Home Tech Support

  • Subscribe to our RSS feed.
  • Twitter
  • StumbleUpon
  • Reddit
  • Facebook
  • Digg

Monday, 6 May 2013

Expected Utility in Quantum States

Posted on 07:12 by Unknown

Last semester, 14.04 — Intermediate Microeconomic Theory covered choice theory under uncertainty; at the same time, I was taking 8.05 — Quantum Physics II, where we had talked about 2-state systems and the formalism of quantum states being complex vectors in a Hilbert space, and as choice under uncertainty discusses how consumers make choices based on states of the world, I thought it would be cool to extend it to quantum states, but I wasn't sure how to do that at that time. Now that 14.03 — Microeconomic Theory and Public Policy is talking about choice under uncertainty as well this semester, and now that I have had some time for the stuff about 2-state systems to simmer in my head, I think I have a slightly better idea of how to think about extended the states of the world to include the superpositions as allowed by quantum mechanics.

One of the simplest examples would be the 2-state system. In the class I am taking now, a typical 2-state system would be a fair coin which can either take on values of heads or tails (each with probability $\frac{1}{2}$). What might this look like in quantum mechanics? We could replace the coin with the spin $S_z$ of an electron. A state of the electron corresponding to being measured as $|\uparrow_z \rangle$ or $|\downarrow_z \rangle$ each with equal probability could be labeled as $|\psi \rangle = \frac{1}{\sqrt{2}} \left(|\uparrow_z \rangle + |\downarrow_z \rangle\right)$. This indeed gives equal probabilities of being measured as spin-up or spin-down in the $z$-direction. The problem is that this state is exactly $|\psi \rangle = |\uparrow_x \rangle$, so the probability of the state being measured in the $x$-direction as spin-up is unity. This leads to issues in trying to reconcile the interpretation of payoffs for different states of the world; this particular state would pay $w(|\uparrow_z \rangle)$ or $w(|\downarrow_z \rangle)$ with equal probabilities until $S_z$ is measured, but would pay $w(|\uparrow_x \rangle)$ with certainty as $S_x$ has essentially already been measured, collapsing a previously unknown state into this one. So there seems to be an issue with trying to stuff the interpretation of probabilistic measurement of a state of the world into the idea of superposing quantum states, as certain measurable states of the world do not commute ($[S_i, S_j] = i\hbar \epsilon_{ijk} S_k$ for instance). So what can be done now?

Recall that each state of the world $j$ has an associated probability $\mathbb{P}_j$. Yet, once a state is measured, those probabilities are meaningless, because a state becomes observed or not observed; it cannot be "observed in fraction $\mathbb{P}_j$" or anything like that. Taking the frequentist view, these probabilities are only meaningful in the large ensemble limit; after a large number of observations, the fraction of those observations corresponding to the state $j$ converges to $\mathbb{P}_j$. This is exactly the same rationale behind the density matrix, which describes the state of an ensemble of systems which are usually not all prepared in the same quantum state, but in the large number limit, the fraction of those prepared in the state $|\psi_j \rangle$ is $\mathbb{P}_j$ such that $\sum_j \mathbb{P}_j = 1$. It is then defined as \[ \rho = \sum_j \mathbb{P}_j |\psi_j \rangle \langle \psi_j | . \] Note that in general $\langle \psi_j | \psi_{j'} \rangle \neq \delta_{jj'}$ because the states do not in general need to be orthogonal. Furthermore, different state preparations can have the same density matrix, in which case the "different" state preparations are actually physically indistinguishable if a quantum mechanical measurement is made.

So what does this have to do with the states of the world? If the example of a fair coin flip is again translated into measuring each eigenvalue of $S_z$ with equal probability, then the density matrix becomes $\rho = \frac{1}{2} \left(|\uparrow_z \rangle \langle \uparrow_z | + |\downarrow_z \rangle \langle \downarrow_z |\right)$. Now it is much more plausible to define a wealth $w(|\psi_j \rangle)$ dependent on the state of the world.

For example, let us now consider an unfair "coin" represented by the density matrix $\rho = \frac{1}{3} \left(|\uparrow_z \rangle \langle \uparrow_z | + 2|\downarrow_z \rangle \langle \downarrow_z |\right)$. What is the probability of measuring this system in the state $|\uparrow_x \rangle$? That would be $\langle \uparrow_x | \rho | \uparrow_x \rangle = \frac{1}{2}$. Similarly, $\langle \downarrow_x | \rho | \downarrow_x \rangle = \frac{1}{2}$. It is interesting to note that equal probabilities are achieved for spin-up and spin-down in the $x$-direction (and $y$-direction as well) for this system; the difference is that now there are off-diagonal terms in the density matrix describing the degree of interference between the states of $S_x$ in the two populations of spins.

