Quantum Physics

Wojciech Hubert Zurek

Theory Division, Los Alamos

Relative States and the Environment (Everett ‘57)

“BEYOND DECOHERENCE”

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

5. Probability of finding an outcome |sk> given states |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

REDUCED DENSITY MATRIX

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ρS t( ) =TrE ΦSE t( ) ΦSE t( ) = α i

2σ i σ i

i

∑

S E

EINSELECTION*, POINTER BASIS,AND DECOHERENCE

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ΦSE 0( ) =ψS ⊗ ε0

= α i σii

∑⎛

⎝ ⎜ ⎞

⎠ ⎟ ⊗ ε

0

Interaction

Entanglement

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α i σ ii

∑ ⊗ εi =ΦSE t( )

EINSELECTION* leads to POINTER STATES(same states appear on the diagonal of for times long compared to the decoherence time; pointer states are effectively classical!) Pointer states left unperturbed by the “environmental monitoring”.

*Environment INduced superSELECTION

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ρS t( )

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, Caldeira, Leggett, Kiefer, Gell-Mann, Hartle, Omnes, Dalvit, Dziarmaga, Cucchietti …

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

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 OF THE TALK:

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

Proof: 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, PROOF DOESN’TRELY ON BORN’S RULE!

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 -- making a “perfect copy”.)

• Proof can be extended to the case when the apparatus 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.

?

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”!

ENVARIANCE (Entanglement-Assisted Invariance)

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

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ψSE

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US = uS ⊗ 1Ebut 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 .

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ψSE

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uS

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UE(US ψSE ) =UE ϕSE =ψSE

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

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ψSE

DEFINITION:

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uS

ENTANGLED STATE AS AN EXAMPLE OF ENVARIANCE:

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ψSE = αkk=1

N

∑ sk εk

Schmidt decomposition:

Above Schmidt states are orthonormal and complex.

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αk

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sk , εk

Lemma 1: Unitary transformations with Schmidt eigenstates:

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uS (sk)= exp(iφkk=1

∑ ) sk sk

leave envariant.

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ψSE

Proof:

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

∑ φk + 2π lk ) sk εk = α k exp(iφk

k=1

∑ ) sk εk = ψSE

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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”:

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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.

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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.

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αk

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

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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;

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± =( sk ± sl )/ 2

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sk , sl

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

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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 of envariantly swappable states

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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.

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αk , sk

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αk

Therefore, by normalization:

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pk =1N

∀k

With additional assumption about probabilities, 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

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sk , εk

€

0 + i 1

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1 + i 0IS ORTHOGONAL TO

Note: Swaps do change unentangled states ! Phases matter!

Probabilities from envariance

Symmetriescan reflect ignorance

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0S 0E + 1S 1Eswap in S ⏐ → ⏐ ⏐ ⏐ 1S 0E + 0S 1E

swap in E ⏐ → ⏐ ⏐ ⏐ 1S 1E + 0S 0E

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p = ψ2

follows!

(Environment-assisted iNVARIANCE)

Special case with unequal coefficientsConsider system S with two states

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0{ , 2}

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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:

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ψSE E'0 =23

0 + + 13

2 2⎛

⎝ ⎜ ⎞

⎠ ⎟ 0 ⇒

23

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

2 2 2 =

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= 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!!!

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0 0, 0 1, 2 2

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ψSE =2

30 + +

1

32 2

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

Probabilities from EnvarianceThe case of commensurate probabilities:

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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:

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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.

S AVon Neumann

S AE

Decoherence & Einselection

S A E

Quantum Darwinism

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 (Cal Tech - IQI)Harold Ollivier (Perimeter Institute)David Poulin (Cal Tech - IQI)