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Uncollapsing the wavefunction by undoing quantum measurements Andrew N. Jordan a; Alexander N. Korotkov b a Department of Physics and Astronomy, University of Rochester, Rochester, New York, USA b Department of Electrical Engineering, University of California, Riverside, CA, USA

First published on: 12 February 2010

To cite this Article Jordan, Andrew N. and Korotkov, Alexander N.(2010) 'Uncollapsing the wavefunction by undoing quantum measurements', Contemporary Physics, 51: 2, 125 — 147, First published on: 12 February 2010 (iFirst) To link to this Article: DOI: 10.1080/00107510903385292 URL: http://dx.doi.org/10.1080/00107510903385292

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Uncollapsing the wavefunction by undoing quantum measurements Andrew N. Jordana* and Alexander N. Korotkovb aDepartment of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA; bDepartment of Electrical Engineering, University of California, Riverside, CA 92521, USA (Received 17 June 2009; final version received 2 October 2009)

We review and expand on recent advances in theory and experiments concerning the problem of wavefunction uncollapse: given an unknown state that has been disturbed by a generalised measurement, restore the state to its initial configuration. We describe how this is probabilistically possible with a subsequent measurement that involves erasing the information extracted about the state in the first measurement. The general theory of abstract measurements is discussed, focusing on quantum information aspects of the problem, in addition to investigating a variety of specific physical situations and explicit measurement strategies. Several systems are considered in detail: the quantum double dot charge qubit measured by a quantum point contact (with and without Hamiltonian dynamics), the superconducting phase qubit monitored by a SQUID detector, and an arbitrary number of entangled charge qubits. Furthermore, uncollapse strategies for the quantum dot electron spin qubit, and the optical polarisation qubit are also reviewed. For each of these systems the physics of the continuous measurement process, the strategy required to ideally uncollapse the wavefunction, as well as the statistical features associated with the measurement are discussed. We also summarise the recent experimental realisation of two of these systems, the phase qubit and the polarisation qubit. Keywords: wavefunction uncollapse; measurement reversal; quantum Bayesianism

1. Introduction questions [16,17]. Wavefunction uncollapse teaches us The irreversibility of quantum measurement is an several things about the fundamentals of quantum axiomatic property of textbook quantum mechanics mechanics. First, there is a notion that wavefunction [1]. In his famous article Law without law, John Wheeler represents many possibilities, but that reality is expresses the idea with poetic flare: ‘We are dealing created by measurement. The fact that the effects of with [a quantum] event that makes itself known by measurement can be undone suggests that this idea an irreversible act of amplification, by an indelible is flawed, or at least too simplistic. If you create record, an act of registration’ [2]. However, it has been reality with quantum measurements, does undoing gradually recognised that the textbook treatment of an them erase the reality you created? Secondly, there instantaneous wavefunction collapse is really a very is a wide-spread belief that quantum measurement special case of what is in general a dynamical process – is nothing more than a decoherence process. This continuous quantum measurement. Continuous mea- suggests that the superposition never really collapses; surements do not project the system immediately into it only appears to collapse. What actually happens, Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 an eigenstate of the observable, but describe a pro- according to this idea, is that all the information cess whereby the collapse happens over a period of time about the system disperses into the environment: [3–13]. The fact that continuous measurement is a when a quantum system interacts with a classical dynamical process with projective measurement as a measuring device, it becomes irreversibly entangled special case, leads us to ask whether the irreversibility with all the particles that make up the measuring of quantum measurement is also a special case. The device and its surroundings. The uncollapse of the purpose of this paper is to review and expand on recent wavefunction demonstrates that decoherence theory developments in this area of research, showing that it is cannot be the whole story, because a true decoherence possible to undo a quantum measurement, thereby process is irreversible [18]. The perspective we take in uncollapsing the wavefunction, and to describe this this paper further advances the ‘quantum Bayesian’ physics in detail for both the abstract and concrete point of view, where the quantum state is nothing physical realisations. more than a reflection of our information about This paper follows our earlier work on the subject the system. When we receive more information about [14,15], as well as other papers investigating similar the system, the state changes or collapses not because

*Corresponding author. Email: [email protected]

ISSN 0010-7514 print/ISSN 1366-5812 online Ó 2010 Taylor & Francis DOI: 10.1080/00107510903385292 http://www.informaworld.com 126 A.N. Jordan and A.N. Korotkov

of any mysterious forces, but simply as a result of procedure [23], which will be briefly discussed in our Bayesian updating. paper. We note that there are many approaches to the The paper is organised as follows. We introduce non-projective (continuous, weak, generalised, etc.) the general subject with a preliminary discussion in quantum measurement, which are essentially equiva- Section 2. We then give some specific examples in lent, in spite of quite different formalisms and Sections 3 (the superconducting phase qubit) and 4 terminology. To name just a few, let us mention mathe- (the double quantum dot qubit, monitored by a matical POVM-type approaches [3,4] (POVM stands quantum point contact). With these physical imple- for positive operator-valued measure), method of mentations, we discuss the uncollapsing strategies and restricted path integrals [5,6], analysis of quantum results. In Section 5 we give a general treatment of the jumps in atomic shelving [7], quantum trajectories physics, using the POVM-based formalism. This is [8], quantum-state diffusion [9], Monte-Carlo wave- done both for pure states (Section 5.1) and mixed function method [10], weak values [11], quantum states (Section 5.2). In Section 5.3 we discuss the filtering [12], and the Bayesian formalism for weak interpretation of wavefunction uncollapse, and what it measurements in solid-state systems [13]. In this paper tells us about quantum information. These general we will use the Bayesian and POVM-type formalisms results are applied to the examples given previously in because they are most suitable for the physical setups of Section 5.4. In Section 6 we further apply the general our main interest. results to the case of the finite-Hamiltonian qubit Our work is indirectly related to the ‘quantum undergoing the uncollapse process. In Section 7, we eraser’ of Scully and Dru¨ hl [19]. There, the which-path generalise to the case of many charge qubits, and information of a particle is encoded in the quantum discuss an explicit procedure to undo any generalised state of an atom, resulting in a destruction of inter- measurement. We discuss recent developments in the ference fringe visibility. On the other hand, if the theory and experiments of measurement reversal in which-path information of the particle is erased, the Section 8 and conclude in Section 9. The Appendix interference fringes are restored. Both the Scully contains results about the statistics of the waiting time proposal and our proposal erase information. How- distribution in the charge qubit example. ever, there is an important difference. In order for the uncollapsing procedure to work, we have to erase the information that was already extracted classically. In 2. Preliminary discussion the ‘quantum eraser’, only potentially extractable Our goal is to restore an initial quantum state information is erased. disturbed by measurement. However, it is important Rather than begin with the abstract idea of to discuss what exactly we mean by that. For example, uncollapse, we first introduce the concept with actual if we start with a known pure state jcini and perform a measurement processes in specific physical contexts. textbook projective measurement, then it is trivial to This brings to mind the saying of Asher Peres: restore the initial state: since we also know the post- ‘Quantum phenomena do not occur in a Hilbert measurement wavefunction jcmi, we just need to apply space. They occur in a laboratory’ [20]. Following a unitary operation which transfers jcmi into jcini.If Peres’ dictum, we discuss measurement processes in we start with a known mixed state, then its restoration a variety of solid state systems, where there has after a projective measurement is a little more Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 been remarkable experimental progress in recent involved;1 however, such a procedure still can be years. Quantum coherence has been demonstrated to easily analysed using standard quantum mechanics and occur in a controllable fashion in systems such as classical probability theory. semiconductor quantum dots and superconducting In this paper we consider a different, non-trivial Josephson junctions. We will discuss the physics of situation: we assume that an arbitrary initial state is measurement in these systems, as well as concrete unknown to us, and we still want to restore it after strategies for uncollapsing the wavefunction. It should the measurement disturbance. To make this idea more be stressed that two of these proposals (a super- precise, we consider a contest between the uncollapse conducting phase qubit and optical polarisation proponent Plato, and an uncollapse skeptic Socrates. qubit) have now been implemented in the laboratory Socrates prepares a quantum system in any state he [21,22], providing conclusive demonstration of wave- likes, but it is unknown to Plato. Socrates sends function uncollapse. It is still too early to predict the state to Plato, who makes some measurement on whether uncollapse can eventually become a really the system, verified by the arbiter Aristotle. Plato then useful tool or it will remain only as a surprising and tries to undo the measurement. If Plato judges that the educating curiosity [18]. One possibly useful applica- attempt succeeded, the system is returned to Socrates, tion is suppression of decoherence by the uncollapse with the claim that it is the original state. Socrates is Contemporary Physics 127

