When quantum systems interact with the environment they lose their quantum
properties, such as coherence. Quantum erasure makes it possible to restore
coherence in a system by measuring its environment, but accessing the whole of
it may be prohibitive: realistically one might have to concentrate only on an
accessible subspace and neglect the rest. If that is the case, how good is
quantum erasure? In this work we compute the largest coherence $\langle
\mathcal C\rangle$ that we can expect to recover in a qubit, as a function of
the dimension of the accessible and of the inaccessible subspaces of its
environment. We then imagine the following game: we are given a uniformly
random pure state of $n+1$ qubits and we are asked to compute the largest
coherence that we can retrieve on one of them by optimally measuring a certain
number $0\leq a\leq n$ of the others. We find a surprising effect around the
value $a\approx n/2$: the recoverable coherence sharply transitions between 0
and 1, indicating that in order to restore full coherence on a qubit we need
access to only half of its physical environment (or in terms of degrees of
freedom to just the square root of them). Moreover, we find that the
recoverable coherence becomes a typical property of the whole ensemble as $n$
grows.Comment: 4 pages, 5 figure

Boyd, Robert W.

Brougham, Thomas

Miatto, Filippo M.

Piché, Kevin

English

arXiv

When a quantum system interacts with its environment it may incur in decoherence. Quantum erasure makes it possible to restore coherence in a system by gaining information about its environment, but measuring the whole of it may be prohibitive: Realistically, one might be forced to address only an accessible subspace and neglect the rest. In such a case, under what conditions will quantum erasure still be effective? In this work we compute analytically the largest recoverable coherence of a random qubit plus environment state and we show that it approaches 100% with overwhelmingly high probability as long as the dimension of the accessible subspace of the environment is larger than D−−√, where D is the dimension of the whole environment. Additionally, we find a sharp transition between a linear behavior and a power-law behavior as soon as the dimension of the inaccessible environment exceeds the dimension of the accessible one. Our results imply that the typical states of a qubit plus environment system admit a measurement spanning only about D−−√ degrees of freedom, any outcome of which projects the qubit on a maximally coherent state. This suggests, for instance, that in the dynamics of open quantum systems, if the interactions are known, it would in principle be possible to gain sufficient information and restore coherence in a qubit by dealing with a fraction of the physical resources

