Theory of Quantum Computation, Communication and Cryptography: 5th Conference, TQC 2010, Leeds, UK, April 13-15, 2010, Revised Selected Papers (Lecture ... Computer Science and General Issues)

194 Pages·2011·3.02 MB·English

by  Wim van Dam

Lecture Notes in Computer Science 6519
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Wim van Dam Vivien M. Kendon
Simone Severini (Eds.)
Theory of Quantum
Computation,
Communication,
and Cryptography
5th Conference, TQC 2010
Leeds, UK, April 13-15, 2010
Revised Selected Papers
1 3
VolumeEditors
WimvanDam
UniversityofCalifornia
DepartmentofComputerScience
SantaBarbara,CA93106-5110,USA
E-mail:[email protected]
VivienM.Kendon
UniversityofLeeds
SchoolofPhysicsandAstronomy
Leeds,LS29JT,UK
E-mail:[email protected]
SimoneSeverini
UniversityCollegeLondon
DepartmentofPhysicsandAstronomy
London,WC1E6BT,UK
E-mail:[email protected]
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ISBN-13 978-3-642-18072-9SpringerBerlinHeidelbergNewYork
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Preface
The Conference on Theory of Quantum Computation, Communication, and
Cryptography(TQC)isanannualmeetingontheoreticalaspectsofquantumin-
formationprocessing.Thegoaloftheconferenceistofosterdevelopmentsinthis
rapidlygrowing,interdisciplinaryfieldbyprovidingaforumforthepresentation
and discussion of original research.
The fifth iteration of TQC was held during April 13–15, 2010, at the Uni-
versity of Leeds, United Kingdom. It included invited talks, contributed talks,
andapostersession,aswellasarumpsessionconsistingofshorttalksonrecent
developments. Authors of selected contributed talks were invited to submit a
paper to these proceedings.
TQC 2010 would not have been possible without the contributions of nu-
merous individuals and organizations, and we sincerely thank them for their
support.
Inputting togetherthescientificprogram,wewereverygratefulforthehard
work and advice of the Program Committee, listed herein. The logistics of the
conference were expertly managed by the Organizing Committee, also listed
herein, and we thank them for their efforts to make the conference a success.
We wouldliketo thanktheinvitedspeakers,Fr´ed´ericMagniez,KaeNemoto,
Frank Verstraete, Ronald de Wolf, and Anton Zeilinger, for their contributions
to the program.
We wouldlike to thank the members ofthe ConferenceSeries SteeringCom-
mittee,YasuhitoKawano,MicheleMosca,andVlatkoVedral,fortheirimportant
advice.
TQC2010wasmadepossiblebyfinancialsupportfromtheBritishComputer
Society,theHeilbronnInstitute,theQuantumInformation,QuantumOpticsand
Quantum ControlGroup of the Institute of Physics,the School of Mathematics
oftheUniversityofLeeds,theSchoolofPhysicsandAstronomyoftheUniversity
of Leeds, the London Mathematical Society, the Sandia National Laboratories,
the Institute for Quantum Computing at the University of Waterloo, and the
Worldwide Universities Network, Leeds; we thank these organizations for their
important contributions.
Finally, we would like to thank Springer for publishing the proceedings of
TQC in the Lecture Notes in Computer Science series.