then allowed to try and find a contradiction in any way he likes, with the whole process monitored by 3. Example 1: the phase qubit Aristotle. If a contradiction can be found, then he can We first give a simple example of erasing information claim to refute the uncollapse claim, but in the absence and uncollapsing the wavefunction for the case of a of contradiction, the uncollapse claim stands. If superconducting phase qubit [24]. The system is Socrates would like to try again to find a contradiction, comprised of a superconducting loop interrupted by or if Plato judges that his undoing attempt was one Josephson junction (Figure 1(a)), which is unsuccessful (and does not return a state), then controlled by an external flux fe in the loop. Two Socrates prepares a new (still unknown to Plato) state, qubit states j0i and j1i (Figure 1(b)) correspond to and the competition continues. If Socrates cannot two lowest states in the quantum well for the find a contradiction after many rounds of the com- potential energy V (f), where f is the superconduct- petition, then Plato will win the contest, and will have ing phase difference across the junction. Transitions successfully demonstrated the uncollapsing of the between the levels j0i and j1i (Rabi oscillations) can quantum state. be induced by applying microwave pulses that are A slightly different but equivalent situation is resonant with the energy level difference. Because of when we know the initial state, but our uncollapsing the energy difference, the two basis states acquire a procedure must be independent of the initial state phase difference between them, which linearly in- (so we can pretend that it is unknown), and there- creases in time. However, this can be eliminated by fore the uncollapsing should restore any initial state going into the rotating frame; we always assume it in in the same way. This formulation is most appro- this section. priate for a real experiment demonstrating the The qubit is measured by lowering the barrier uncollapsing. Finally, we may consider the more (which depends on fe), so that the upper state j1i general case where the measured system is entangled tunnels into the continuum with the rate , while the with another system, and we wish to restore the initial state j0i does not tunnel out. The tunnelling event is state of the compound system without any access to sensed by a two-junction detector SQUID inductively its second part. coupled to the qubit (Figure 1(a)). The traditional statement of irreversibility of a quantum measurement can be traced to the fact that it may be described as a mathematical projection. 3.1. Partial collapse Projection is a many-to-one mapping in the Hilbert For sufficiently long tunnelling time t, t 1, space, and therefore the same post-measurement state the measurement is a (partially destructive) projective generally corresponds to (infinitely) many initial measurement [24]: the system is destroyed if the states.2 It is therefore impossible to undo a projective tunnelling occurs, while if there is no record of measurement. tunnelling, then the state is projected onto the lower However, the situation is different for a general state j0i. This measurement technique is remarkable [4] (POVM-type) measurement, which typically cor- in the fact that the wavefunction is collapsed if responds to a one-to-one mapping jcini!jcmi in the nothing happens. A more subtle situation arises if the Hilbert space of wavefunctions (in this paper we barrier is raised after a finite time t * 71 [25]. consider only ‘ideal’ measurements which do not The system is still destroyed if tunnelling happens, Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 introduce extra decoherence). In this case the post- while in the case of no tunnelling (which we refer measurement wavefunction jcmi can still be asso- to as a null-result measurement) the state is partially ciated with the unique initial state jcini, and a collapsed because the inference from the measure- well-defined inverse mapping exists mathematically. ment is ambiguous. This is due to the fact that This makes the uncollapsing possible in principle. Since the inverse mapping is typically non-unitary, it cannot be realised as an evolution with a suit- able Hamiltonian. However, it can be realised using another POVM-type measurement with a specific (‘lucky’) result. Rather than giving the most general, abstract case first, we start with a couple of specific examples of realistic systems where such a measurement either has Figure 1. (a) Schematic of a phase qubit controlled by an external flux fe and inductively coupled to the detector been done, or could be done in the near future. Once SQUID. (b) Energy profile V(f) with quantised levels the reader has the basic idea, we will proceed to representing qubit states. The tunnelling event through the generalise the process and develop its characteristics. barrier is sensed by the SQUID. 128 A.N. Jordan and A.N. Korotkov

classically a null-result measurement could be be- null-result cases, because it was not possible to cause the system is in the lower energy state (and distinguish if a tunnelling event happened during thus would not ever tunnel), or because it is in the measurement or during tomography. However, a upper energy level (and we simply didn’t give it simple trick of comparing the protocols with and enough time). without tomography made it possible to separate the The conditional probability for decaying if the null-result cases. qubit is in state j1i is given by 1 7 exp (7t). Given some initial density matrix rin, we can interpret the 3.2. Uncollapsing diagonal density matrix elements (rin,00 and rin,11) as classical probabilities of being in states j0i or We will now describe how to undo the state j1i. We can then invoke the Bayes rule [26,27] to disturbance ((1), (2), (3)) caused by the partial collapse update the probabilities,3 given the data of ‘nothing resulting from the null-result measurement. The un- happened’. It is not possible to use a simple general doing of this measurement consists of three steps [14]: rule for updating the off-diagonal element r01; how- (i) exchange the amplitudes for the states j0i and j1i ever, analysis of the full dynamics in the extended by application of a microwave p-pulse, (ii) perform Hilbert space [25,28] gives a simple result as well, so another measurement by lowering the barrier, identical that the qubit density matrix changes after the null- to the first measurement, (iii) apply a second p -pulse. result measurement as If the tunnelling did not happen during the second r measurement, then the information about the initial r ðtÞ¼ in;00 ; ð1Þ qubit state is cancelled (both basis states have equal 00 r r exp t in;00 þ in;11 ð Þ likelihood for two null-result measurements). Corre- r exp ðtÞ spondingly, according to Equations (1), (2) and (3) r ðtÞ¼ in;11 ; ð2Þ (which are applied for the second time with exchanged 11 r þ r exp ðtÞ in;00 in;11 indices 0 $ 1), any initial qubit state is fully restored. An added benefit to this strategy is that the phase j is rin;01exp ½ijðtÞexp ðt=2Þ also cancelled automatically; the physics of this phase r01ðtÞ¼ : ð3Þ rin;00 þ rin;11exp ðtÞ cancellation is the same as in the spin-echo technique for qubits. Such a measurement is ideal and does not decohere It is easy to mistake the above pulse-sequence as the qubit. Notice that in a simple model [10,28] simply the well known spin-echo technique alone. We there is no relative phase j that is added during the stress that this is not the case: spin-echo deterministi- measurement process, if one uses the rotating frame. cally reverses an unknown unitary transformation In the real experiment [25], however, the energy (arising, for example, from a slowly varying magnetic difference between the states j0i and j1i changes in field) without gaining or losing any information about the process of measurement because it is affected which state the system is in. Our strategy is probabil- by the changing external flux fe, and therefore even istic and requires erasing the classical information that in the (constantly) rotating frame the phase j is non- one extracts from the system to begin with. It is a zero. (probabilistic) reversal of a known non-unitary trans- Actually, in the experiment [25] the situation is formation – and therefore quite different from spin Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 even more complex because the tunnelling rate echo. gradually changes in time; also, instead of controlling While both measurements should give null results the measurement time t, it is much easier to control for the success of our procedure, let us define the the tunnelling rate. As a result, the measurement uncollapsing success probability PS as the null-result should be characterisedR by the overall strength probability of only the second measurement, assuming t 0 0 pt ¼ 1 exp ½ 0 ðt Þ dt . Nevertheless, for simpli- that the first measurement already gave the null result. city, we use here the physically transparent language Such a definition originates from our intention to of Equations (1), (2) and (3) with exp(7t) understood characterise the probability of undoing the first as 1 7 pt. measurement. (For the joint probability of two null Up to such changes of notation, the coherent results we will later use notation P˜S). To calculate PS, non-unitary evolution ((1), (2), (3)) has been experi- let us start with the initial qubit state rin. Then after the mentally verified in [25] using tomography of the post- first null-result measurement the qubit state is given by measurement state. The state tomography consisted Equations (1), (2) and (3), and after the p-pulse of three types of rotations of the qubit Bloch sphere, the occupation of the upper level is r~11 ¼ rin;00= followed by complete (projective) measurement. In ½rin;00 þ rin;11 exp ðtÞ. The success probability is the experiment it was not possible to select only the simply the probability that the second tunnelling Contemporary Physics 129