What can we do with this? Payoffs now need to be defined over every plausible noncommuting observable in the Hilbert space (because commuting observables yield the same state, and the payoff is really defined for the state). Thankfully this is fairly simple for spins, as the noncommuting observables in question are the components of $\vec{S}$.

Let us return to the previous unfair "coin". Classically (so it would be a coin), the result of the coin flip would be the end of the story. Let us suppose that the consumer had a risk-neutral utility $V(w) = w$. Let us also suppose that there was a game which payed off 20 if heads and 0 if tails and cost 10 to enter. The expected payoff, accounting for the cost of entering, would be $\mathbb{E}(w) = -\frac{10}{3}$, so the consumer would prefer to not play.
Classically that would be the whole story. Quantum mechanically, though, there is more to the story. If the owner of the game decided to measure $S_z$, then the result would be the same as the classical result. If the owner decided instead to measure $S_x$ or $S_y$, then $\mathbb{E}(w) = 0$ so the consumer would be indifferent between playing and not playing, which is certainly a different outcome from preferring not to play. In fact, if the consumer does not know for sure what the owner decides to measure, then probabilities could be assigned to the measurement choice itself, and these could then be used to find the expectation value of payoff or utility over all possible choices, where each of those choices will have an expected payoff or utility as well.

One thing to look further at is how to extend this to include continuous ensembles. The example in class was how a state of the world might be the outdoor temperature measured in some range. The quantum mechanical equivalent might be having a continuum of systems, each infinitesimal one having a position inside a given range of allowed positions; the only issue with this is that matter is not continuous, so I don't think it is possible to have a density matrix in the form of $\rho = \int |\psi(s) \rangle \langle \psi(s)| \, ds$ for some continuous index $s$. However, that can certainly be further investigated, and maybe I'll do that next time. The point is that states of the world in analyzing expected utility can easily be generalized to include quantum states through density matrices, and the question of which observable to measure in a noncommuting set brings out some interesting behavior not seen in classical states.
Email ThisBlogThis!Share to XShare to FacebookShare to Pinterest
Posted in class, college, economics, MIT, physics, quantum mechanics, science | No comments
Newer Post Older Post Home

0 comments:

Post a Comment

Subscribe to: Post Comments (Atom)

Popular Posts

  • Long-Term Review: openSUSE 12.2 KDE
    I did this long-term review on my normal UROP desktop computer with the 64-bit edition of the OS. Follow the jump to see how it fared. Also ...
  • How-To: Make Xfce Like Unity
    This is more or less the sequel to this post. It came about because I wanted to see if it would be easy to make Xfce look like Apple's ...
  • SourceForge, Pages, and Respins
    I may have mentioned this in a previous post, but I have added new static pages to this blog. I wanted to mention this again as I will proba...
  • Review: Trisquel 4.0.1 LTS "Taranis"
    Main Screen + Main Menu I've read a couple of reviews of Trisquel GNU/Linux, an Ubuntu-based distribution which aims to remove as much n...
  • Review: Fedora 18 "Spherical Cow" GNOME
    Although I have reviewed a number of Fedora remixes, I haven't reviewed proper Fedora since the very first review/comparison test I post...
  • Review: KDE 4.6
    A couple days ago, KDE 4.6 was released for the world to enjoy. It boasts myriad bug fixes, new features for applications like Dolphin and M...
  • A Disappointing Review of #! 10 "Statler"
    Before I say anything else, I'd just like to say that the reason why I haven't posted anything in 2 weeks has been due to me being q...
  • Review: Linux Mint 14.1 "Nadia" MATE + GNOME 3/Cinnamon
    Wow. It's been a really long time since I've had the time to sit down and do a review like this. The reason for that is because this...
  • Review: Pardus 2013 KDE
    My spring break is coming to an end (I only have 1.5 more days), so I figured it might be nice to do another review while I still can. Today...
  • Review: Slackware 13.1
    KDE Main Screen I never envisioned myself trying out any of the more advanced distributions like Slackware, Arch, or Gentoo, but having trie...