October 2010 Wim van Dam
Vivien Kendon
Simone Severini
Organization
Program Committee
Wim van Dam University of California, Santa Barbara (Chair), USA
Simone Severini University College London (Co-chair), UK
Dagmar Bruß Heinrich Heine University, Germany
Andrew Childs University of Waterloo, Canada
Matthias Christandl Ludwig Maximilians University, Germany
Nilanjana Datta University of Cambridge, UK
Aram Harrow University of Bristol, UK
Peter Høyer University of Calgary, Canada
Rahul Jain National University of Singapore
Elham Kashefi University of Edinburgh, UK
Debbie Leung University of Waterloo, Canada
Hoi-Kwong Lo University of Toronto, Canada
Juan Pablo Paz University of Buenos Aires, Argentina
Francesco Petruccione University of KwaZulu-Natal, South Africa
David Poulin Universit´e de Sherbrooke, Canada
Martin Ro¨tteler NEC, Princeton, USA
Miklos Santha Universit´e Paris Sud, France
Seiichiro Tani NTT, Tokyo, Japan
Jean-Pierre Tillich INRIA, Rocquencourt, France
Pawel Wocjan University of Central Florida, USA
Organizing Committee
Vivien Kendon University of Leeds (Chair), UK
Martin Aulbach University of Leeds, UK
Dave Bacon University of Washington, USA
Stephen Bartlett University of Sydney, Australia
Katie Barr University of Leeds, UK
Stephen Brierley University of York, UK
Katherine Brown University of Leeds, UK
Barry Cooper University of Leeds, UK
Peter Crompton University of Leeds, UK
Vladimir V. Kisil University of Leeds, UK
Neil Lovett University of Leeds, UK
Stefano Pirandola University of York, UK
Mike Stannett University of Sheffield, UK
Rob Wagner University of Leeds, UK
Table of Contents
Asymptotically Optimal Discrimination between Pure Quantum
States .......................................................... 1
Michael Nussbaum and Arleta Szko(cid:2)la
On Quantum Estimation, Quantum Cloning and Finite Quantum de
Finetti Theorems ................................................ 9
Giulio Chiribella
Simple Sets of Measurements for Universal Quantum Computation and
Graph State Preparation.......................................... 26
Yasuhiro Takahashi
Computational Depth Complexity of Measurement-Based Quantum
Computation .................................................... 35
Dan Browne, Elham Kashefi, and Simon Perdrix
Local Equivalence of Surface Code States ........................... 47
Pradeep Sarvepalli and Robert Raussendorf
Testing Non-isometry Is QMA-Complete ............................ 63
Bill Rosgen
Quantum Search with Advice...................................... 77
Ashley Montanaro
Simulating Sparse Hamiltonians with Star Decompositions ............ 94
Andrew M. Childs and Robin Kothari
The PolynomialDegree of Recursive Fourier Sampling ................ 104
Benjamin Johnson
Generalized Self-testing and the Security of the 6-State Protocol ....... 113
Matthew McKague and Michele Mosca
A Conceptually Simple Proof of the Quantum Reverse Shannon
Theorem........................................................ 131
Mario Berta, Matthias Christandl, and Renato Renner
Geometric Entanglement of Symmetric States and the Majorana
Representation .................................................. 141
Martin Aulbach, Damian Markham, and Mio Murao
Monogamy of Multi-qubit Entanglement in Terms of R´enyi and Tsallis
Entropies ....................................................... 159
Jeong San Kim and Barry C. Sanders
VIII Table of Contents
Bypassing State Initialisation in Perfect State Transfer Protocols on
Spin-Chains ..................................................... 168
C. Di Franco, M. Paternostro, and M.S. Kim
Teleportation of a Quantum State of a Spatial Mode with a Single
Massive Particle ................................................. 175
Libby Heaney
Author Index.................................................. 187
Asymptotically Optimal Discrimination between
Pure Quantum States
Michael Nussbaum1,(cid:2) and Arleta Szkol(cid:2)a2
1 Department of Mathematics, Cornell University,Ithaca NY,USA
2 Max Planck Institutefor Mathematics in theSciences, Leipzig, Germany
Abstract. We consider the decision problem between a finite number
ofstatesofafinitequantumsystem,whenanarbitrarilylargenumberof
copiesofthesystemisavailableformeasurements.Weprovideanupper
bound on the exponential rate of decay of the averaged probability of
rejecting the true state. It represents a generalized quantum Chernoff
distance of a finite set of states. As our main result we prove that the
bound is sharp in the case of purestates.