will not occur, PS ¼ r~00 þ r~11exp ðtÞ¼1 r~11ð1 null-result selection preferentially selects the cases exp ðtÞÞ, which can be expressed as with energy relaxation events, so that the procedure should no longer work well when 1 7 pt becomes et P : 4 comparable to the probability of energy relaxation. (It S ¼ t ð Þ rin;00 þ e rin;11 is interesting that in some range of parameters, the same procedure preferentially selects the cases without Notice that if t ¼ 0 (i.e. the measurement has not energy relaxation events, thus effectively suppressing started), then the success probability is 1 because the qubit decoherence [23] – see Section 8.3.) state never changed, and consequently does not need Exact uncollapsing requires an ideal detection, to be changed back. This is unsurprising. In the other which does not decohere a quantum state; Equations limit, given a sufficiently long time of no tunnelling, (1), (2) and (3) correspond to such an ideal detection. t 1, the qubit state j0i is indicated with high However, if various decoherence mechanisms are confidence after the first null-result measurement, and taken into account [28], then only imperfect uncollap- consequently the uncollapse success probability be- sing is possible. The theory of imperfect uncollapsing is comes very small, recovering the traditional statement a subject of further research. of irreversibility for a projective measurement. We stress that the possibility of uncollapsing as well as our formalism requires a quantum-limited detector, i.e. one 4. Example 2: double-quantum-dot charge qubit that introduces no additional dephasing to the system. The next example we consider is illustrated in Figure 2: For such a detector measuring a pure state, the state a charge qubit made of a double-quantum-dot (DQD), remains pure throughout the partial collapse, and the populated by a single electron, which is measured uncollapse. We also note that if the qubit is entangled continuously by a symmetric quantum point contact with other qubits, the uncollapsing restores the state of (QPC). This setup has been extensively studied in the whole system. earlier papers, both theoretically [29] and experimen- Uncollapsing of the phase qubit state has recently tally [30]. In contrast to the previous example, we will been experimentally realised by Nadav Katz and denote the basis states of the DQD charge qubit as j1i colleagues in the lab of John Martinis, at UC Santa and j2i (the electron being in one dot, or the other), to Barbara [21]. The experimental protocol was slightly be consistent with previous papers on this setup. The shorter than that described above: it was missing the measurement is characterised by the average currents second p-pulse, so the uncollapsed state was actually I1 and I2 corresponding to the states j1i and j2i, and by 5 the p-rotation of the initial state. Shortening of the the shot noise spectral density SI. We treat the protocol helped in decreasing the duration of the pulse additive detector shot noise as a Gaussian, white, sequence, which was about 45 ns, including the state stochastic process, and assume the detector is in the tomography. Since the qubit energy relaxation and weakly responding regime, jDIjI0, where DI ¼ I1 7 dephasing times were significantly longer, T1 ¼ 450 ns I2 and I0 ¼ (I1 þ I2)/2, with QPC voltage much larger and T2 ¼ 350 ns, the simple theory described above was sufficiently accurate. The same trick as for the partial-collapse experiment [25] was used to separate tunnelling events during the first, second, and tomo- Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 graphy measurements, because the detector SQUID was too slow to distinguish them directly. The uncollapse procedure should restore any initial state. However, instead of examining all initial states to check this fact, it is sufficient to choose four initial states with linearly independent density matrices and use the linearity of quantum operations [4]. In the experiment [21] the uncollapse procedure was applied 1/2 1/2 to the initial states (j0iþj1i)/2 ,(j0i 7 ij1i)/2 , j0i, Figure 2. Illustration of the uncollapsing procedure for the and j1i, and then the results were expressed via the charge qubit. The slanted lines indicate the deterministic language of the quantum process tomography [4] output of the detector in the absence of noise, if the qubit is 4 (QPT). The experimental [21] QPT fidelity of the in state j1i or j2i. The initial measurement yields the result r0. The detector is again turned on, hoping that at some future uncollapsing procedure was above 70% for pt 5 0.6. time the measurement result r(t) ¼ r0 þ ru(t) crosses the A significant decrease of the uncollapsing fidelity for origin, at which time the detector is turned off, successfully larger measurement strength pt, especially for pt 4 0.8, erasing the information obtained in the first measurement, was due to finite T1 time and the fact that the and restoring the initial qubit state. 130 A.N. Jordan and A.N. Korotkov

than all other energy scales, so that the measurement average current I tends to either I1 or I2 because the process can be described by the quantum Bayesian probability density P(I) of a particular I is formalism [13]. X PðIÞ¼ rii ð0ÞPiðIÞ; ð11Þ i¼1;2 4.1. Measurement dynamics for a non-evolving qubit and Pi (I) ! d(I–Ii) for t/TM ! ? Therefore, r(t) tends We begin for simplicity with the assumption that there to +?, continuously collapsing the state to either j1i is no qubit Hamiltonian evolution, so that the qubit (for r ! ?)orj2i (for r ! –?). Importantly, for the state evolves due to the measurement only (this can special case when the initial state is pure, the state also be effectively done using ‘kicked’ quantum remains pure during the entire process. Notice that the nondemolition (QND) measurements [31]). As was density matrix Equations (5), (6) and (7) formally shown in [13], at low temperature the QPC is an ideal coincide with Equations (1), (2) and (3) in the phase quantum detector (which does not decohere the qubit example, if the qubit states are renumbered, the measured qubit), so that the evolution of the qubit phase j is neglected, and t is replaced with 2r. density matrix r due to continuous measurement 1/2 preserves the ‘murity’ [32] M:r12/(r11r22) while the diagonal matrix elements evolve according to the 4.2. Uncollapsing for the charge qubit classical Bayes rule [26,27].3 We define the electrical In order to describe how to uncollapse the charge qubit currentR through the QPC averaged in a time t as state, we note that if r(t) ¼ 0 at some moment t, then t 0 0 IðtÞ¼½ 0 Iðt Þ dt =t, and the quantum Bayesian equa- the qubit state becomes exactly the same as it was tions read [13] initially, r(t) ¼ r(0), as follows from Equations (9) and (7). This of course must be the case if t ¼ 0, i.e. before r11ð0ÞP1ðIÞ the measurement began, but is equally valid for some r11ðtÞ¼ ; ð5Þ r11ð0ÞP1ðIÞþr22ð0ÞP2ðIÞ later time. To see why this is so from the informational point of view, we note that in the absence of noise, r ð0ÞP ðIÞ the measurement result from states j1i, j2i would r ðtÞ¼ 22 2 ; ð6Þ 22 simply be r1,2(t) ¼ +t/TM. With the noise present, the r11ð0ÞP1ðIÞþr22ð0ÞP2ðIÞ measurement outcome r(t) ¼ 0 splits in half the difference between states j1i and j2i. Such an outcome 1=2 M¼r12=ðr11r22Þ ¼ const; ð7Þ corresponds to an equal statistical likelihood of the states j1i and j2i, and therefore provides no informa- where the conditional (Gaussian) probability densities tion about the state of the qubit. of a current I realisation, given that the qubit is in j1i, Suppose the outcome of a measurement is r0, j2i are partially collapsing the qubit state toward either state j1i (if r0 4 0), or state j2i (if r0 5 0). The previous ‘no 1=2 P1;2ðIÞ¼ðt=pSIÞ information’ observation suggests the following strat- 2 egy for uncollapsing [14]: continue measuring, with the exp ½ðI I1;2Þ t=SI: ð8Þ hope that after some time t the stochastic result of Equations (5) and (6) may be simplified by noting the second measurement ru(t) becomes equal to 7r0, Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 so the total result r(t) ¼ r þ r (t) is zero, and therefore r ðtÞ r ð0Þ 0 u 11 ¼ 11 exp ½2rðtÞ; ð9Þ the initial qubit state is fully restored. If this happens, r22ðtÞ r22ð0Þ the measuring device is immediately switched off where we define the dimensionless measurement result and the uncollapsing procedure is successful (Figure 2). as However, r(t) may never cross the origin, and then the uncollapsing attempt fails. The probability of success tDI rðtÞ¼ ½IðtÞI0 PS is therefore the probability that r(t) crosses the SI Z origin (at least once) after it starts from r0. t DI 0 0 This strategy requires the observation of a parti- ¼ ½Iðt ÞI0 dt : ð10Þ SI 0 cular measurement result that may never materialise. The strategy shifts the randomness to the amount of Notice that r(t) is closely related to the total charge time that needs to elapse in order to find the desired passed through the QPC, and therefore r(t) accumu- measurement result. Of course, in a given realisation lates in time. For times much longer than the the measurement result could take on the desired value 2 ‘measurement time’ [33] TM ¼ 2SI/(DI) (the time scale multiple times, so we will take as our strategy to turn required to obtain a signal-to-noise ratio of 1), the off the detector the first time the measurement result Contemporary Physics 131

