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The Uncertainty Principle in the Presence of Quantum Memory Matthias Christandl, ETH Zurich based on joint work with M. Berta, R. Colbeck, J. Renes and R. Renner arXiv:0909.0950 Nature Physics 6, 659 - 662 (2010) The Uncertainty Principle • „In quantum mechanics, the Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision.“ (Wikipedia) The Uncertainty Principle • „In quantum mechanics, the Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known to predicted with arbitrary precision.“ (Wikipedia) 1 any she chooses, there is a measure quantified by Heisenberg using the standard deviation. rt of measurement fferFor two observables, R and S, the resulting bound on the 1 as memory which gives thestandard same deviation. outcome quantified by Heisenberg using the ton ofBob’s o the uncertainty can be expressed in terms of the commutafferFor two observables, R andobservables, S, the resulting bound on the obtains. Hence, for any R and S, th ciple tor [5]: the uncertainty can be expressed in terms of the commutarcertainties inH(S)1 vanish, in clear violati iple tor Robertson [5]: H(R) and form h be (1) ∆R generally, · ∆S ≥ |�[R, S]�|. r,Equation in(2). More the violation depen 21 the h be S]�|. the system (1) an ∆R · ∆S ≥ |�[R, servthe amount of entanglement between Variance of Inspired by information theory, uncertainty has since 2 the ured measurement result beenmemory. quantified using the Shannon Entropy [6]. The quantum erved in Inspired by information theory, uncertainty has since first uncertainty relations of this type were by Bia� l ynickiured mutabeen quantified using the Shannon Entropy [6]. The Birula and Mycielski [7] and Deutsch [8]. Later, Maassen dthis in Entropic form first uncertainty relationsDeutsch’s of this type were by Bia�that lynickiand Uffink [9] improved result to show utaeigenbasis of S the Birula and Mycielski [7] and Deutsch [8]. Later, Maassen eigenbasis of R this per-Shannonand entropy of 1 to show that Uffink [9] improved Deutsch’s result 1 the , (2) H(R) + H(S) ≥its log2standard measurement result HowFor observable R, we denote deviation ∆ c p perrver �R2 � − �R�2 , where �R� denotes the1 expectation value o , of the proba(2) H(R) + H(S) ≥ log Bialynicki-Birula, Mycielski, Deutsch, 2 Howwhere H(R) denotes the Shannon entropy unre2 2 For c max Kraus, Maassen and Uffink non-degenerate observables, c := |�ψ |φ �| j j,k rver bility distribution of the outcomes when R is measured. k logi1are � and |φ � the eigenvectors of R and S, respectivel |ψ j k where H(R) denotes the Shannon entropy of the probanrequantifies complementarity of the observThe term n the c bility 2distribution of the outcomes when R is measured. ogi- The Uncertainty Principle • • 1 nbergbymeasured tified Heisenberg using the standard deviation. been quantified using the Shannon Entropy [6] n be boundedRinand S, the resulting bound on the wo observables, first uncertainty relations of this type were by Bia of the can commutatainty be expressed in terms of the commutaBirula and Mycielski [7] and Deutsch [8]. Later, M plication of this ]: and Uffink [9] improved Deutsch’s result to show nnot predict the easurements per1 1 (1) ≥ log2 , · ∆S ≥ |�[R, S]�|. H(R) + H(S) precision. ∆R How2 c d if the observer red by information theory, uncertainty has whereonH(R) theissince Shannon entropy of the emory, an unre-depends Variance valuedenotes Entropy good measure quantified using Shannon Entropyof[6]. The bility distribution the outcomes when R is me cent technologiof the observable of ignorance 1 uncertainty relations of this type by Bia�the lynickiquantifies complementarity of the The term cwere e strengthen the 2 at and Mycielski Deutsch [8]. Later, Maassen applies even[7] if andables, similarly to the commutator in Equation ( State independent lower Uffink improved resulttotothink showabout that uncertainty relations BoundDeutsch’s can be zero mory.