Piche, Kevin

Enlighten

     Miatto, F. M., Piche, K., Brougham, T., and Boyd, R. W. (2015) Recovering full coherence in a qubit by measuring half of its environment. Physical Review A: Atomic, Molecular and Optical Physics, 92(6), 062331. (doi:10.1103/PhysRevA.92.062331)   There may be differences between this version and the published version. You are advised to consult the publisher’s version if you wish to cite from it.   http://eprints.gla.ac.uk/129757/            Deposited on: 11 October 2016                       Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk Recovering full coherence in a qubit by measuring half of its environmentFilippo M. Miatto1, Kevin Piche´1, Thomas Brougham2 and Robert W. Boyd1,2,31Dept. of Physics, University of Ottawa, Ottawa, Canada2School of Physics and Astronomy, University of Glasgow, Glasgow (UK) and3Institute of Optics, University of Rochester, Rochester, USA(Dated: March 20, 2015)When quantum systems interact with the environment they lose their quantum properties, suchas coherence. Quantum erasure makes it possible to restore coherence in a system by measuringits environment, but accessing the whole of it may be prohibitive: realistically one might have toconcentrate only on an accessible subspace and neglect the rest. If that is the case, how good isquantum erasure? In this work we compute the largest coherence 〈C〉 that we can expect to recoverin a qubit, as a function of the dimension of the accessible and of the inaccessible subspaces of itsenvironment. We then imagine the following game: we are given a uniformly random pure state ofn+ 1 qubits and we are asked to compute the largest coherence that we can retrieve on one of themby optimally measuring a certain number 0 ≤ a ≤ n of the others. We find a surprising effect aroundthe value a ≈ n/2: the recoverable coherence sharply transitions between 0 and 1, indicating thatin order to restore full coherence on a qubit we need access to only half of its physical environment(or in terms of degrees of freedom to just the square root of them). Moreover, we find that therecoverable coherence becomes a typical property of the whole ensemble as n grows.INTRODUCTIONDecoherence is a physical process that interests the sci-entific community from a fundamental point of view (howdoes the quantum-to-classical transition occur?) and alsofrom a technical one (how can we maintain a system co-herent enough throughout a quantum protocol?) [1–6].One of the techniques for restoring coherence is known as“quantum erasure”, which consists in measuring the en-vironment in the most appropriate basis in order to erasethe information that it stores and thereby recover coher-ence [7]. An example would be in a Young double-slitexperiment where two orthogonal polarizers have beenput in front of the slits and the interference fringes havedisappeared. Quantum erasure (in this case with post-selection) would consist of orienting a polarizer diago-nally before the screen to erase the which-slit informa-tion stored in the Hilbert space of polarization (whichwas acting as the environment) and restore the fringes.Quantum erasure relies on an optimal measurement ofthe environment of a qubit Q, consisting of a numberof measurement operators {pˆij}, to restore its coherence.The j-th measurement operator pˆij projects the state ofQonto the conditional state ρˆj , which displays a coherenceCj (defined below) [8]. We stress that quantum erasuredoes not rely on postselection [9–11], as the coherencethat we maximize is the average over all the outcomes:〈C〉 = ∑j pjCk, where pj is the probability of observ-ing the j-th outcome. Recall that ρˆj does depend onthe outcome j, and that the incoherent sum∑j pj ρˆk isthe same as tracing over the environment: this preventsus from measuring coherence directly and violate the nosignalling principle [? ]. It is clear that the more informa-tion one is able to erase from the environment, the moreof the original coherence one is able to restore. In ourmodel we consider one qubit Q, immersed in an environ-Accessible qubits (a)Inaccessible qubits (k)QFIG. 1. We imagine a qubit Q within an ensemble of n en-vironment qubits, where a of them are accessible. The restk = n − a are inaccessible. We find that if a ≥ k (i.e. ifwe can access at least half of them) there exists an optimalmeasurement on the accessible qubits whose outcomes projectQ onto states with coherence 〈C〉 ∼ 1 − 2k2a+2, which quicklyapproaches 1 as a increases past n/2.ment with Hilbert space