takes on r ¼ 0. In the classical stochastic physics this is these new coordinates will have its own probability known as a first passage process [34], the theory of of eventually crossing the origin, PC(r+). Because which is well developed and is detailed in Appendix 1. the diffusive dynamics is generated by choosing either In order to analyse the uncollapsing strategy rþ or r7 with equal weighting, it follows in the limit performance, in particular to find the success prob- dt ! 0 that ability P , it is important to notice that the off- S X 1 diagonal elements of the qubit density matrix r do not P ðr Þ¼ P ðr Þ: ð12Þ C 0 2 C come into play when we consider the detector output I(t) (this is true only in the case of zero or QND- eliminated qubit Hamiltonian; in Section 6 we will Expanding PC (r+) in this relation in a Taylor series, generalise to the finite qubit Hamiltonian case). As a we find from the linear in dt term that result, the quantum problem can be exactly reduced to 2 D @ PC ¼v1 @r PC; ð13Þ a classical problem by substituting r11(t)andr22(t) r0 0 with classical probabilities, evolving in the course of measurement according to the classical Bayes rule, where @ and @2 denote the first and second r0 r0 while evolution of the off-diagonal elements r12 ¼ r21 derivatives with respect to r0. Taking into account can be found automatically from the murity conserva- that PC ¼ 1 for r0 ¼ 0 and PC ¼ 0 for r0 ¼ ?, the tion law (7). We therefore model the qubit by a above differential equation may be easily solved to classical bit with probability p1 ¼ r11(0) of being find PC ¼ exp (7v1r0/D) ¼ exp (72r0). Now collect- prepared in state ‘1’ and probability p2 ¼ r22(0) of ing the probabilities of the zero line crossing for both being in state ‘2’. If the bit is in state ‘1’, the bit states, we find dimensionless measurement result r(t) evolves as a random walk with diffusion coefficient D ¼ (DI)2/ PS ¼ p~1PC þ p~2 ¼ exp ðr0Þ 2 4SI ¼ 1/2TM and drift velocity v1 ¼ (DI) /2SI ¼ =ðp1 exp ðr0Þþp2 exp ðr0ÞÞ: ð14Þ 1/TM (see Equations (8) and (10)). For the bit state ‘2’ the random walk of r(t) has the same diffusion The derivation for r0 5 0 is similar and leads to the 2 coefficient but the opposite drift velocity v2 ¼ 7(DI) / extra factor exp (2r0), so that the crossing probability 2SI. We are given the fact that the first part of the in both cases can be written as PS ¼ exp (7jr0j)/ measurement had the result r0 (i.e. we select only (p1 exp (r0) þ p2 exp (7r0)). such realisations). We need to analyse the stochastic Thus, using the trick of reducing the quantum behaviour of the total measurement result r(t) during dynamics to the classical problem, we have found the second part of measurement, with most attention the probability of successful uncollapsing for a DQD to the crossing of the zero line r(t) ¼ 0 (for conve- charge qubit with no Hamiltonian evolution [14]: nience we shift t ¼ 0 to the beginning of the second exp ðjr0jÞ measurement, so that r(0) ¼ r0). P ¼ ; ð15Þ S exp ðr Þr þ exp ðr Þr Let us find the probability PS of such a crossing. 0 in;11 0 in;22 Here we obtain it in a simple way, while in Appendix 1 we reproduce the result in a more complicated way, where rin characterises the qubit state before the first which also allows us to analyse the statistics of the measurement. Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 waiting time. For definiteness take r0 4 0 (this will be This result formally corresponds to Equation (4) extended later). Then the result r(t) will necessarily with substitution t ! 2r0 (for positive r0), and its cross 0 if the bit is actually in the state ‘2’, because in physical meaning is also similar. When the first this case r(t ¼ ?) ¼ 7? while r(t ¼ 0) ¼ r0 4 0. measurement result indicates either qubit state with The probability of being in the state ‘2’ is p~2 ¼ p2 exp good confidence (jr0 j1), the probability of success (7r0)/(p1 exp (r0) þ p2 exp (7r0)) from (9), which PS given by Equation (15) becomes very small, 3 differs from p2 because of the Bayesian update. If eventually becoming PS ¼ 0 for a projective measure- the bit is in state ‘1’ (this happens with probability ment, realised for r0 ¼ +?. In the other limit of p~1 ¼ 1 7 p~2), then r(t ¼ ?) ¼þ? and the crossing r0 ¼ 0, the success probability is unity because no time of 0 may never happen; however, it is still possible with needs to elapse – the state is already undisturbed. some probability PC, which depends on r0, and also on While the above method allows us to find the D and v1. To find PC, let us consider an infinitesimal success probability PS, there are other important time step dt and model the diffusion by discrete jumps characteristics of the uncollapsing strategy, which we in r of magnitude Dr ¼ + (2D dt)1/2. After a step dt, have not discussed, such as the mean waiting time the coordinate will then shift to one of two positions, required to uncollapse the state. This and other charac- 1/2 r ¼ r+, where r+ ¼ r0 þ v1dt + (2D dt) . Each of teristics of the strategy can be calculated through the 132 A.N. Jordan and A.N. Korotkov

1 more powerful methods of first-passage theory. This corresponding to the nonunitary inverse operator Mm , method and the results obtained from it are discussed multiplied by an arbitrary constant (which is not in Appendix 1. important because of the normalisation). This can be accomplished with another measurement, possibly together with unitary operations. As shown below, 5. General formalism we can realise measurement undoing if the second The previous two examples were fairly straightforward measurement realises a Krauss operator of the form to analyse because the unitary dynamics was effectively 1=2 turned off, and only the measurement dynamics needed L ¼ CULEm VL; ð19Þ to be accounted for. In general the situation is more complicated. For example, if the DQD qubit had a where UL and VL are any unitary operators, and C is finite Hamiltonian while the QPC was on, there would an unspecified constant that will be discussed later. be both measurement dynamics and Hamiltonian (The operator L also has a decomposition of the form dynamics. This situation leads us to discuss first the UE1/2, but with a different POVM element E.) The general formalism in its most abstract form before uncollapse then consists of three steps: first, the unitary turning our attention to this more complicated example. y y operator VLUm is applied to reverse the unitary part of Mm and prepare for the second measurement. Next the measurement operator L is applied. Finally the unitary 5.1. Formalism for wavefunctions y operator U is applied to reverse the remaining unitary Let us first consider a pure initial state jc i, and L in part of L. We can now see the effect of the uncollapsing postpone a generalisation to mixed states until Section operation on the state jc i by applying Equation (17) 5.2. In the formalism of a general (ideal) quantum m and the unitaries to find measurement [4] which transfers pure states into pure states, the measurement with result m is associated y y y ULLVLUmjcmi with the linear Kraus operator Mm, so that the jc i¼ ¼jc i; ð20Þ f y y y in probability of result m is jjULLVLUmjcmijj

2 y y Pmðjc iÞ ¼ jjMmjc ijj ; ð16Þ thus restoring the original state, because ULLVL in in y UmMm ¼ C, which is removed by the normalisation. where jj... jj denotes the norm of the state, and the (The phase of C is not important, since it affects only (conditioned) state after measurement is the overall phase of the wavefunction.) However, in order for the operator L to be physi- M jc i { c m in ; 17 cally realisable, the operator L L must belong to j mi¼ 1=2 ð Þ ½PmðjciniÞ another complete set of POVM elements, and therefore all its eigenvalues must not exceed unity (otherwise where the denominator makes jcmi properly normal- some states will be assigned probabilities that are ised. (Very often people prefer to omit this denomi- above unity; notice that the eigenvalues are non- nator and work with non-normalised states; this makes y 2 y 1 y negative automatically). Since L L ¼jCj VLEm VL,its the mapping linear.) The operators Em ¼ MmMm ðmÞ Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 eigenvalues are directly related to the eigenvalues p (called POVM elements [4]) are Hermitian and positive P i ðmÞ semidefinite by construction; these operators must of the operator Em. Expressing Em ¼ ipi jiihij, obey the completeness relation Sm Em ¼ 1, which where the eigenvectors jii form an orthonormalE basis, ensures that the total probability of all measurement { y the eigenvectors of L L are obviously VLji , and the results is unity. A measurement operator M can 2 ðmÞ m corresponding eigenvalues are jCj =p . Since all these always be written as i eigenvalues must not exceed 1, we find the following 1=2 2 Mm ¼ UmEm ; ð18Þ inequality on jCj , 2 ðmÞ where Um is a unitary operator (an important special jCj min p ¼ min Pm; ð21Þ 1=2 i i case is when Mm ¼ Em ; this corresponds to the ‘quantum Bayes theorem’ [6,35]). where min Pm is the probability of the result m, Now let us discuss wavefunction uncollapse in this minimised over all possible states jcini in the Hilbert ðmÞ general and abstract context [14]. The state disturbance space. The equality of min Pm to mini pi follows from rule (17) is typically a nonunitary one-to-one map in Equation (16). the Hilbert space. To undo the measurement with In general, the uncollapse cannot be accomplished known result m, we have to realise a physical process deterministically, since we rely on a measurement with Contemporary Physics 133