[9]QuantifyOne way bound ropy, we provide the following game (the uncertainty game) betwe 1 and Bob. Before the game comm nty of the outplayers, Alice (2) H(R) + H(S) ≥ log2 , Based on measurements advanced c hich depends on Alice and Bob agree on two R and Elementary to prove ween the system game proceeds astheorem follows: from Bob analysis creates a quantum eart H(R) denotes the Shannon entropy of the probafrom its sigof his choosing and sends it to Alice. Alice then pe distribution of the outcomes when R is measured. Goal of this talk: uantum physics, one of the two measurements and announces her ch 1 quantifies the complementarity of the observerm c generalisation of the entropic form ted the first promeasurement to Bob. Bob’s task is then to minim 2 similarly to the commutator in Equation (1). The Uncertainty Principle • I. FORMULAE R = σ Examplez RR==σσz|ψz0� = |0� • Measurement of spin 1/2 particle 2 |ψ1 � = |1� S = σ S=σ x R = σz S = σxx 1 √ |φ0 � = (|0� + |1�) 2 1 |φ1 � = √ (|0� − |1�) 2 |ψ0 � = |0� |ψ1 � = |1� S = σx H(R) H(S) ≥ 1 1+H(S) 1 H(R) + ≥ 1 √ (|0� |φ0 � = √ (|0� |1�) |φ1 � =≥ H(R) ++ H(S) 1 − |1�) 1 2 2 √ |φ0 � = (|0� + |1�) 2 ForII. mutually bases UNCERTAINTY (e.g. Fourier) OF THE ENTROPIC RELATION 1STATEMENTunbiased |φ1 � = √ (|0� − |1�) H(R) H(S)≥≥log logdd H(R) H(R) + 2H(S) ≥ 1++H(S) • H(R) + H(S) ≥ log d The statement of our uncertainty relation involves two measurements described by orth **insert introduction** mal bases {|ψj �} and {|φk �} in a d-dimensional Hilbert space (note that the observables ar uter Science, ETH Zurich, 8092 Zurich, Switzerland. nische Universität Darmstadt, 64289 Darmstadt, Germany. Dated: October 31, 2009) The Uncertainty Game Alice‘s quantified by Heisenberg using the standard deviation.1 result? For two observables, R and S, the resulting bound on the 2 heart of c differk to the uncertainty can be expressed in terms of the commutaprinciple tor [5]: , for in1 both be (1) ∆R · ∆S ≥ |�[R, S]�|. 2 sely, the observInspired by information theory, uncertainty has since easured been quantified using the Shannon Entropy [6]. The Minimise nded in first uncertainty relations of this type were by Bia�lynickimmuta-FIG. 1: Illustration of the uncertainty game. (1) Bob sends a state to Alice, which may, in general,uncertainty be entangled with his Birula and Mycielski [7] and Deutsch [8]. Later, Maassen memory. (2) Alice measures either R or S and notes her outcome. (3) Alice announces her measurement choice to of thisquantum Bob. Bob’s goal is then to minimize his uncertainty about Alice’s measurement outcome. her and Uffink [9] improved Deutsch’s result to about show that dict the result We now proceed to state our uncertainty relation. It given the quantum memory is always greater than nts per-holds in the presence of quantum memory and provides a 0. As discussed above, 1 Bob can guess both R and on the uncertainties of the measurement outcomes S perfectly with such a, strategy. (2) H(R) + H(S) ≥ log Limit: 2 n. How-bound which depends on the amount of entanglement between c 2. If A and B are not � entangled, i.e. the state takes the system, A, and the quantum memory, B. Mathematthe form ρ = p ρ ⊗ ρ , for probabilities p observerically, it is the following relation: 3 AB j j j A j B j � = |0� The|ψ Uncertainty Game 0 |ψ1 � = |1� Alice‘s result? 