A⊗K, where the A-dimensionalsubspace A is accessible and the K-dimensional subspaceK is inaccessible. Therefore, we consider only measure-ments that span A, but ignore K, i.e. pˆij = pˆiAj ⊗ 1ˆK.Given measurements of such type, we study how wellthey perform and we find that as long as A >∼ K, one canrestore almost perfect coherence on Q. This phenomenonmight be closely related to quantum darwinism [4].The rest of this Letter is organized as follows: in thenext section we define the coherence of a qubit and weshow how it is influenced by a measurement on its en-vironment. Then we split the environment into an ac-cessible and an inaccessible part and we prove the mainresult. In the final section we supply examples and weshow that the average coherence becomes typical as theenvironment grows in size.arXiv:1502.07030v2  [quant-ph]  19 Mar 20152QUANTIFYING AND RESTORINGCOHERENCEOur first task is to identify the quantity that we aregoing to study, namely the coherence of Q. In contrast topurity, coherence depends on the basis that we choose.In the Bloch sphere (where the poles are the preferredbasis elements |0〉 and |1〉, which we will also call “alter-natives”, implying a measurement of σˆz) the coherence isthe distance of the Bloch vector from the imaginary lineconnecting the North pole to the South pole, i.e. givena Bloch vector with coordinates v = (x, y, z), the coher-ence is√x2 + y2, also known as “visibility” in view ofthe analogy of a qubit with a Mach-Zender interferome-ter (see below) [12]. It is clear that if we picked a differentpair of opposite points on the surface as the new Northand South poles, the coherence would generally change.We can understand the meaning of coherence by mak-ing an analogy with a photon travelling through a Mach-Zender interferometer: the two arms of the interferome-ter constitute the two possible alternatives and so we canrepresent the path of the photon with a qubit. As onevaries the phase of one arm with respect to the other,the Bloch vector rotates around the z axis in a circularmotion of radius√x2 + y2. This radius is precisely thevisibility of the fringes at the output of the interferome-ter: if we had all the which-arm information, the Blochvector would have coordinates either (0, 0, 1) or (0, 0,−1),and in neither case we would see any fringes, while if thetwo alternatives were in a balanced coherent superposi-tion, the Bloch vector would be on the equator and thevisibility would be at its largest. Note that in order tolose the fringes we don’t need to actually “know” whicharm the photon is in, we just need such information tobe “knowable”, as in such case the qubit would be max-imally entangled with some other system and its statewould be maximally mixed (i.e. v = (0, 0, 0)). In termsof the density matrix of the qubit, the coherence is givenby twice the absolute value of either of the off-diagonalelements.Now, consider a qubitQ that is entangled with anothersystem (which we call S and which is not necessarilyanother qubit):|ψ〉 = α|0, s0〉+ β|1, s1〉, (1)where |s0〉 and |s1〉 are the states of S corresponding tothe states ofQ. The density matrix ofQ alone is obtainedby tracing over S:ρˆQ =( |α|2 α∗β〈s1|s0〉αβ∗〈s0|s1〉 |β|2)(2)The coherence is therefore given by C = 2|αβ∗〈s0|s1〉|,which is proportional to the overlap between the statesof S. This happens because the more the states |s0〉 and|s1〉 are orthogonal, the better one can distinguish themand learn about the qubit, i.e. the more informationabout the alternatives of Q is stored in S, which is actingas an “environment”. This is the essence of the dualityprinciple.A way of erasing such information would be to measureS in a basis that is unbiased with respect to |s0〉 and |s1〉,by way of an optimal measurement with elements pˆij [8]defined by maximizing the mean coherence over all theprobability operator measures on S:C = sup{pˆij}∈POM(S)∑j|2αβ∗〈s0|pˆij |s1〉| (3)= 2Tr∣∣αβ∗|s1〉〈s0|∣∣ (4)where Tr|x| is the trace norm of x. In this way, regardlessof the outcome, we would not learn which of the states|s0〉 or |s1〉 the environment is in, and after such measure-ment the qubit is in a state with coherence C = 2|αβ∗|.However, in realistic situations such flexibility may notbe possible, i.e. we may not be able to access all the nec-essary degrees of freedom of S and we will have to splitit into an accessible part and an inaccessible one. Howmuch coherence can we expect to recover in that case?Motivated by the importance of the question, we nowcompute how much coherence