# Presentation: 587781

## Quantum Originsof 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,...

**Notes:**
The general goal -- whether & how can one understand the Universe around us in terms of quantum theory. 
Colloquium -- a bit less detailed, “general audience”.
Will try to be as rigorous as time permits.
Invite questions -- will stop twice & ask for q’s before the end

## Textbook* Quantum Theory

- Textbook* Quantum Theory

- Quantum states of a system are represented by vectors in its Hilbert space. **(“Quantum** **Superposition Principle”**)
- Evolutions are unitary (e.g. generated by Schroedinger equation). **(“Unitarity”)**
- Immediate repetition of a measurement yields the same outcome. **(“Predictability”)**
- (a) Outcomes are restricted to orthonormal states {|sk>} (eigenstates of the measured observable). (b) Just one outcome is seen each time**. (“Collapse Postulate”)**
- 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!!!**

*******Dirac**

**Notes:**
Note: 3 “on the border”. Not controversial: Need it to even have a concept of a state.
4(&5?) outside of “just quantum” -- require classical component to the Universe.

## DECOHERENCE AND EINSELECTION 

-    “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...)

## EINSELECTION*, POINTER BASIS,

**REDUCED DENSITY MATRIX**

**EINSELECTION*********,**** POINTER BASIS,**
**AND DECOHERENCE**

**EINSELECTION********* leads to POINTER STATES**
- Stable 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**

**Notes:**
Pointer states are preferred, effectively states that define “events”.
In relative state context they define “branches”.
They are needed to discuss probabilities -- the anser question “probability of WHAT”?

## Goal

**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 real
- states that can be found out by us arise from quantum substrate.

**Notes:**
Cannot justify “real” collapse -- inconsistent with 1&2.
But can justify “most of what’s needed”.

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

**PREDICTABILITY IMPLIES RESTRICTION TO ORTHOGONAL STATES!!!**

- “collapse”

**Notes:**
Preferred observable (which defines preferred states). 
EVENTS THAT CAN BE COMMUNICATED HAVE TO BE ORTHOGONAL!!!

## 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](http://arxiv.org/abs/quant-ph/0703160)_

**Notes:**
Axiom 5 not used!!!
“preferred states” emerge -- how does this compare with decoherence?

## Summary: Observables are Hermitean

**Theorem**: 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:

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

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

- “information gain
- implies disturbance”
- (Bennett, Brassard,& Mermin)

- DISTINGUISHABLE
-  “**EVENTS”**

**Notes:**
Cannot “find out” quantum states as a result!
Connection with cloning!

## 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 real
- states that can be found out by us arise from quantum substrate.

- 

- WE HAVE “EVENTS”!

## {

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

## {

## !{

## ........Depends on Born’s Rule!!!

**EINSELECTION, POINTER BASIS,**
**AND DECOHERENCE**

**......****..Depends on Born’s Rule!!!**

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

- REDUCED DENSITY MATRIX

- WE HAVE
- “EVENTS”!

**Notes:**
Cannot derive axiom (v) while using axiom (v) -- CIRCULARITY!!!

**Notes:**
We have a PHYSICAL argument why probabilities of the outcome states have same Schmidt coeff’s have the same probabilities.
This is an OBJECTIVE property of a global entangled pure state!!!!!
ENVARIANCE does not work classically!!!

## ENVARIANCE (Entanglement-Assisted Invariance)

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

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

**then        is ENVARIANT with respect to      . **

** **

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

**DEFINITION:**

**Notes:**
Symmtery!!! (“ASSISTED SYMMETRY”)

## ENTANGLED  STATE AS AN EXAMPLE OF ENVARIANCE:

**Schmidt decomposition:**

-  Above Schmidt states                are orthonormal and       complex.

- Lemma 1: Unitary transformations with Schmidt eigenstates:

- leave         envariant.

- Proof:

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

**Notes:**
Local phase of Schmidt coeff. has no physical significance!
Explain (“domesticate”) Schmidt basis / decomposition.

## PHASE  ENVARIANCE  THEOREM

**Fact 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”:

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

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

**Notes:**
Phases lose local physical significance .... Decoherence!!!

## Envariance of entangled states:the case of equal coefficients

- In this case ANY orthonormal basis is Schmidt. In particular, in the
- Hilbert subspace spanned by any two {             } one can define a
- Hadamard basis;

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

**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

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

- 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.
- “Pedantic assumption”; when states get swapped, so do probabilitites;
- 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**

## Special case with unequal coefficients

- Special case with **unequal **coefficients

- Consider system S with two states

-  The environment E has three states               and

- An auxilliary environment E’ interacts with E so that:

- 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!!!

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

## Probabilities from Envariance

- The case of commensurate probabilities:

- 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:

- General case -- by continuity. QED.

## ENVARIANCE* -- SUMMARY

- New symmetry - ENVARIANCE - of joint states of quantum
-       systems. It is related to causality.
- 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).

**Notes:**
From now on, we can use the usual tools of decoherence (reduced density matrices, trace, etc.).

## 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 real
- states 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** !!!

- How many copies of the information about S can be extracted from E? (Redundancy)
- What is this information about? (i.e, what observable of S gets redundantly imprinted in E?)
- 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?

## Measuring what E knows about S

- Measuring what *E* knows about *S*

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

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

- NOTE: This shape of
**P**artial  **I**nformation Plot
- is NOT ”typical” !!!!

## Partial Information Plots(a visual characterization of information storage)

- 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.

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

## 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!

- ✵ Ollivier, Poulin, & Zurek, *PRL 93, 220401* (2004).
- ✵ Ollivier, Poulin, & Zurek, PRA 72, 042113 (2005)..

- angle with
- pointer 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.