2 max. entangled 1 |φ0 � = √ (|0� + |1�) 2 1 |φ1 � = √ (|0� − |1�) 2 Ui ⊗ Ūi ( � |i�|i�) = � |i�|i� guesses correctly FIG. 1: Illustration of the uncertainty game. (1) Bob sends a state to Alice, which may, in general, be entangled with his conditional entropy quantum memory. (2) Aliceimeasures either R or S iand notes her outcome. (3) Alice announces her measurement choice to Bob. Bob’s goal is then to minimize his uncertainty about Alice’s measurement outcome. Alice-Bob system after her measurement We now given the quantum memory is always greater than with Rproceed to state our uncertainty relation. It holds in the presence of quantum memory and provides a bound on the uncertainties of the measurement outcomes which depends on the amount of entanglement between the system, A, and the quantum memory, B. Mathematically, it is the following relation:3 H(R|B) + H(S|B) = 0 0. As discussed above, Bob can guess both R and S perfectly with such a strategy. 2. If A and B are not i.e. the state takes � entangled, j j the form ρAB = j pj ρA ⊗ ρB , for probabilities pj The Uncertainty Game 2 partially entangled FIG. 1: Illustration of the uncertainty game. (1) Bob sends a sta quantum memory. (2) Alice measures either R or S and notes her Bob. Bob’s goal is then to minimize his uncertainty about Alice’s m We now proceed to state our uncertainty relation. It holds in the presence of quantum memory and provides a bound on the uncertainties of the measurement outcomes which depends on the amount of entanglement between conditional entropy the system, A, and the quantum memory, B. MathematFIG. 1: Illustration of the uncertainty game. (1) Bob sends a state to Alice, which may, in general, be entangled with his Alice-Bob system 3 Alice announces her measurement quantum memory. (2) Alice measures either R or S and notes her outcome. (3) choice to ically, it is the following relation: Bob. Bob’s goal is then to minimize his uncertainty about Alice’s measurement outcome. before her measurement New Limit: 1 + H(A|B). H(R|B) + H(S|B) ≥ log c We now proceed to state our uncertainty relation. It holds in the presence of quantum memory and provides a bound on the uncertainties of the measurement outcomes which depends on the amount of entanglement between the system, A, and the quantum memory, B. Mathematically, it is the following relation:3 given the quantum memory is always greater than 0. As discussed above, Bob can guess both R and 2 S perfectly with such a strategy. (3) 2. If A and B are not i.e. the state takes � entangled, j j the form ρAB = j pj ρA ⊗ ρB , for probabilities pj are not entangled, i.e. the 1 2. If A and B � both be memory, quantum memory, B. Mathemat� m,,e cannot A, and the quantum B. Mathematj(1) j j j |�[R, S]�|. ∆Ramount · ∆S ≥ The Uncertainty Principle in the P the form ρ = p ρ ⊗ ρ , for pro which depends on the of entanglement between the form ρ = p ρ ⊗ ρ , for proba 3 3 AB j AB j 2 A B j ore precisely, the A B j Bob’s uncertainty about the mea therelation: following relation: ng 2. j j j j the system, A, and the quantum memory, B. has Mathematf any two observand quantum states ρvon and ρthen , then H andBerta, quantum states ρ and ρ , H( 1the 1 by conditional Neumann e A B Inspired by informationMario theory, uncertainty since A B Matthias Christandl, Roger Colbeck 3 Since H(R|B) ≤ H(R) and H(S|B) 1is the following relation: enberg measured ically, it 1 Since H(R|B) ≤ H(R) and H(S|B) ≤ 1 directly been quantified using the Shannon Entropy [6]. The generalizes the Shannon + H(A|B). (3) (R|B) + H(S|B) ≥ log Faculty of Physics, Ludwig-Maximilians-Univer 2 (3) (S|B) ≥ log2 + an be bounded in H(A|B). all states, weBia� recover Maassen and Uffi c 2 first uncertainty relations of