Keywords: multiple quantum state discrimination, generalized quan-
tum Chernoff distance, quantum hypothesistesting, error exponents.
1 Introduction
Invariousbranchesofquantumtheorysuchasquantuminformationprocessing,
quantumcommunicationtheoryorquantumstatisticsoneofthe basicproblems
istodeterminethestateofagivenquantumsystem.Inthesimplestcasethereis
a finite set of states specifying the possible preparationof the quantum system.
In the Bayesian approach of quantum statistics, the likelihood of the different
statesisdeterminedbyanaprioriprobabilitydistribution.Onemakesadecision
in favor of one of the states following a specified rule based on the outcomes of
a generalized measurement -called a quantum test. In the binary case optimal
tests, i.e. tests minimizing the averaged probability of rejecting the true state,
are known to be given by Holevo-Helstromprojections [5], [4]. These generalize
the classical likelihood ratio tests. Here we consider the scenario where there is
an arbitrarily large finite number n of copies of the quantum system available
for performing a measurement. The corresponding state is then described by
an n-fold tensor product of one of the associated density operators. There are
two main goals: firstly, to construct a sequence of quantum tests in n which
maximizetheasymptotic(exponential)rateofdecayoftheaveragedprobability
of rejecting the true state. The second goal is to determine the corresponding
optimalerrorexponent.Ithasbeenshownthatinthebinarycaseasymptotically
optimalquantumtests, thus in particularthe Holevo-Helstromtests,achievean
exponentialrateofdecaywhichisequaltothe quantumChernoffbound,cf.[8],
[1] and [2]. Surprisingly, the corresponding questions in the case of r >2 states
(cid:2) Supportedin part byNSFgrant DMS-08-05632.
W.vanDametal.(Eds.):TQC2010,LNCS6519,pp.1–8,2011.
(cid:2)c Springer-VerlagBerlinHeidelberg2011
2 M. Nussbaum and A. Szko(cid:3)la
have not yet received a final answer, despite a number of efforts and numerous
strongresultsobtainedinrelationtomultiple quantumstatediscrimination,see
[11], [7], [3], [10] and references therein.
We define a generalizedquantum Chernoff distance of a finite set of states as
the minimum of the binary quantum Chernoff distances over all possible pairs
of different states. The binary quantum Chernoff distance has been introduced
in the context of binary quantum hypothesis testing in [8]. Relying on [8] we
prove that the generalized quantum Chernoff distance specifies a bound on the
achievableasymptoticerrorexponentsinmultiplequantumstatediscrimination.
Thisisinlinewithresultsobtainedinthecontextofclassicalmultiplehypothesis
testing, cf. [9]. As our main result we prove that in the special case of pure
quantumstatesthisbound,indeed,isachievableandhencespecifiestheoptimal
asymptoticerrorexponent.The correspondingasymptoticallyoptimalquantum
tests rely on a Gram-Schmidt orthonormalization procedure of the associated
state vectors.Similar quantum tests were already consideredby Holevo in [6] in
the contextofquantumminimal errordecisionproblems.However,the question
of the corresponding asymptotic error exponent is not addressed in [6].
2 Notations and the Main Results
LetS beafinitequantumsystemandHbetheassociatedcomplexHilbertspace
with dimH = d<∞. Further denote by A the algebra of observables of S, i.e.
A is the algebra of linear operators on H. For each n ∈ N denote by A(n) the
algebra of linear operators on the n-fold tensor product Hilbert space H⊗n. It
represents the algebra of observables of a compound quantum system Sn with
its n unit systems being of the same type S.
For eachn∈N the setofdensity operatorsinA(n) correspondsone-to-oneto
the state space S(A(n)) of A(n). Recall that a density operator is defined to be
a self-adjoint, positive linear operator of trace 1.