a specific result, corresponding to the operator L. measurement decreases. Mathematically, this means We can calculate the uncollapse success probability that some eigenvalues of Em become closer to 0. As a PS from Equation (16), with Mm ! L and cini ! consequence, min Pm becomes smaller (see Equation y y (21)), therefore lowering the upper bound for PS. For VLUmjcmi, a projective measurement PS ¼ min Pm ¼ 0, thus y y 2 making the state uncollapse impossible. PS ¼jjLV U jcmijj L m It is interesting to discuss the case when the initial 2 2 Cjc i jCj state jcini is known to belong to a certain subspace of ¼ in ¼ : ð22Þ 1=2 P c the Hilbert space, and we therefore wish to restore ½PmðjciniÞ mðj iniÞ states only in this subspace. In this case, the calculation 2 Now using the bound (21) for jCj , we find the bound of min Pm should be limited to this subspace, which for the success probability of uncollapsing after the may increase the bound (23) for the success probability first measurement with result m [14]: PS. A trivial example of such a situation is when the initial state jcini is known to Plato. Then it is not min Pm necessary to minimise Pm over all possible initial states PS ; ð23Þ PmðjciniÞ in Equation (23), because the set of possible states consists of only one (known) state, thus allowing where the denominator is the probability of the result uncollapsing with 100% probability. This is exactly the m for a given initial state, while the numerator is this case discussed at the beginning of Section 2. probability minimised over all possible initial states. The bound (23) is one of the most important results (notice a similar result in [16]) and deserves 5.2. Formalism for density matrices discussion. First, this bound is exact in the sense that it So far we have dealt only with pure states; however, it is achievable by an optimal uncollapsing procedure. is very simple to generalise the uncollapsing formalism This is because the uncollapsing operator with jCj¼ to include density matrices. In this case the initial 1/2 (min Pm) is still a physically allowed operator. As we density matrix rin is transformed by the first measure- will see later, the upper bound (23) is achievable in real ment into the state [4] experimental setups (in particular, in the examples y discussed in the two previous sections). However, non- MmrinMm rm ¼ ; ð24Þ optimal uncollapsing procedures, especially involving a Pm sequence of measurements, can lead to smaller success probabilities (an example of non-optimal uncollapsing where the probability Pm of the measurement result has been discussed in [36]; another example will be m is discussed at the end of Section 6). An analysis of the y procedures with an arbitrary sequence of measure- PmðrinÞ¼TrðMmMmrinÞ: ð25Þ ments and unitary operations is similar to the above: the corresponding measurement and unitary operators Using the uncollapsing procedure previously discussed should simply be multiplied. and using the same measurement operator L given by Notice that the success probability (22) and the Equation (19), we find that the uncollapsed state Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 inequality (23) depends on the initial state, which is y y y y y unknown to the person performing the uncollapsing ULLVLUmrmUmVLL UL rf ¼ ¼ rin ð26Þ (Plato, see description in Section 2). Therefore, the y y y TrðL LVLUmrmUmVLÞ success probability PS can be calculated by the man who knows what the initial state is (Socrates), while coincides with the initial state. The uncollapsing Plato can only estimate PS; for example, he can success probability PS is equal to the denominator in calculate the worst-case scenario (the minimum of PS Equation (26), and satisfies the relation over the accessible Hilbert space) or can calculate 2 the average of PS over all possible initial states (this PS ¼jCj =PmðrinÞð27Þ procedure will be discussed in the next subsection). Recalling the fact that it is not possible to undo a (as in Equation (22)). The constant jCj2 is still limited fully collapsed state due to the nature of projective by the inequality (21), and therefore the probability of measurement, the uncollapsing probability PS should success has the upper bound [14] decrease with increasing strength of the first measure- min Pm ment. Qualitatively, a stronger measurement is one PS ; ð28Þ that tends to a projection, as the uncertainty in the PmðrinÞ 134 A.N. Jordan and A.N. Korotkov

which is the same as the bound (23), except for the new In the general case the above statement, that both notation in the denominator, which reminds us of the ways of computing the averaged uncollapsing prob- possibly mixed initial state. The minimisation of Pm in ability are equivalent, can be proven both logically and the numerator should now be performed over the space explicitly. For the logical proof we notice that Plato’s of all possible initial mixed states; however, the result judgment of successful uncollapsing does not depend obviously coincides with the minimisation over the on whether or not Socrates knows the randomly pure states only. Similar to the case discussed in picked state; therefore, the average probability of the Section 5.1, the inequality (28) is the exact bound; it is cases judged to be successful should be the same in achieved by an optimal uncollapsing procedure, which both situations (whether or not Socrates knows what maximises jCj. the state is). Now let us also prove this statement If the initial state is pure, then the formalism of this explicitly, thus checking that our formalism is self- subsection is trivially equivalent to the formalism of consistent. Suppose the initial state is prepared by Section 5.1. It becomes more general in the case when Socrates by choosing randomly from a set of initial (k) the ‘actual’ initial state is mixed; for example, this states r with probabilities Pk (the most natural (k) happens when the initial state has been in contact with case is when initial states are pure, r ¼jckihckj; an unmonitored environment or Socrates prepares a however, this is not necessary). Then the bound for ðavÞ state by a blind random choice from a set of pure the average probability of uncollapsing success PS is states. A more interesting case for the result (28) is the average of the bounds (28): when the measured system is entangled with another X system, which does not evolve by itself. Then the ðavÞ min Pm P P0 : ð29Þ S k ðkÞ k formalism can be applied to the compound system; Pmðr Þ however, the measurement probability Pm depends 0 only on the reduced initial density matrix, traced over Notice, however, that Pk is the posterior probability the entangled second part. Therefore, in the entangled distribution given the result m, which is different from 3 bipartite case the uncollapsing procedure restores the Pk. We may now invoke the classical Bayes rule 0 state of the whole system, while the success probability [26,27] to relate the posterior Pk to the prior Pk and (k) PS is given by Equation (28) with rin being the reduced the conditional probability Pm(r ) to have result m density matrix. given state k, so that Another advantage of Equation (28) in comparison ðkÞ 0 P ðr ÞP with Equation (23) can be seen by referring back to the P ¼ P m k : ð30Þ k ðk~Þ preliminary discussion with the contest between Plato k~Pmðr ÞPk~ and Socrates. In the derivation of both results the initial state is the ‘actual’ initial state, which is known SubstitutingP Equation (30) into Equation (29) and to Socrates, but typically unknown to Plato. However, using kPk ¼ 1 in the numerator, we obtain as we will prove below, Equation (28) can still be used by Plato in a somewhat different sense: with r being ðavÞ min Pm min Pm in P P ¼ ; S ðkÞ ðavÞ 31 understood as an averaged density matrix representing Pmðr ÞPk Pmðr Þ ð Þ a distribution of possible initial states. In this case, P k ðavÞ ðkÞ Equation (28) gives the uncollapsing probability where r ¼ kr Pk is the averaged initial state. Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 averaged over this distribution. For example, if Plato This ends the proof that Equation (28) can be used for knew that Socrates’ strategy is to prepare one of two an unknown initial state, with rin being understood as possible (nonorthogonal) states jc1i, jc2i, with prob- the average of all possible initial states. abilities P and 1 P, then he could find the average uncollapsing probability in two ways. The first method is that he could simply average the uncollapsing 5.3. Uncollapsing probability, information, and an probabilities of the two states (also taking into account irreversibility measure the information acquired in the first measurement, see We defined the success probability PS as a probability below). Alternatively, he could recall that the random to uncollapse the post-measurement state rm. We now state preparation described above is equivalent to wish to start counting the overall success probability ˜ considering the initial density matrix rin ¼Pjc1ihc1jþ PS from the time before the first measurement, so that ˜ ð1 PÞjc2ihc2j, and then apply the result (28) to this PS is the probability of the pair of events: measurement density matrix.6 In this way, in the absence of any with result m and then successful uncollapsing. Using information, Plato could estimate his typical success Equations (27) and (28) we easily find the relation rate by calculating (28) for a fully mixed state, ~ 2 invoking the principle of indifference [27]. PS ¼ PmðrinÞPS ¼jCj ð32Þ Contemporary Physics 135

2 and the upper bound PS ¼jCj /Pm(rin) also depends on rin. However, the collapse–uncollapse probability P˜S of observing P~S min Pm: ð33Þ both the result m followed by a successful uncollapse is independent of rin – see Equation (32). Therefore, the Notice that P˜S is independent of the initial state. While combined effect of partial collapse and uncollapse this property may seem somewhat surprising, we will brings no information about the initial state. see later why it is rather obvious. More quantitatively, we can use the same frame- If we now wish to consider all possible results of the work as at the end of Section 5.2 in order to track the first measurement, and perform different uncollapsing information gain during the procedure. Suppose Plato procedures for each measurement result, then the total assigns an initial distribution Pk of possible initial ~total probability of uncollapsing PS is bounded as states as a statistical prior, to be updated as more X information comes in. The measurement with result m total P~ min Pm; ð34Þ brings in this information, so Plato updates his prior to S m 0 the posterior distribution P (see Equation (30)). k 00 and is also independent of the initial state. The bounds Calculating in a similar way the distribution Pk after (33) and (34) are exact and reachable by optimal the pair of the measurement and unmeasurement uncollapsing procedures. The bound (34) indicates results, we find that 1–Sm min Pm can be used as a measure of irrever- ðkÞ 0 ðkÞ 00 P ðr ÞP P~ ðr ÞP sibility (collapse strength) due to the measurement P S k P S k Pk ¼ ~ 0 ¼ ; 35 operation. P rðkÞ P~ ðrðkÞÞP ð Þ k~ Sð ÞPk~ k~ S k~ Now let us discuss the relationship between the uncollapsing procedure and our knowledge of the where PS(r) and P˜S(r) denote, respectively, the prob- initial state. While uncollapsing is possible even if we abilities of uncollapsing (28) and a combined collapse– know nothing about the initial state of the system, at uncollapse pair (32) for the initial state r. However, as first glance it seems like we gain some knowledge about we have already stressed, P˜S(r) is independent of the initial state in the process. This leads to the the initial state r, and therefore cancels out of the following interesting paradox, initially considered expression (35). This fact (and the normalisation of the by Royer [37]. By doing both a measurement and prior {Pk}) restores the initial prior distribution, 00 unmeasurement, one can seemingly learn something Pk ¼Pk, and therefore Plato has learned nothing, about the initial state without disturbing it. Then by thus avoiding the paradox. Reversing the logic, in order repeating the measurement þ unmeasurement process to avoid the paradox, P˜S must be independent of the many times, even though the probability of such an initial state, as found in Equations (32) and (33). event rapidly decreases to zero, the successful event would lead to essentially perfect knowledge of the initial state, leaving the state itself perfectly intact! One 5.4. Revisiting the previous uncollapse strategies could then violate a host of known results, such as the With these general and abstract results in hand, we can no-cloning theorem. now revisit the two examples discussed in Sections 3 The resolution of the paradox lies in the fact that and 4. Notice though that the principle drawback of the pair of measurement and unmeasurement actually the general results is that while they give the ideal Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 brings exactly zero information. Uncollapsing the state success probability, they do not provide the strategy as can only occur when the information in the second to how achieve the upper bound. measurement exactly contradicts the information The measurement situation in the case of the phase gained in the first measurement, thus nullifying it. qubit (Section 3.1) may be described with a two- This can happen in weak quantum measurements outcome POVM, with elements En and Ey, where n because there is uncertainty about the system in the denotes the null result, and y denotes the affirmative measurement result. It is to the extent that this (tunnelling) result. The POVM elements, given in the ambiguity exists that it is possible to undo the weak j0i, j1i basis are measurement. Let us examine this in more detail. We learn something about a pre-measurement state 10 00 En ¼ ; Ey ¼ ; ð36Þ when the measurement result depends on the state. The 0 expðtÞ 01expðtÞ measurement with result m brings some information about r because the probability P (r ) depends with the obvious completeness relation E þ E 1. in m in n y ¼y on the initial state rin. The ability to successfully un- It is interesting to notice that while En ¼ Mn Mn collapse the state also brings some information about corresponds to a Kraus operator Mn (see below), no the initial state because the uncollapsing probability meaningful Kraus operator My can be introduced for 136 A.N. Jordan and A.N. Korotkov