this type were by l ynickiall states, we recover Maassen and Uffink Institute for Theoretical Physics, ETH c case that(2). Bob has a quantum m of the commuta3 Equation Institute of Theoretical Computer Science, E 1 Birula and Mycielski [7] and Deutsch [8]. Maassen 4 Later, Equation (2). The additional term H(A for S. ertainty about the measurement R+ is H(S|B) denoted4 Perimeter mplication of this + H(A|B). (3) H(R|B) ≥ log Institute for Theoretical Physics, 31 Caroline 2Inresult and Uffink [9] improved Deutsch’s to show that out the measurement R is denoted 3. the absence of quantifies theTechnische quantum mem 5 c ditional von Neumann nnot predict the entropy, H(R|B), which 3. In Institute for hand Applied side Physics, Univers right the am the absence of the quantum memor neneralizes Neumann can 1reduce the bound (3) to H(R) (Dated: June 2 theentropy, Shannon entropy, which H(R), to the easurements per- H(R|B), between the system and the mem 1 can reduce the bound (3) to H(R) + log +H(A). If the state of the system , (2) H(R) + H(S) ≥ he Shannon entropy, H(R), to the 2 yBob precision. Howhas a quantum B, andabout likewisethe measurement Bob’smemory, uncertainty Rexamples: is denoted cc 1 instructive log2 c then +H(A). state the system, H(A)If=the 0 and weofagain recover3.tA quantum memory, B, and likewise id if the observer he additional term H(A|B) appearing on the by the conditional von Neumann entropy, H(R|B), which Maassen Uffink, Equation (2). the How then H(A) = 0and we again recover t where of H(R) denotes the Shannon entropy ofand the probamemory, an unredalside quantifies theappearing amount entanglement term H(A|B) on the directly generalizes the Shannon entropy, H(R), to the 1. If the state, A, and memo system, A, is in a mixed state then H(I bility distribution of the outcomes when R is measured. Maassen and Uffink, Equation (2). Howe ecent technologihe system and the memory. We discuss some 1 tifies the amount of entanglement 1 case that a quantum memory, B, and p H(R|B) + H(S|B) ≥ loglikewise theA, resulting bound isstate stronger than E quantifies the complementarity of athe observTheBob term has 2 then entangled, H(A|B) = we strengthen the eand examples: system, is in mixed then H(A c c the memory. We some 4 discuss t 2 even when there is no quantum memor The similarly additional term H(A|B) appearing on the forifS. ables, at applies even to the commutator in Equation (1). dimension of the system the resultingthe bound is stronger than Equ c 1 0 emory. One way to think about uncertainty relations is via he state, QuantifyA, and memory, B, are maximally right hand side quantifies the amount of2h(�) entanglement log exceed logmost even is cannot noapplications, quantum memory. 4. when In terms of the = 1there − 2 cnew 2 d,w ropy, we following game uncertainty game) between two ngled, thenprovide H(A|B) =the− log2system d, where d(the isthe between the and memory. We discuss case is when A and Bsome are entangled, bs to H(R|B)+H(S|B) ≥ 0, w ,dimension and memory, B, are maximally inty of the outplayers, andSince Bob.4. Before theofgame commences, of the system sent toAlice Alice. In terms new applications, the most iu imally so. The resulting bound is then instructive examples: 1 conditional entropy of a sys K ≥ H(R|E)R−and H(R|B) nhich H(A|B) = −on log d, where d(3)isagree depends Alice and Bob on two measurements S. are Theentangled, butq cannot exceed log d, the bound reduces 2 2 case isand, when A and B c since the conditional entropy H(A tween the system of the system sent to Alice. Since game proceeds as follows: Bob creates a quantum state (R|B)+H(S|B) ≥ 0, which is trivial, since the