after the swap they are equal to the prob’s

of state in E that were not affected)

COUNTERSWAP on S

|1>|1> + |0>|0> (prob’s in E must be the same as they were to begin with -- the global state is back to the “original”)

Probabilities can “stay the same” and also “get swapped” only when they are equal!!! (p(0)=p(1)) (Barnum, Schlosshauer & Fine, WHZ)

Probability of envariantly swappable states

€

ψSE ∝ exp(iφk)k=1

N

∑ sk εk

By the Phase Envariance Theorem the set of pairs provides a complete description of S. But all are equal.

€

αk , sk

€

αk

With additional assumption about probabilities (e.g., perfect correlation as on the previous transparency), one can prove

THEOREM 2: Probabilities of envariantly swappable states are equal. (a)“Pedantic assumption”; when states get

swapped, so do probabilitites;(b)When the state of the system does not change

under any unitary in a part of its Hilbert space, probabilities of any

set of basis states are equal.(c) Because there is one-to-one correlation

between

Therefore, by normalization:

€

pk =1N

∀k

€

sk , εk

Special case with unequal coefficientsConsider system S with two states

€

0{ , 2}

€

0{ ,1, 2 } The environment E has three states and

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+ = 0( + 1)/ 2

An auxilliary environment E’ interacts with E so that:

€

ψSE E'0 =23

0 + + 13

2 2⎛

⎝ ⎜ ⎞

⎠ ⎟ 0 ⇒

23

0 0 0 +1 1( )/ 2 + 13

2 2 2 =

€

= 0 0 0 + 0 1 1 +2 2 2( )/ 3

States have equal coefficients. Therefore,Each of them has probability of 1/3. Consequently: p(0) = p(0,0)+p(0,1) = 2/3, and p(2) = 1/3.

….. BORN’s RULE!!!

€

0 0, 0 1, 2 2

€

ψSE =2

30 + +

1

32 2

no need to assume additivity! (p(0)=1-p(2))!

Probabilities from EnvarianceThe case of commensurate probabilities:

€

ψSE

′ e 0

= mkM

α k

1 2 4 3 4 sk

1

mkjk =1

mk

∑ ejk

ε k

1 2 4 4 3 4 4

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

k=1

N

∑

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

c0

⇒

⇒ 1

Msk( j)

j=1

M

∑ ej

cj

Attach the auxiliary “counter” environment C:

THEOREM 3: The case with commensurate probabilities can be reduced to the case with equal probabilities. BORN’s RULE follows:

€

pj =1M

, pk = pjkjk =1

mk

∑ =mk

M=αk

2

General case -- by continuity. QED.

ENVARIANCE* -- SUMMARY1. New symmetry - ENVARIANCE - of joint states

of quantum systems. It is related to causality.

2. In quantum physics perfect knowledge of the whole may imply

complete ignorance of a part.

3. BORN’s RULE follows as a consequence of envariance.

4. Relative frequency interpretation of probabilities naturally

follows.

5. Envariance supplies a new foundation for environment - induced

superselection, decoherence, quantum statistical physics, etc., by

justifying the form and interpretation of reduced density matrices.

*WHZ, PRL 90, 120404; RMP 75, 715 (2003); PRA 71, 052105 (2005).

Plan

• Why the measurement outcomes are limited to an orthogonal subset of all the possible states in the Hilbert states?

• Why does “Born’s rule” yield probabilities?• How can “objective classical reality” -- states

we can find out -- arise from the fragile quantum states that are perturbed by measurements? (“Quantum Darwinism”)

Derive controversial axioms 4&5 from the noncontroversial 0-3.Understand emergence of “objective classical reality” -- how realstates that can be found out by us arise from quantum substrate.

Quantum Darwinism

• The imprint left by the system S in the environment E is the cause of decoherence

• The focus of decoherence is the information that is left in S in spite of E. (“reduced density matrix” of S)

• Quantum Darwinism is focused on the information about S that can be found out indirectly from E !!!

(i) How many copies of the information about S can be extracted from E? (Redundancy)

(ii) What is this information about? (i.e, what observable of S gets redundantly imprinted in E?)

(iii) Why does this matter? Robin Blume-Kohout (PI, Waterloo)Harold Ollivier (INRIA, France)David Poulin (IQC, Waterloo)

Quantum Darwinism -- “The Big Picture”

• Certain “fittest” information about S proliferates -- it is recorded in E in many copies

• Complementary information is diluted -- effectively obliterated

• The fittest information is about the einselected pointer states that can “survive” decoherence

• Information that can be obtained indirectly and independently by many is in effect objective

• Its acquisition does not endanger preexisting state of the system (which can be “found out” as if it were classical) (Environment as a witness)

How do we analyze this?

Mutual Information

where

● Measures the increase in entropy from eliminating correlations between S and E.

● No reference to observables or measurements.

● Bounded by .

Measuring what E knows about S

1. Partial Info: How much does a fragment supply?2. Redundancy: How many disjoint fragments supply ?

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1−δ( )HS

PIP = How info is distributed in E(Partial Information Plot)

MaximumClassicalInformation“all but a small fraction δ” ofclassical info

NOTE: This shape ofPartial Information Plotis NOT ”typical” !!!!

cc

Partial Information Plots(a visual characterization of information storage)

● Plot size of a fragment (F) vs. amount of info it provides.

● 3 info profiles:- redundant (decoherence) - distributed- TYPICAL (encoded)

● We average over all fragments of a given size.

● = Redundancy of “all but δ” of the available info

(1-δ)HS

}

✵ Blume-Kohout & Zurek, Foundations of Physics (~Nov. 2005), quant-ph/0408147

€

Rδ

What does E know about?

Q: What observables can be inferred from E?- Consider GHZ:- “Ising” model -> singly branching states

A: Only the pointer observable recorded redundantly!

cc

a)

✵ Ollivier, Poulin, & Zurek, PRL 93, 220401 (2004).

✵ Ollivier, Poulin, & Zurek, PRA 72, 042113 (2005)..

angle withpointer observable

Quantum Darwinism -- brief summary of results

• Environment knows about the system. • Every fragment of the environment knows the same

thing about the system.• They all know about the pointer observable.• Many observers can extract that information without

disturbing the state of the system.• Pointer states / observables become “objective”.• The ability of a state to survive decoherence without

getting perturbed (1st part of the talk) is the key.