the POVM element Ey, because in the case of a by comparing the general upper bound (28) for the tunnelling event the system leaves its two-dimensional success probability PS with the result (15). Substituting Hilbert space and becomes incoherent (so that a the probabilities in the bound (28) with probability single Kraus operator cannot be introduced even in densities, we find the extended Hilbert space). However, this is not important for us because we are interested in the null- min fP1ðIÞ; P2ðIÞg PS ; ð38Þ result case only. The null-result Kraus operator Mn P1ðIÞrin;11 þ P2ðIÞrin;22 can be found by comparing the phase qubit dynami- cal Equations (1), (2) and (3) with the general where I corresponds to the measurement result r0. This state disturbance rule (24); this gives Mn ¼ diag{1, bound coincides with Equation (15) because P1(I)/ exp (7t/2) exp (7ij)}. P2(I) ¼ exp(2r0), which proves the optimality of the It is easy to find that the uncollapsing strategy of discussed ‘wait and stop’ strategy. (Surely, there are Section 3.2 based on two null-result measurements is numerous non-optimal uncollapsing strategies; for optimal in the sense that its success probability PS example, by stopping the measurement after the reaches the upper bound (28). The numerator of (28) second crossing of the origin by r(t).) is the smallest eigenvalue of En, which is exp (7t), It is also instructive to not specify the result of the while the denominator is Tr(Enrin) ¼ rin,00 þ first measurement, but to find the total probability ~total exp (7t)rin,11. Therefore, the bound (28) coincides PS (see Equation (34)) that the initial qubit state can with Equation (4), thus confirming the optimality of be restored after a measurement for time t. This is the analysed uncollapsing procedure. given by averaging PS in Equation (15) over the results The overall success probability P˜S, which is the r0 with the corresponding weights (11). This averaging joint probability of two null results is is technically easier using the form of Equation (38) and gives ~ PS ¼ðrin;00 þ exp ðÞtrin;11ÞPS ¼ exp ðtÞ: ð37Þ ! Z 1=2 ~total t PS ¼ dIminfP1ðIÞ;P2ðIÞg ¼ 1 erf ; As expected (see Section 5.3) this probability does not 2TM depend on the initial state. It is easy to see that it also ð39Þ coincides with the upper bound (33) for P˜S. 2 While the analysed double-null-result uncollapsing which depends only on the ‘strength’ t/TM ¼ t(DI) / strategy is optimal, an example of a non-optimal 2SI of the first measurement, but not on the initial uncollapsing for a phase qubit was considered in [36]. state, as expected from the discussion in Section 5.3. It was shown that if the measurement process is Notice that the result (39) reaches the upper bound performed simultaneously with Rabi oscillations, then (34) because (15) reaches the upper bound (28). in the null-result case the initial state is periodically restored. The non-optimality of uncollapsing for such a procedure is due to measurement of an evolving 6. Evolving charge qubit qubit, which corresponds to a sequence of many Armed with the general uncollapsing formalism measurements; a similar reason for the non-optimality of Section 5, we now turn to the case of the will be discussed at the end of Section 6. finite-Hamiltonian DQD measured by QPC, by Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 Now let us turn to the DQD charge qubit example including internal evolution of the qubit via a qubit of Section 4. First, the qubit measurement dynamics Hamiltonian, [(5), (6) and (7)] can be related to the general POVM- HQB ¼ðe=2Þsz þ Hsx; ð40Þ type measurement formalism in the following way (similar to [32]). For a fixed time t the measurement where e is the energy asymmetry between the quantum result m can be associated with the averaged QPC dot levels, and H is the tunnel coupling between the current I (or, equivalently, with the dimensionless dots. In this case Equations (5)–(7) are no longer valid quantity r). The Kraus operator Mm in the measure- and should be replaced by the Bayesian equations [13] ment basis j1i and j2i then should be chosen as (in Stratonovich form [38]) 1/2 1/2 Mm ¼ diag{[P1(I)] ,[P2(I)] } in order to reproduce 2DI Equations (5), (6) and (7). Notice that Pm in Equation r_ 11 ¼r_ 22 ¼2HImr12 þ r11r22 ½IðtÞI0; ð41Þ (25) now describes the probability density of the result SI I instead of probability, because the measurement r_ ¼ ier þ iHðr r Þðr r Þ result becomes a continuous variable. 12 12 11 22 11 22 We can now check to see if the ‘wait and stop’ DI ½IðtÞI0 r12; ð42Þ uncollapsing strategy of Section 4.2 is the optimal one SI Contemporary Physics 137

where I(t) is the QPC current, transformation may be chosen to be the short corre- lation time T discussed above, thus cancelling the

IðtÞ¼r11ðtÞI1 þ r22ðtÞI2 þ xðtÞ; ð43Þ large term and making Equations (45)–(47) well defined. The only price to be paid for this transforma- containing white noise x(t) with spectral density SI, tion is an altered normalisation, that will cancel in the and we use h ¼ 1. These evolution equations are normalised density matrix (44). nonlinear and not very simple to deal with. To discuss For a particular realisation of the detector output the undoing of a continuous measurement, it is more I(t0), 0 t0 t, Equations (45)–(47) define a linear convenient to use a non-normalised density matrix s, map s(0) ! s (t), corresponding to a particular Kraus which has an advantage of dealing with linear operator Mm (which, therefore, can be denoted as equations. M{I}). For the uncollapsing we have to realise the map, 1 We rewrite Equations (41)–(42) in the form corresponding to the inverse Kraus operator CMfIg (see Section 5.1). It is obvious that in contrast to the r ¼ s=Trs; ð44Þ case of the non-evolving qubit discussed in Section 4,