imallyative, so. The resulting boundfrom is then the bound is different the ent part from its sigIfofthe state, A, sends and itmemory, B, are maximally his choosing and to Alice. then performs a itional entropy of a1. system after measurement 1 Alice xceed log bound (3) reduces 2 d, the 4. sical ones. and, the conditional entropy H(A|B ≥and logsince − H(R|B) −choice H(S|B) 2announces quantum physics, one of the two measurements her of c− log2 d, where d is entangled, then H(A|B) = S|B) ≥ 0, which is trivial, since the ative, the bound isfundamental different from the exis ated the first promeasurement to Bob. Bob’s task is then to minimize his Aside from its significance, 3 separable ? system entangledsent opy of a system after measurement the dimension of the to Alice. Since A related uncertainty relation has be sical ones. y [3, 4]. However, uncertainty about Alice’s measurement outcome. This is to the develop also has potential application 1 1 ature [10], which is �implied by the re � log cannot exceed log d, the bound (3) reduces ≥ log − H(R|R ) − H(S|S ) 2 2 a quantum memillustrated in in Figure 1. 24 c entanglement ture quantum technologies. Oneisobvious can uncertainty relation has been conjectured the literc More precisely, H(R|B) the conditi from its fundamental significance, o P P to H(R|B)+H(S|B) ≥ the 0,uncertainty which is trivial, since the inciple bethe relation Equation (2) boundsAside Bob’s in the case that , which is cannot implied by we derive. fieldofofthequantum cryptography. In the |ψ ��ψ | ⊗ 1 1 )ρ state ( j j AB ( j also has potential application to the developm isely, H(R|B) is the conditional von Neumann entropy proposals are sememory. However, if hemeasurement does have and Brassard conditional entropy of a system after 80s, Wiesner [3], and Bennett Phe has no quantum P applying the completely positive map H(A|B) ≈ − log2 d |ψj ��ψj | ⊗ 11)ρAB ( |ψj ��ψj | ⊗ 11) formed by te ( The New Entropic Uncertainty Relation bound on the uncertainties of the measurement outcomes which depends on the amount of entanglement between the system, A, and the quantum memory, B. Mathematically, it is the following relation:3 Proof to show 1 H(R|B) + H(S|B) ≥ log2 + H(A|B). c (3) method: smooth entropy calculus (Renner, Bob’s uncertainty about the measurement R is 2005) denoted eigenvalue 2 by the conditional von Neumann entropy, H(R|B), which directly generalizes the Shannon entropy, H(R), to the case that Bob has a quantum memory, B, and likewise for S.4 The additional term H(A|B) appearing on the right hand side quantifies the amount of entanglement between the system and the memory. We discuss some instructive examples: 1. If the state, A, and memory, discontinuous B, are maximally entangled, then H(A|B) = − smooth log2 d, where d is entropies the dimension of the system sent to Alice. Since log2 1c cannot exceed log2 d, the bound (3) reduces to H(R|B)+H(S|B) ≥ 0, which is trivial, since the 3 4 bound on the uncertainties of the measurement outcomes which depends on the amount of entanglement between the system, A, and the quantum memory, B. Mathematically, it is the following relation:3 Proof 2 1 H(R|B) + H(S|B) ≥ log2 + H(A|B). c (3) to show threeBob’s stepsuncertainty about the measurement R is denoted by the conditional von Neumann entropy, H(R|B), which 1.Inequality for min and max-entropies directly generalizes the Shannon entropy, H(R), to the case that Bob has a quantum memory, B, and likewise for S.4 The additional term H(A|B) appearing on the right hand side quantifies the amount of entanglement between the system and theentropies memory. We discuss some 2.Convert to smooth instructive examples: 3 1. If the state, A, and memory, B, are maximally entangled, then H(A|B) = − log2 d, where d is 3.Asymptotic limit von Neumann entropy the dimension of the system sent to Alice. Since log2 1c cannot exceed log2 d, the bound (3) reduces to H(R|B)+H(S|B) ≥ 0, which is trivial, since the 4 Proof to show proof Christandl and Winter, 2005; Boileau and Renes, 2009 Application: Entanglement Witness Alice‘s result? 