Categories

  • 11
  • 13
  • 1st birthday
  • 200th post
  • 2010
  • 2011
  • 2012
  • 2D
  • 3 Idiots
  • 3D
  • 4
  • 600-series
  • 600C
  • 670C
  • 7
  • 7z
  • 8 glasses every day
  • A Short History of Nearly Everything
  • Abiword
  • abuse of copyright
  • Acer
  • ACTA
  • Activities
  • Adafruit
  • admission
  • Adobe
  • Adobe Flash
  • advertisement
  • Afghanistan
  • agricultural company
  • airport security
  • Albert-Laszlo Barabasi
  • amarok
  • amateur
  • amazon
  • Amy Chua
  • anaconda
  • android
  • AP
  • apology
  • apple
  • applications
  • April fools
  • aptosid
  • Arch
  • ArchBang
  • Arizona
  • asana
  • asthma
  • asus
  • Athena
  • ati
  • ATT
  • AUSTRUMI
  • autofailblog
  • autonomy
  • avatar
  • ayurveda
  • bad experience
  • ban
  • basmati rice
  • Ben Kevan
  • bias
  • Big Bang
  • big brother
  • Bill Bryson
  • biography
  • birthday
  • blackbox
  • blind
  • blog
  • blog catalog
  • Blogger
  • Blogilo
  • Blogspot
  • BMW
  • Bodhi Linux
  • bombing in russia
  • Book Review
  • bootloader
  • boson
  • brand name
  • break
  • breakfast cereal
  • Bridge Linux
  • British Chiropractic Association
  • broadcast
  • browser
  • BSD
  • Burj Khalifa
  • Bursts
  • bus
  • cable
  • calculus
  • cambridge
  • canonical
  • capitalism
  • care
  • Carolus Linnaeus
  • cell
  • cell phone
  • CentOS
  • central planning
  • CGS
  • Chak De India
  • Chakra
  • Chamber of Commerce
  • chat
  • cheese webcam booth
  • chemistry
  • chicken
  • chicken tax
  • china
  • choice
  • choqok
  • Chrome OS
  • Chromium
  • chrysler
  • Cinnamon
  • Cinnarch
  • City ID
  • class
  • codecs
  • coffee
  • college
  • commodore
  • Commonwealth Games
  • comparison
  • compatibility
  • competition
  • compositing
  • conference
  • congress
  • copyright
  • copyright infringement
  • corruption
  • counterfeiting
  • courts
  • Creative Commons
  • crunchbang linux
  • cryptography
  • crystal
  • CSS
  • CTKArchLive
  • custom linux spin
  • CwF + RtB
  • Daniel Craig
  • Das U-Blog by Prashanth
  • DEB
  • debian
  • debt
  • Dedoimedo
  • deficit
  • democrat
  • denial
  • Department of Justice
  • derivative
  • desktop effects
  • Die Another Day
  • disability
  • disappointment
  • disney
  • distribution
  • DMCA
  • DNA
  • dolphin
  • donation
  • dormitory
  • dream
  • DreamWorks
  • driver
  • DRM
  • Dubai
  • dvd
  • earthquake
  • Ease
  • ebook
  • economics
  • Edmunds
  • Edubuntu
  • education
  • educational
  • EFF
  • electricity
  • elementary
  • empathy
  • Enlightenment
  • enzo tedeschi
  • EPDFView
  • epiphany
  • essay
  • Evince
  • exam
  • excitement
  • eye of gnome
  • F-Spot
  • facebook
  • Faenza
  • familiarity
  • family
  • FBI
  • Featured Comments
  • fedora
  • Fedora Core
  • Feedbooks
  • felicia
  • Fermat's Last Theorem
  • Ferris Bueller's Day Off
  • fifa
  • file sharing
  • first
  • First Amendment
  • first sale
  • Fluxbox
  • Folder View
  • FOLLOW-UP
  • football
  • ford
  • free software
  • FreeTechie
  • frequency
  • FreshOS
  • frisk
  • frivolous
  • Fuduntu
  • Fusion
  • future
  • FVWM
  • Gabrielle Giffords
  • Gauss
  • GDM
  • gentoo
  • George Lucas
  • GhostBSD
  • GIMP
  • Gloobus
  • gloria
  • glyn moody
  • gm
  • Gnash
  • gnome
  • GNOME 3
  • GNOME Activities
  • GNOME Shell
  • gnu
  • Gnumeric
  • google
  • Google Docs
  • Gottfried Leibniz
  • government intervention
  • gparted
  • graduation
  • graphics card
  • GRUB
  • gtk+
  • GUI
  • gwenview
  • gwibber
  • Hackers
  • happy new year
  • hardware
  • Harry Potter
  • health
  • heartbeat
  • Higgs
  • high speed rail
  • hollywood
  • homeland security
  • homeless
  • honda
  • How To Train Your Dragon
  • How-To
  • hp
  • HTC
  • HTML
  • i386
  • ibm
  • Ice
  • Iceweasel
  • identity
  • In Defense of Food
  • incentives
  • Inception
  • india
  • Infinite Monkey Theorem
  • Inside Line
  • installation
  • Intel
  • intellectual monopoly
  • intellectual property
  • internet explorer
  • internship
  • Investopedia
  • ipad
  • iphone
  • iphone OS
  • ipod touch
  • Iraq
  • iron man 2
  • Isaac Newton
  • isadora
  • issues
  • ITworld
  • jailbreak
  • James Bond
  • james cameron
  • japanese
  • jill sobule
  • jim lynch
  • jon
  • Julia
  • Julian Assange
  • justice
  • KahelOS
  • Katya
  • KDE
  • kde 3.5
  • KDE 4
  • kde 4.4
  • KDE 4.5
  • KDE 4.6
  • KDE 4.7
  • KDE Activities
  • KevJumba
  • keyboard
  • Kinect
  • KOffice
  • kolourpaint
  • Kongoni
  • konqueror
  • Kopete
  • Kororaa
  • kpackagekit
  • KPresenter
  • kubuntu
  • kwin
  • Lage Raho Munna Bhai
  • laptop
  • last week of school
  • Latvia
  • law
  • lawsuit
  • learning experience
  • LED
  • legal fees
  • lenny
  • Leonard Mlodinow
  • LG
  • liar
  • libel
  • liberal
  • LibreOffice
  • LILO
  • linux
  • linux live cd
  • Linux Mint
  • Linux Today