1 2 this cannot be done by simply continuing the s_ 11 ¼2H Ims12 s11 ½IðtÞI1 ; ð45Þ SI measurement and waiting for a specific result. The reason is that now the map is characterised by six real 1 2 parameters (eight parameters for a linear operator s_ 22 ¼ 2H Ims12 s22 ½IðtÞI2 ; ð46Þ 1 SI CMfIg with neglected overall phase and normalisa- tion), instead of one parameter for the non-evolving s_ ¼ ies þ iHðs s Þ case (see Equations (7) and (9)). We will discuss a little 12 12 ()11 22 2 2 later how the six-parameter uncollapsing procedure ½IðtÞI0 ðDIÞ s12 þ ; ð47Þ can be realised explicitly. Before that we discuss how to SI 4SI find the operator M{I} in a more straightforward way, from Equations (45)–(47). so that s(0) ¼ r(0), while the ratio s(t)/r(t) decreases Let us consider only the evolution of (unnorma- with time and is equal to the normalised probability lised) pure states jc(t)i¼a(t)j1iþb(t)j2i, so that density of the corresponding realisation of the detector s ¼jcihcj. Then Equations (45)–(47) can be rewritten output I(t0), 0 t0 t.7 In the language of general as quantum measurement this formulation corresponds e 1 2 to omitting the denominator in Equation (24). Notice a_ ¼þi a iHb a ½IðtÞI1 ; ð48Þ 2 2S that we still consider an ideal detector, so an initially I pure state remains pure, s 2 s s .7 j 12 j ¼ 11 22 _ e 1 2 b ¼i b iHa b ½IðtÞI2 ; ð49Þ A casual inspection of Equations (45)–(47) shows 2 2SI that they are seemingly not well defined because the 2 2 2 terms [I(t)–I0,1,2] contain the term x(t) ¼ ? (from where the infinite part of I (t) can be cancelled in the the relation hx(t)x(0)i¼(SI/2)d(t)). This divergence is same way as discussed above. The linearity of these artificial because there will always be a small correla- equations guarantees that for any given realisation of tion time T of the noise and/or a finite detector I(t), it is sufficient to solve (48) and (49) for the initial Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 bandwidth B (corresponding to T ¼ 1/4B), so there states j1i and j2i in order to find the solution for an 2 will be a large but finite constant C¼hx(t) i¼SI/4T arbitrary initial state of the qubit. Defining v1 ¼ 2 contained in terms of the form [I(t) 7 I0,1,2] .Itis a1(t)j1iþb1(t)j2i as the solution of (48) and (49) for easy to see that Equations (45)–(47) do not change initial state j1i, and v2 ¼ a2(t)j1iþb2(t)j2i as the if we subtract the same constant from these terms solution of (48) and (49) for initial state j2i, we can 2 2 [I(t) 7 I0,1,2] ! [I(t) 7 I0,1,2] 7 C. This can be shown write the solution for an arbitrary initial state jcini¼ by considering another unnormalised density matrix jc(0)i¼aj1iþbj2i as jc(t)i¼av1 þ bv2. Therefore, Z ¼ s exp (t/T). Writing the linear Bayesian equations the Kraus operator M{I} for a given realisation of I(t), (45)–(47) in the form s_ ¼ f ½s, the equations trans- in the j1i, j2i basis is ij ij form to Z_ ¼ f ½Z exp ðt=TÞ exp ðt=TÞþZ =T under ij ij ij a ðtÞ a ðtÞ the change of variables. The unnormalised Bayesian M ¼ 1 2 : ð50Þ fIg b ðtÞ b ðtÞ equations are linear in the density matrix elements 1 2 sij, so the exponential factors cancel out. The new For the uncollapsing we need to apply the Kraus 1 equations are thus the same as the old ones with a operator CMfIg, which maps the state v1 onto Cj1i and 2 constant C¼SI/4T subtracted from the [I(t) 7 I0,1,2] the state v2 onto Cj2i. The reason why we need a non- terms. The unspecified constant T in the density matrix unitary transformation is that the vectors v1 and v2 are 138 A.N. Jordan and A.N. Korotkov

y 1 1 in general non-orthogonal and have different norms. V ¼ U CL MfIg. For brevity we will not show the Geometrically, such a transformation can be done matrices U and V explicitly. by using a relative shrinking or stretching of two In the physical realisation of the uncollapsing orthogonal axes (found for a given M{I}), which would procedure the measurement step L can be performed make v1 and v2 orthogonal and equal in norm, in exactly the same way as in Section 4.2. Comparing followed by a unitary transformation (this would Equation (53) with Equations (9) and (7), we see that correspond to the decomposition of the form UE1/2 – the continuous measurement by the QPC should be see Section 5.1). However, for a practical realisation of stopped when the dimensionless measurement result uncollapsing it is most natural to use the shrinking or r(t) reaches the value stretching of the axes j1i and j2i via a continuous 1=2 1=2 QND measurement with the QPC in the way con- r1 ¼ lnðlþ=lÞ > 0orr2 ¼ lnðl=lþÞ < 0; ð55Þ sidered above for a non-evolving qubit. In this case the uncollapsing procedure can be done in three steps (see for the first and second choice in Equation (53), Section 5.1): unitary evolution V, continuous QND respectively (the choice should be made beforehand, measurement (where the qubit Hamiltonian is turned since it determines operation V). As previously men- off, e ¼ H ¼ 0) described by a diagonal matrix L, and tioned, the constant C is not important here because a final unitary operation U (in the notation of Section the physical state is always normalised. The procedure y y y fails if the desired result is not reached during the 5.1, V corresponds to VLUm, and U corresponds to UL). These operators should satisfy continuous measurement. The unitary operations V and U can be practically 1 realised in three substeps each: z-rotation on the Bloch ULV ¼ CMfIg; ð51Þ sphere by applying non-zero energy asymmetry e for and it is easy to find U, L,andV explicitly by some time, y-rotation by applying non-zero tunnelling recognising Equation (51) as a singular value decom- H, and then one more z-rotation. However, the last 1 position of the operator CMfIg (recall here that L is z-rotation of V and the first z-rotation of U are simply diagonal; also notice that the standard form for the added to each other (since L does not change the singular value decomposition is slightly different, with relative phase of the state components or, equivalently, V denoted as V{). the azimuth angle on the Bloch sphere). The corre- To find L explicitly, we notice that sponding trivial degree of freedom can be eliminated, for example, by realising the operation V in only two 2 y 2 y substeps, without the second z-rotation. C MfIgMfIg ¼ UL U ; ð52Þ Let us count the number of real parameters, 2 y which is simply the diagonalisation of C MfIgMfIg. characterising the uncollapsing procedure. Since V Therefore, and U together provide 2 6 3–1¼ 5 parameters, and ! !the desired result r in the measurement step adds one l1=2 0 l1=2 0 more parameter, the overall number of parameters is 6. L ¼ C or L ¼ C þ ; ð53Þ 1=2 1=2 As expected, this is exactly the needed number of 0 lþ 0 l parameters characterising an arbitrary Kraus operator where for the qubit (neglecting normalisation and overall Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 2 2 phase). Let us also mention the fact from linear algebra jjv1jj þjjv2jj that the singular value decomposition (51) is unique l ¼ 2 2 3 in the non-degenerate case (l 4 l 7 4 0), up to the !1=2 þ 2 2 2 permutation of singular values (corresponding to the 4 jjv1jj jjv2jj 25 þjv1 v j ð54Þ choice in Equation (53)) and arbitrary phase factors in 2 2 columns of U, with compensating changes in V (this corresponds to the previously discussed compensation y are the eigenvalues of the operator MfIgMfIg and the of the z-rotations). vectors vi are defined above Equation (50). The Now let us discuss the probability PS of the 2 2 2 Cauchy–Schwartz inequality, jv1 v2j jjv1jj jjv2jj , successful uncollapsing. From the general theory of guarantees the non-negativity of l7. (The notation Section 5 (Equation (23)), it is bounded from above v1 v2 is used for the inner product hv2jv1i of v2 and v1.) by a fraction, PS minP{I}/P{I}(jcini), in which the To find U, we use Equation (52) again and see that denominator is the probability density of the given the columns of U are composed of the eigenvectors of realisation I(t) for the initial state jcini¼aj1iþbj2i y 2 2 MfIgMfIg (the sequence of columns depends on the (with jaj þjbj ¼ 1), while the numerator is this choice in Equation (53)). Finally, V is given by probability minimised over all initial states. So, the Contemporary Physics 139