2 entangled ? error probability FIG. 1: Illustration of the uncertainty game. (1) Bob sends a state to Alice, which may, in general, be entangled with his quantum memory. (2) Alice measures either R or S and notes her outcome. (3) Alice announces her measurement choice to Bob. Bob’s goal is then to minimize his uncertainty about Alice’s measurement outcome. We now proceed to state our uncertainty relation. It holds in the presence of quantum memory and provides a bound on the uncertainties of the measurement outcomes which depends on the amount of entanglement between the system, A, and the quantum memory, B. Mathematically, it is the following relation:3 given the quantum memory is always greater than 0. As discussed above, Bob can guess both R and S perfectly with such a strategy. 2. If A and B are not i.e. the state takes � entangled, j j the form ρAB = j pj ρA ⊗ ρB , for probabilities pj 4 Perimeter Institute for Theoretical Physics, 3 Bob. Bob’s goal is then to minimize his uncertainty about Alic Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, 5 5 Institute Darmstadt, for Applied Institute for Applied Physics, pTechnische Universität 64289Physics, Darmstadt,Technis German pS R (Dated: June 25, 2010) (D The Uncertainty Principle in the Presenc Application: Entanglement Witness h(p ) + p Matthias log (d 1) ≥Christandl, H(R|B) We now proceed to− state our uncertainty relation. Mario Berta, Roger Colbeck, R 1R 2 1 11 2, 3, 4It 11 Jos holds in theFaculty presence of quantum memory and provides a ⊗ of Physics, Ludwig-Maximilians-Universität Mün d d 2log2 (d − 1) ≥ H(S|B) ) + p h(p S S Institute forof Theoretical Physics, ETH Zurich, 8 boundpRonpSthe uncertainties the measurement outcomes 3 Institute of amount TheoreticalofComputer Science,between ETH Zuri which depends on the entanglement λi � � 4 log d Perimeter Institute for Theoretical Physics,231 Caroline Street Nor ⊗ Ū ( |i�|i�) = ( |i�|i�) U R R and the quantum memory, B. Mathemath(pR )the + pRsystem, log2 (d ≥ H(R|B) 5 − 1) A, Institute for Applied iPhysics,3Technische Universität Darm i Hm ically, it is the following relation: i (Dated: June 25, 2010) 1 � � h(pS ) + pS log2 (d − 1) ≥ H(S|B) p R pS |i�|i�) = ( |i�|i�) US ⊗ ŪS ( 1 λmax H i i � � + H(S|B) ≥ log + H(A|B). (3) H(R|B) 2 |i�|i�) = ( |i�|i�) UR ⊗ ŪR ( cheart The iuncertaintyi principle [?h(p ] lies at the 1 ≥ H(R|B) R ) + pR log2 (d − 1) Hma Hmin (ρ) = log of quantum theory, illuminating a dramatic difλmax ference with classical mechanics. The principle H(A|B) ≈ − log Bob’s about the 2 d measurement R is denoted � uncertainty � bounds the = uncertainties the outcomes of any ŪS (the|i�|i�) ( |i�|i�)vonofNeumann US ⊗by � conditional entropy, H(R|B), which H minρ(ρ) two observables on a system in terms of the exH(ρ) = −trρ log h(p ) + p log (d − 1) ≥ H(S|B) i i S S 2 directly Shannon H(R), to the pectation generalizes value of their commutator. It entropy, implies d logthe 2 heart uncertaintythat principle [? ] lies at the Uncertainty relations an observer with a classical memory cannot case that Bob has a quantum memory, B, and likewise 2 H (ρ) = rankhave ρ uantum theory, illuminating a dramatic difmax an observer � mea� logcan predict4 the outcomes of two incompatible The additional term H(A|B) appearing