Let r ∈ N and Σ be a set of density operators ρi ∈ S(A), i = 1,...,r,
representingthe possible states of the quantumsystem S. Assume that for each
n ∈ N there is a compound quantum system Sn being an n-fold copy of S.
This means, in particular, that the corresponding quantum state is in Σ⊗n :=
{ρ⊗n}r , i.e. it is uniquely determined by the index i∈{1,...,r}.
i i=1
Further,letE(n) ={E(n)}r beapositiveoperatorvaluedmeasure(POVM)
i i=1
inA(n),i.e.eachE(n),i=1,...,r,isaself-adjointelementofA(n) withE(n) ≥0
(cid:2) i i
and r E(n) =1.ThePOVMsE(n) describequantumtestsfordiscrimination
i=1 i
betweenther statesfromΣ⊗n,orsimplyquantum tests for Σ⊗n,byidentifying
the measurement outcome corresponding to E(n), i=1,...,r, with the density
i
operator ρ⊗i n, respectively. If ρi happens to describe the true state of S, and
correspondingly ρ⊗i n determines the state of Sn, then the associated individual
success probability is given by
Succi(E(n)):=tr [ρ⊗i nEi(n)] .
Asymptotically Optimal Discrimination between Pure QuantumStates 3
Theindidvidualerrorprobabilityreferstothesituationwhenthedensityoperator
ρi is discarded as possible preparation of S; it is given by the formula
Erri(E(n)):=tr [ρ⊗i n(1−Ei(n))] .
(cid:2)
Assuming 0 < pi < 1, i = 1,...,r, with ri=1pi = 1 to be the a priori distri-
bution of the r quantum states from Σ the averaged error probability is defined
by
(cid:3)r
Err(E(n))= pitr [ρ⊗i n(1−Ei(n))] .
i=1
Ifthelimitlimn→∞−n1 logErr(E(n))exists,werefertoitastheasymptoticerror
exponent. Otherwise we have to consider the corresponding limsup and liminf
expressions.
Fortwodensityoperatorsρ andρ thequantum Chernoff distance isdefined
1 2
by
ξQCB(ρ1,ρ2):=−log0≤ins≤f1tr [ρ11−sρs2] . (1)
It specifies the optimal achievable asymptotic error exponent in discriminating
between ρ and ρ , compare [8], [1], [2]. Quantum tests with minimal averaged
1 2
error probability for a pair of different density operatorsρ and ρ on the same
1 2
Hilbert space H are well-known to be given by the respective Holevo-Helstrom
projectors
Π :=supp (ρ −ρ ) , Π :=supp (ρ −ρ ) =1−Π .
1 1 2 + 2 2 1 + 1
Here supp a denotesthe supportprojectorofa self-adjointoperatora,while a
+
means its positive part, i.e. a = (|a|+a)/2 for |a| := (a∗a)1/2, see [5], [4]. As
+
mentioned in the introduction, the Holevo-Helstrom projectors generalize the
likelihood ratio tests for two probability distributions. This can be verified by
letting ρ and ρ be two commuting density matrices, cf. [8].
1 2
For a set Σ = {ρi}ri=1 of density operators on H, where r > 2, we introduce
the generalized quantum Chernoff distance
ξQCB(Σ):=min{ξQCB(ρi,ρj): 1≤i<j ≤r} . (2)
This is in full analogy to the definition of the generalized Chernoff distance in
classical multiple hypothesis testing, where the density operators are replaced
by probability distributions on a finite sample space, cf. [9].
Our first theorem is an implication of Theorem 2.2 in [8].
Theorem 1. Let r ∈ N and Σ = {ρi}ri=1 be a set of pairwise different density
operators on H with corresponding a priori probability distribution {pi}ri=1. For
any sequence E(n), n∈N, of quantum tests for Σ⊗n, respectively, it holds
1
limsup−nlogErr(E(n))≤ξQCB(Σ) ,
n→∞
where ξQCB(Σ) is the generalized quantum Chernoff distance defined by (2).