is recoverable on averagewhen Q is immersed in an environment of which we canonly access a subspace. The main difference with the ex-ample given above, is thatQ and the environment probedby our measurement are generally not in a pure state suchas the one in Eq. (1). To do this, we consider a randompure state of a qubit in an AK-dimensional environmentwith Hilbert space A ⊗ K which is split into an accessi-ble subspace of dimension A and an inaccessible one ofdimension K. Such a pure state is therefore sampled uni-formly in a Hilbert space Q⊗A⊗K of dimension 2AK.After tracing away the inaccessible environment K, weare left with a 2A-dimensional state in Q⊗A which canalways be written as a 2A× 2A density matrix:ρˆ =(Rˆ0 XˆXˆ† Rˆ1)(5)where Tr(Rˆ0) and Tr(Rˆ1) are the probabilities of measur-ing the qubit in the alternatives |0〉 and |1〉 and Xˆ is thecross-term. The largest coherence of the qubit that wecan obtain by optimally measuring the accessible spaceA is given by twice the trace norm of the cross-term:C = 2Tr|Xˆ|, see Eq. (4) [8]. We recall that the tracenorm of Xˆ can be computed as the sum of the squareroot of the A eigenvalues of the matrix Xˆ†Xˆ, i.e.Tr|Xˆ| =A∑i=1√λi(Xˆ†Xˆ). (6)The random 2A-dimensional states ρˆ are statisti-cally distributed according to the induced trace measure3P2A,K(ρˆ) and constitute a Ginibre ensemble [13]. A wayof sampling uniformly from such ensemble is to generate a2A×K complex gaussian random matrix µ (with entriessampled from the complex normal distribution centredon the origin and with unit variance) and then build the2A× 2A density matrixρˆ =µ†µTr(µ†µ). (7)However, when we calculate C we don’t need the wholematrix ρˆ, but only the A×A off-diagonal block Xˆ whichis proportional to the product M = µ†1µ2 of two indepen-dent A × K complex gaussian random matrices µ1 andµ2 (see Fig. 2). We find the proportionality factor byrecalling that we are averaging over the whole ensembleand that the mean is linear, so we can take the averagevalue of the denominator in (7): 〈Tr(µ†µ)〉 = 4AK. Toµ†1µ†2µ2µ1XX†R0R1AKA /FIG. 2. The A × A cross-term X in the random matrix ρ ofEq. (5) and (7) is proportional to the product between twoindependent random matrices µ1 and µ2 that make up µ.find the average of Tr|M |, we use the moments m` of themarginal distribution of eigenvalues of M . In particular,the average square root of the eigenvalues of M is themoment of order ` = 1/2, i.e. 〈Tr|M |〉 = Am 12. We cancompute such a moment by applying Eq. (57) of Ref. [14]to our matrices, and we find:m 12=4pi5/2(−1)K 4F˜3(12 ,1−A,1−A,1−K12−A, 12−A, 12−K∣∣∣∣1)A! Γ(A)Γ(K)(8)where the function 4F˜3 is a regularized Hypergeometricfunction. It is now straightforward to obtain the finalresult:〈C〉 = 2〈Tr|Xˆ|〉 = 2 〈Tr|Mˆ |〉4AK=m 122K. (9)EXAMPLESAn expression with regularized Hypergeometric func-tions like Eq. (8) can be rather obscure. For this reason,we evaluate it explicitly for some values of A and we showthe two distinct behaviours that it exhibits, for K →∞and for 1 ≤ K ≤ A. Let’s pick a uniformly randomstate of a qubit immersed in an entirely inaccessible K-dimensional environment. What is the average coherenceof the qubit? As we are not performing any operation onthe environment, this is equivalent to setting A = 1 inEq. (8) and (9) and the answer is〈C1〉 = pi3/2(−1)K2K!Γ(12 −K) ∼ √pi2√K(as K →∞) (10)In other words, decoherence in absence of any interven-tion scales like O(1/√K) at a rate of√pi/2. By control-ling a two-dimensional space (i.e. A = 2) we obtain〈C2〉 = pi3/2(−1)K(13− 22K)32K!Γ(32 −K) ∼ 11√pi16√K(as K →∞)(11)By controlling a three-dimensional space we obtain〈C3〉 ∼ 107√pi128√K(as K →∞), (12)and so on. It turns out that the high-K scaling is alwaysO(1/√K). If instead we look at the scaling for K → 0we find a linear behaviour: 〈C〉 ∼ 1 − K4A , the transitionhappening rather sharply at A = K (see inset in Fig. 3).We give an explicit example for A = 100 in Fig. 3.0 100 200 300 400 5000.00.20.40.60.81.0K1  K4008.489pK  KhC100i50 100 150-0.0015-0.002-0.0025FIG. 3. The average coherence 〈C〉 for A = 100 (solid or-ange), together with the high-K and low-K approximations(dashed). Up to K = A, the behaviour of 〈C〉 is purely lin-ear (1 − K4A, see inset). For K > A the behaviour changesdramatically, and is asymptotic to O(1/√K). The shadingindicates the linear region.We now give an example in terms of ensembles ofqubits: let us consider our qubit Q immersed in an en-semble of n other qubits, and let us pick a random stateof all n + 1 of them. We wish to compute how muchcoherence we can expect to recover on Q on average aswe gain control of more and more qubits in the ensem-ble. In this case A = 2a and K = 2n−a = 2k. We seethat 〈C〉 is close to zero for a <∼ k and it approaches 1as a >∼ k (see Fig. 4 and 5). Note that such variation40 1 2 33 qubits0 1 2 3 44 qubits0 1 2 3 4 55 qubits0 1 2 3 4 5 66 qubits0 1 2 3 4 5 6 77 qubits0 1 2 3 4 5 6 7 88 qubits0 1 2 3 4 5 6 7 8 99 qubits0 1 2 3 4 5 6 7 8 9 1010 qubits0 1 2 3 4 5 6 7 8 9 10 1111 qubitsFIG. 