on the for S. mechanics. nce with classical The precision. principle a(system. Although surements to arbitrary However, this U ⊗ Ū ( |i�|i�) = |i�|i�) R R pR pof S any nds the uncertainties of the outcomes knowledge an observ implication no longer holds if the observer instead � right hand side quantifies the amount of entanglement Hmin (ρ) =i max Hmin i observableshas on aaquantum system inmemory, terms ofa the exarbitrary properties o σ:�−close to ρ scenario which is likely implies implies entanglement • Only need to estimate two parameters. • Full tomography needs D parameters. • Different from traditional entanglement witnesses. between the system and the memory. We discuss some Application: Quantum Key Distribution Application: Quantum Key Distribution • Intuition for security of BB84: uncertainty principle • Proofs of security: entanglement distillation privacy amplification • Problem: Eve has quantum memory • New uncertainty principle leads to a formula for the key rate generalises Shor & Preskill‘s formula The Uncertainty Game be his th to wi e c i led ho l ng t c ng ta en ta en rem en em u be ur .#,/ (' %" FIG. 1: Illustration of the uncertainty game. (1) Bob 2sends a sta quantum memory. (2) Alice measures either R or S and notes her 2 Bob. Bob’s goal is then to minimize his uncertainty about Alice’s m !" !" Application:!"Quantum Key Distribution !" !" We now proceed to state our uncertainty "#$! relation. It holds in the presence of quantum memory and"#$! provides a bound on the uncertainties %" of the measurement outcomes %" which depends on the amount of entanglement between $" the%"system, A, and the quantum memory,!"#$%"&' B. Mathemat$" #" 3 ically, it is the following relation: &(&)*+' (*.#,/(' )"%0)&(' !" &' %" ' + #$ !" &)* $" !" "#$ $" "#$! " # $ ! "#$ ! ! &( &( ' &' "& % %" +' +' #$ #$ !" &)* !" &)* $" "#$ ! %" !" &( $" $" !"# $ % " & ' & ( & !) " * + # ' $ %" &(& &' ) * +' !"#$%"&' ),-(*.#,/(' )"%0)&(' %" &(&)*+' ),-(*.#,/(' )"%0)&(' "%$! 1 (3) H(R|B) !+ H(S|B) ≥ log2 + H(A|B). #""%$! #" c 3 #" %" "#" ! "%$! %" "#" 2 2 "%$! 2 2 rstag, 25. März 2010 erstag, 25. März 2010 2 "&$! )"%0)&(' %" is equivalent to ),-(*.#,/(' ),-(*.#,/(' )"%0)&(' Bob’s uncertainty about the measurement R is denoted ! "&$! 1 "#" by the conditional von Neumann entropy, H(R|B), which n result, (3), as H(R|E) + H(S|B) ≥ log2 c , a form tion of directly the uncertainty generalizes game. (1) Bob sends athe particleShannon to Alice, which may, in general, be H(R), entangled with his the ),-(*.#, entropy, to (2) Alice measures either R or S and notes herBoileau outcome. (3) Aliceand announces her measurement choice (see to iously conjectured by Renes [23] !ry. tainty case relation provides a lower bound on Bob’s resulting uncertainty about Alice’s outcome. Bob has a sends quantum memory, B, and likewise : Illustration of that the uncertainty game. (1) Bob a particle to Alice, which may, in general, be entangled with his %" conjectured by Boileau and Renes, 2009 ),-(*.#,/(' ),-(*.#,/(' "%$! "&$! Supplementary Information). Together these imply um memory. (2) Alice R or S and notes her outcome. (3) Alice announces her measurement choice to 4 measures either The Uncertainty Game Institute of Theoretical Computer Scien Faculty of Physics, Ludwig-Maximilians-Uni be his th to wi e c i led ho l ng t c ng ta en ta en rem en em u be ur %" .#,/ (' 2 Mario Berta, Matthias Christandl, Roger Colbe Institute for Theoretical Physics, E 1 1 Mario Berta, Matthias Christandl, Roger2 Colbeck, 3 1 !" 2 2 Faculty of Physics, Ludwig-Maximilians-Univers Application: Quantum Key Institute forDistribution Theoretical Physics, ET Perimeter Institute for Theoretical Physics, 31 Caro 23 !" 