  • Lisa
  • live cd
  • live dvd
  • live usb
  • long
  • Lubuntu
  • lunatic
  • LXAppearance
  • lxde
  • LXPanel
  • mac
  • mac os x
  • Madbox
  • madurai
  • Mageia
  • mainstream tech press
  • malware
  • mandriva
  • Manjaro Linux
  • marginal cost
  • mark shuttleworth
  • Mark Zuckerberg
  • market
  • market share
  • massacre
  • mastery
  • MATE
  • mathematics
  • Mayans
  • MBodhi Linux
  • mcps
  • meat
  • mebibyte
  • media
  • media companies
  • medicine
  • MEEP
  • Megabus
  • megabyte
  • mepis
  • Metacity
  • metric system
  • MGSE
  • Michael Nielsen
  • Michael Pollan
  • mickey mouse
  • microsoft
  • microsoft office
  • middle
  • Midori
  • misconceptions
  • misrepresentation
  • MIT
  • MLB
  • Mokshagundam
  • money
  • monopoly
  • mouse
  • movie
  • Movie Review
  • Mozilla
  • Mozilla Firefox
  • Mozilla Prism
  • mpaa
  • multiboot
  • MultiSystem
  • MWM
  • national health service
  • national security
  • nautilus
  • NCAA
  • ncurses
  • netbook
  • Netrunner
  • neutrino
  • new computer
  • new york
  • new york city
  • new zealand
  • newbie
  • news corp
  • NFL
  • NHS
  • NIST
  • normal distribution
  • novell
  • numbers
  • nutrition science
  • nutritionism
  • NVidia
  • NZCS
  • obama
  • okular
  • One
  • open standards
  • open-source
  • openbox
  • openoffice.org
  • opensolaris
  • openSUSE
  • Opera
  • oracle
  • Oxidized Trinity
  • P. W. Singer
  • panel
  • paramount
  • Pardus
  • parenting
  • parody
  • particle
  • patent
  • pay-to-pirate
  • PC-BSD
  • pclinuxos
  • pcmanfm
  • Pear OS
  • pearson education
  • Peppermint OS
  • Peter Pan
  • philosophy
  • Photograph 51
  • photonic
  • PHP
  • physics
  • pidgin
  • Pierce Brosnan
  • Pinguy OS
  • pink
  • Pink Floyd
  • piracy
  • plasma
  • plasmoid
  • poll
  • Porteus
  • power
  • power law
  • prejudice
  • presentation
  • president
  • president obama
  • presumption of innocence
  • Princeton
  • printing
  • prisoner
  • privacy
  • profit
  • programming
  • progress
  • Project Natal
  • promotion
  • proprietary
  • public domain
  • purpose
  • qed
  • QEMU
  • qt
  • quantum electrodynamics
  • quantum mechanics
  • radio
  • rape
  • Rawhide
  • Razor-Qt
  • red hat
  • Reflection
  • Rekonq
  • religion
  • Remastersys
  • rent
  • repossession
  • republican
  • retroactive copyright
  • review
  • rewards
  • RHEL
  • RIAA
  • Righthaven
  • RMA
  • robotics
  • rolling release
  • rootkit
  • ROSA
  • royalty
  • RPM
  • RSS
  • rule
  • rupert murdoch
  • sabayon
  • safari
  • saints
  • Salix OS
  • Samsung
  • sarah palin
  • Saudi Arabia
  • scanner
  • school
  • school network
  • science
  • Scientific Linux
  • security theater
  • selection
  • semester
  • Semplice
  • senior
  • Shiki
  • shooting
  • Shotwell
  • shut down
  • SI
  • sidux
  • Simon Singh
  • simplymepis
  • Skype
  • skyscraper
  • Slackware
  • slander
  • slashdot
  • social media
  • social policy
  • socialism
  • software patents
  • solar
  • SolusOS
  • SolydXK
  • sony
  • sony-bmg
  • SOPA
  • Source Code
  • SourceForge
  • SPARC
  • special effects
  • spying
  • spyware
  • Squeeze
  • SSH
  • Star Wars
  • State Department
  • statin
  • statistical mechanics
  • Statler
  • Stella
  • steve jobs
  • stewart
  • Stuxnet
  • subscriber
  • subsidy
  • substitute
  • sun
  • Sun Tzu
  • super bowl
  • Super Bowl XLV
  • super user
  • Suresh Kalmadi
  • survey
  • Symbicort
  • synaptic
  • tablet
  • Talledega Nights
  • tax
  • tech company
  • Tech Drive-in
  • techdirt
  • Technorati
  • Ted Williams
  • terrorist
  • thanksgiving
  • The Adjustment Bureau
  • The Amazing Race
  • The Art of War
  • The Code Book
  • The Drunkard's Walk
  • The King's Speech
  • The Social Network
  • the tunnel
  • the undercover economist
  • thermophotovoltaic
  • thunar
  • tim
  • Tim Harford
  • tint2
  • torrent
  • Toy Story 3
  • toyota
  • tracking device
  • trademark
  • train
  • treason
  • Trinity
  • Trisquel
  • trivial
  • troll
  • TSA
  • TuxMachines
  • Twitter
  • TWM
  • UberBang
  • ubuntu
  • ubuntu one
  • UK
  • unetbootin
  • unintended acceleration
  • units
  • Unity
  • Unixoid Review
  • UROP
  • US
  • utopia
  • V. S. Narayana Rao
  • VectorLinux
  • vegan
  • vegetarian
  • Verizon
  • vesa
  • Viewnior
  • ViewPad
  • ViewSonic
  • violation
  • virtual desktop
  • VirtualBox
  • virus
  • Visvesvaraya
  • vlc
  • warfare
  • water
  • WattOS
  • wavelength
  • Wayland
  • web-connected printer
  • webcam
  • WebOS
  • weekly
  • whistle
  • widget
  • wifi
  • wiki
  • Wikileaks
  • William Shakespeare
  • windowing system
  • WindowMaker
  • windows
  • windows 7
  • windows vista
  • windows xp
  • Wired for War
  • word
  • WordPress
  • world cup
  • Wubi
  • x11
  • XBMC
  • xbox360
  • xfce
  • xkcd
  • xp
  • yahoo
  • yoga
  • YouTube
  • YSA
  • Zenwalk
  • Zorin OS