4. Graphs of the recoverable coherence 〈C〉 of a qubit Qas a function of the number a of qubits of the environmentthat we can control, where the total, n, is displayed on top.In blue we show the 50, 90 and 99 percentiles around themean (red line) and all plots are from 0 to 1. We can seethat 〈C〉 transitions from a value close to 0 to a value closeto 1 as we gain access to more than half of the environmentqubits. Notice that as the total number of environment qubitsgrows, the mean becomes a better representative of the wholeensemble.always happens across the same number of qubits. Infact, for a ≥ dn/2e, the linear scaling makes it possibleto compute the asymptotic behaviour 〈C〉 ∼ 1− 2k2a+2 (asn → ∞). This law allows us to plot graphs for largenumbers of qubits (see Fig. 5).0 50 100 150 200200 qubitsFIG. 5. The recoverable coherence 〈C〉 of a qubit Q for anenvironment of n = 200 qubits displays a very sharp increasefrom 0 to 1 as soon as one can control more than 100 ofthem, meaning that there exists an environment observablewith elements {pˆiAj ⊗ 1ˆK} which project Q onto maximallycoherent states.How typical is the value of 〈C〉? To answer this ques-tion we produced thousands of random states from Gini-bre ensembles of several qubits, for n = 3 to 11. Theresults are shown in Fig. 4: already with an environmentmade of a handful of qubits, the average coherence is atypical value of the system, as all the states have a valueof C that falls extremely close to 〈C〉, the more so as ngrows.CONCLUSIONSIn this Letter we showed that quantum erasure can re-store full coherence in a qubit even by addressing onlyabout half of its environment. We also observed that theaverage recoverable coherence is a typical property of anensemble of random states, i.e. C ∼ 〈C〉 as n→∞. Thismeans that the existence of such optimal measurementis almost always guaranteed. Our result is even moresurprising if restated in terms of degrees of freedom, asthe optimal measurement needs only address about√Dof the total number D of degrees of freedom of the envi-ronment. Moreover, typicality assures us that this resultholds for any such partition of the environment.ACKNOWLEDGEMENTSF.M.M. thanks Carlos Gonza´lez Guille´n for a fruit-ful consultation, Karol Z˙yczkowski for encouraging com-ments and Mike McDonell for illuminating discussions.This work was supported by the Canada Excellence Re-search Chairs (CERC) Program, the Natural Sciencesand Engineering Research Council of Canada (NSERC)and the UK EPSRC.[1] M. Scully and H. Walther, Phys. Rev. A 39, 5229 (1989).[2] M. Brune, E. Hagley, J. Dreyer, X. Maˆıtre, A. Maali,C. Wunderlich, J. M. Raimond, and S. Haroche, Phys.Rev. Lett. 77, 4887 (1996).[3] C. J. Myatt, B. E. King, Q. A. Turchette, C. A. Sack-ett, D. Kielpinski, W. M. Itano, C. Monroe, and D. J.Wineland, Nature 403, 269 (2000).[4] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).[5] A. D’Arrigo, R. L. Franco, G. Benenti, E. Paladino, andG. Falci, Annals of Physics 350, 211 (2014).[6] A. Orieux, A. D’Arrigo, G. Ferranti, R. L. Franco, G. Be-nenti, E. Paladino, G. Falci, F. Sciarrino, and P. Mat-aloni, Sci. Rep. 5 (2015).[7] D. M. Greenberger and A. Yasin, Phys. Lett. A 128, 391(1988).[8] B.-G. Englert and J. A. Bergou, Optics communications179, 337 (2000).[9] R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich,Proceedings of the National Academy of Sciences 109,9314 (2012).[10] E. Bolduc, J. Leach, F. M. Miatto, G. Leuchs, and R. W.Boyd, Proceedings of the National Academy of Sciences111, 12337 (2014).[11] J. Leach, E. Bolduc, F. Miatto, K. Piche, G. Leuchs,R. W. Boyd, et al., arXiv preprint arXiv:1406.4300(2014).[12] B. Englert, Phys. Rev. Lett. 77, 2154 (1996).[13] K. Z˙yczkowski and I. Bengtsson, Geometry of QuantumStates (Cambridge University Press, Cambridge, 2006).[14] G. Akemann, J. R. Ipsen, and M. Kieburg, Phys. Rev.E 88, 052118 (2013).

Recovering full coherence in a qubit by measuring half of its environment

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