5 Institute for Theoretical Physics, ETH Institute of Theoretical Computer Science Institute for Applied Physics, Technische UnZ 3 4 !" of Theoretical Institute Computer E Perimeter Institute for Theoretical Physics,Science, 31 Carolin (Dated: J 4 5 Perimeter forfor Theoretical Physics, 31the Caroline S The Institute Uncertainty Principle in Pres Institute Applied Physics, Technische Unive "#$! 5 (Dated: Jun Institute for Applied Physics, Technische Universit "#$! 1 1 2, 3,25 4 (Dated: June Mario Berta, Matthias Christandl, Roger Colbeck, !" !" $" !" &( &' %" ' + #$ !" &)* $" &( &( ' &' "& % %" +' +' #$ #$ !" &)* !" &)* $" $" $" !"# $ % " & ' & ( & !) " * + # ' $ %" &(& &' ) * +' 1 %" !" 1 !" 4 "#$ "#$! " # $ ! Faculty of Physics, Ludwig-Maximilians-Universität $" 3 2 Institute for Theoretical Physics, ETH Zuric K ≥ H(R|E) − H(R|B) %" !"#$%"&' $" #" 3 1 Institute of Theoretical Computer Science, ETH Z ,(*.#,/(' as H(R|E) + H(S|B) ≥ log , a form &(&)*+' )"%0)&(' 2 c 4 Kfor ≥(see H(R|E) −%"H(R|B) #" Institute Theoretical Physics, !"#$%"&' 31 Caroline Street ectured byPerimeter Boileau and Renes [23] ),-(*.#,/(' )"%0)&(' %" &(&)*+' 5 Together these imply )"%0)&(' ary ),-(*.#,/(' Information). 1 Institute for Applied−Physics, Technische "%$! K ≥≥ H(R|E) H(R|B) DevetakUniversität and Winter, 2005 D log %" 2the − H(R|B) − H(S|B) R|B) − H(S|B). Furthermore, using (Dated: June 25, 201 ! c "%$! "#$ ! ! "#$ ! %" #" "%$! ≥ log − H(R|B) − H(S|B) ! #" 1 ements cannot decrease entropy,"#" we have 2 %" 2 "&$! )"%0)&('c"#" ),-(*.#,/(' 1 ),-(*.#,/(' � H(A|B)(4) ≈ − log2 d Pauli log2 qubits, − H(R|R ) −X&Z H(S|S � ). ),-(*.#,/(' 1 "&$! c � ),-(*.#,/(' 2 )"%0)&('! � − H(R|R ) − H(S|S ) %" 2 2 2 "#" ≥ log2 this c tion ofto the uncertainty game. (1) Bob sends particle 1 tocAlice, which may, �in general, be entangled ),-(*.#, ds a generalization of aShor and � with his certai (2) Alice measures either R or S and ≥ notes her 2 outcome. Alice announces measurement) choice to log − (3)H(R|R ) −herH(S|S !ry. stainty result [17], which is recovered in the relation provides a lower bound on Bob’s1 resulting uncertainty about Alice’s outcome. log − ac2h(�) : Illustration of the uncertainty game. (1)= Bob sends particle to Alice, which may, in general, be entangled 2d %" with hiswhich rstag, 25. März 2010 erstag, 25. März 2010 "%$! "%$! "&$! e observables to qubits Shor and Preskill, um memory. (2) Aliceapplied measures either R or S andand notesasher outcome. (3) Alice announces her measurement choice to2000 We now proceed to state our uncertainty relation. It holds in the presence of quantum memory and provides a bound on the uncertainties of the measurement outcomes which depends on the amount of entanglement between the system, A, and the quantum memory, B. MathematUncertainty principlerelation: in the3 presence of ically, it is the following Summary • quantum memory 1 H(R|B) + H(S|B) ≥ log2 + H(A|B). c (3) Fundamental relevance • Bob’s uncertainty about the measurement R is denoted by the conditional von Neumann entropy, H(R|B), which Applications: • directly generalizes the Shannon entropy, H(R), to the • Entanglement Witness case that Bob has a quantum memory, B, and likewise Key Distribution The additional term H(A|B) appearing on the for S.4Quantum right hand side quantifies the amount of entanglement Generalisation to smooth entropies between the system and the memory. We discuss some operational meaning, quantum information theory instructive examples: andstate, Computer to 1. If the A, and Science memory, relative B, are maximally • Physics entangled, then H(A|B) quantum knowledge = − log2 d, where d is