Blog Archive

  • ▼  2013 (63)
    • ►  September (4)
    • ►  August (9)
    • ►  July (3)
    • ►  June (9)
    • ▼  May (7)
      • Review: SolydXK 2013.04.06
      • Featured Comments: Week of 2013 May 26
      • Done with 6th Semester!
      • Review: Korora 18 "Flo" KDE
      • Featured Comments: Week of 2013 May 12
      • Review: CrunchBang ("#!") Linux 11 "Waldorf"
      • Expected Utility in Quantum States
    • ►  April (10)
    • ►  March (11)
    • ►  February (6)
    • ►  January (4)
  • ►  2012 (85)
    • ►  December (4)
    • ►  November (3)
    • ►  October (2)
    • ►  September (6)
    • ►  August (8)
    • ►  July (12)
    • ►  June (12)
    • ►  May (7)
    • ►  April (6)
    • ►  March (6)
    • ►  February (9)
    • ►  January (10)
  • ►  2011 (179)
    • ►  December (5)
    • ►  November (8)
    • ►  October (9)
    • ►  September (12)
    • ►  August (15)
    • ►  July (15)
    • ►  June (15)
    • ►  May (16)
    • ►  April (15)
    • ►  March (19)
    • ►  February (21)
    • ►  January (29)
  • ►  2010 (173)
    • ►  December (24)
    • ►  November (23)
    • ►  October (34)
    • ►  September (36)
    • ►  August (15)
    • ►  July (18)
    • ►  June (13)
    • ►  May (8)
    • ►  April (2)
Powered by Blogger.

About Me

Unknown
View my complete profile