into an orthogonal pair of vectors, while the goal of the denominator is given by the squared norm of the final second measurement is to equalise their norms, state jc(t)i¼av þ bv , 1 2 keeping them orthogonal. The first goal can be 2 achieved by stretching/shrinking of the Hilbert space PfIgðjciniÞ ¼ jjav1 þ bv2jj ; ð56Þ along any axis u of the form v1 þ cv2 (v1v2)/jv1v2j with an arbitrary positive real number c (it is easy to while the numerator is given by minimising (56) over visualise this procedure of making two vectors all normalised initial states. It is easy to see that this orthogonal by assuming the space of real vectors, for minimum is equal to the minimum eigenvalue l y 7 which the axis u is geometrically in between v and v ; of the operator M M , given by Equation (54); 1 2 fIg fIg the same geometrical idea works for complex vectors). therefore, We recall that measurement for a non-evolving qubit l (Section 4.1) stretches (squeezes) the j1i axis, while PS 2 : ð57Þ jjav1 þ bv2jj squeezing (stretching) the j2i axis. Therefore, the first goal can be achieved by a unitary operation which Converting this result into the language of density rotates u into 1 , followed by a continuous measure- matrices and simultaneously generalising it to an j i ment (with a QPC) of a non-evolving qubit, to be arbitrary initial state r , we obtain the bound in stopped when the mapped vectors become orthogonal. 1=2 2 2 2 2 2 After the vectors {v1, v2} are transformed into an jjv1jj þjjv2jj jjv1jj jjv2jj 2 2 2 þjv1 v2j orthogonal pair by the (successful) first measurement, PS ; ð58Þ the second part of the procedure should stretch/shrink 2 2 rin;11jjv1jj þ rin;22jjv2jj þ 2Re½rin;12 v1 v2 the 2D Hilbert space along the resulting vectors to make them equal in norm. This can be done similarly, in which the numerator is the explicit expression (54) by a unitary rotation and partial measurement of a for l . It is easy to check that this result reduces to 7 non-evolving qubit. Finally, another unitary operation the bound (38) in the non-evolving case, in which can be used to map the resulting pair of vectors into v ¼ ([P (I)]1/2,0)T and v ¼ (0,[P (I)1/2)T. 1 1 2 2 {C 1 ,C2 }, thus completing the uncollapsing proce- The uncollapsing procedure discussed in this j i j i dure. Notice that both measurements are performed in section is optimal in the sense that it corresponds the ‘wait and stop’ manner, and both measurements to the upper bound of Equation (58). To prove this should be successful to realise the uncollapsing. While statement, instead of calculating P explicitly, let us S the successfully uncollapsed state is still perfect in this use the fact (see Section 5.3) that the product P P S {I} procedure, the probability of success is lower than the cannot depend on the initial state. Therefore, it is bound (58). To prove this non-optimality, let us again sufficient to prove the optimality of PS only for one use the initial eigenstate jcini, which corresponds to initial state. Let us choose the state jcini that is the y eigenvalue l7, so that the bound (58) is 100%. Then eigenvector of MfIgMfIg, corresponding to the eigen- the probability of success for the first measurement is value l–. Then after the first measurement (operator in general less than 100% (it is 100% only for one M{I}) and the unitary operation V it is transformed specific axis discussed previously, while here we into one of the basis states (j1i or j2i for the first or consider a range of possible axes by allowing c to second choice in (53), respectively). Recall that for the vary). Thus, the success probability is less than 100% Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 non-evolving (QND) measurement case, r(t) ! ? for for this special state, and therefore PS is below the the initial state j1i, while r(t) ! 7? for the initial bound (58) for any initial state. state j2i. The crossing thresholds (55) indicate that the measurement L is always successful because r(t) necessarily crosses the desired value (which is positive 7. General procedure for entangled charge qubits for j1i and negative for j2i, as discussed above). Let us present an explicit procedure [14] which can be Therefore, the uncollapsing success probability for used in principle to undo an arbitrary measurement this special state is 100%, that is equal to the upper Mm of any number N of entangled qubits with maxi- bound (58). As mentioned above, the optimality of mum probability of success PS. For simplicity we the procedure for this special state also proves the consider double-quantum-dot charge qubits and optimality for any initial state. assume that any unitary transformation can be used Obviously, an uncollapsing procedure can also be in the procedure. If the operator Mm was produced by non-optimal. As an example, let us consider a proce- a one-qubit measurement, and other entangled qubits dure which realises the desired mapping {v1, v2} ! were not experiencing a Hamiltonian evolution, then {Cj1i,Cj2i} using two measurements instead of one. the formalism of Section 4.1 is essentially unchanged The goal of the first measurement is to map {v1, v2} [40], and uncollapsing of the measured qubit leads to 140 A.N. Jordan and A.N. Korotkov

the restoration of the whole entangled state. If the even for the N-qubit state j1i, the rate g of electron operator Mm was produced by a one-qubit measure- tunnelling through the QPC is rather low, so that we ment, while other qubits were evolving in a unitary can distinguish single tunnelling events (technically, way but not interacting with the measured qubit, then this would require an additional single-electron tran- uncollapsing is also easy: we should uncollapse the sistor). If we perform the measurement during time measured qubit in the usual way (Section 4.2) and t and see no tunnelling through the QPC, then should apply the inverse unitary transformation for the similarly to the case of Section 3.1, the corresponding other qubits. In this section, however, we do not null-result Kraus operator Mn shrinks the j1i axis of consider these simple special cases; the goal is to undo the Hilbert space by the factor exp(7gt/2), while an arbitrary Kraus operator Mm. leaving all perpendicular axes unchanged. For the 1=2 Let us decompose Mm as Mm ¼ UmEm (see Equa- matrix elements this means h1jMnj1i¼exp ðgt=2Þ; ? ? 0 ? ? tion (18)). Reversing the unitary operation Um can be hcj jMnjcj 0 i¼djj ; hcj jMnj1i¼h1jMn þ cj i¼0, N ? done in the regular Hamiltonian way, so the nontrivial where we introduced a set of 2 7 1 states jcj i 1=2 part is undoing the Em operator. WeP recall the spanning the subspace orthogonal to j1i. ðmÞ ~ diagonalisation of Em is given by Em ¼ i pi jiihij The general strategy to implement the operator L is with vectors jii forming an orthonormal basis. As the following. We first note that in the basis jii that discussed in Section 5.1, for the optimal uncollapsing diagonalises L~, this diagonal matrix can be represented which maximises the success probability, we have to as a product of 2N diagonal matrices, where each term perform a procedure corresponding to the measure- in the product has all diagonal entries as 1, except ~ ðmÞ 1=2 1=2 ~ ment operator L ¼ðmin jpj Þ Em which is also the ith entry: diag {1, 1,...,Lii,...,1}.Each of these diagonal in the basis jii with corresponding matrices may be interpreted as a separate Kraus ~ ~ ðmÞ ðmÞ 1=2 matrix elements Lii ¼hijLjii¼½ðmin jpj Þ=pi , operator that can be sequentially implemented. Thus, all of which are between 0 and 1. Given N qubits, i the explicit physical procedure consists of 2N steps, ranges from 1 to 2N. Notice that L is obviously each of which has three substeps. First, we apply a Hermitian. unitary transformation U1 which transforms the first Our procedure is to realise L~ with a sequence of basis vector ji ¼ 1i into the state j1i. Then the null-result measurements and unitary operations. evolution of all qubits is stopped, and the detector is Shown in Figure 3 is an illustration of the physical turned on for a time t1. This time is chosen so that set-up that is used for the measurements: a QPC the null-result Kraus operator L(1) has the desired (tunnel junction) capacitively coupled to N non- matrix element h1jL(1)j1i¼L11; this condition yields –1 evolving DQD charge qubits. We assume that the t1¼ –2g ln L~ii. The measurement is then followed by QPC is tuned to a highly nonlinear regime, for which y the reverse unitary, Ui , to take the state j1i back to no electron can tunnel across the QPC barrier on state ji ¼ 1i. This three-substep procedure is then experimentally relevant time-scales unless all qubits are repeated for i ¼ 2,3,...,2N, sequentially transforming in the state j1i. We name this multi-qubit state the state jii to j1i with unitary Ui, and performing –1 j1i:j1,1,...,1i. Such a regime is possible because of measurement with the detector for a time ti ¼ –2g ln the exponential dependence of the tunnelling rate on ~ y Lii, followed by the reverse unitary, Ui . This sequence QPC barrier height, while the barrier height depends of steps decomposes the uncollapsing operator L~ as linearly on the states of the coupled qubits. Of course, Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 this regime is not quite realistic; however, we discuss ~ y ð2NÞ y ð2Þ y ð1Þ ð59Þ L ¼ U N L U2N ...U L U2U L U1: the procedure in principle. We also assume that 2 2 1 The uncollapsing procedure is successful only if there were no tunnelling events in the QPC. By construction, the success probability PS for this procedure maximises the general bound (28). The success probability PS for the uncollapsing ~ ~y ~y ~ process rm ! rin with rin ¼ Lr~L =TrðL Lr~Þ and y r~ ¼ UmrmUm, can be calculated as X X ~y ~ ~2 PS ¼ TrðL Lr~Þ¼ Liir~ii ¼ r~ii expðgtiÞ; ð60Þ i i Figure 3. Schematic set-up for uncollapsing of N entangled qubits. The tunnel junction detector (QPC) is in a strongly where r~ij are the matrix elements of r~ in the basis jii, nonlinear regime, so that an electron can tunnel through it which diagonalises L~. We may also find this result with rate g only when all qubits are in state j1i. from another perspective by realising that the success Contemporary Physics 141

probability is simply the product of the null-result through the charge degree of freedom. This technique ðiÞ N probabilities pS of all 2 measurements, circumvents the otherwise difficult problem of control- P P ling the weakly interacting spin. While we refer the Y2N i 2N r~jj exp ðgtjÞþ r~jj reader to [41–45] for the details, we will give a P ¼ PðiÞ; pðiÞ ¼ P j¼1 Pj¼iþ1 ; S S S i1 2N simplified thumb-nail sketch of the physics here. i j¼1 r~jj exp ðgtjÞþ j¼iþ1 r~jj The qubit is encoded with two electron spins, 61 ð Þ where each electron is confined in a separate quantum ðiÞ where the expression for pS comes from comparing dot. In contrast to our charge qubit discussion, these the traces of unnormalised density matrices after each dots are open, with electrons able to enter and leave. of 2N steps of the procedure. It is instructive to show Electrical bias is applied across this double quantum ðiÞ explicitly that this expression for pS is equal to the dot leading to charge transport. Electrons can tunnel expected expression sequentially, but spin blockade [46] restricts transport to situations where the two electrons form a spin pðiÞ ¼ 1 ½1 exp ðgt Þr~ðiÞ; ð62Þ S i ii singlet (0,2)S on the right dot while the spin triplet in which r~ðiÞ is the normalised density matrix before the (0,2)T is outside the transport energy window due to ith step of the procedure (after i 7 1 null-result steps). the large single quantum dot exchange energy (here This can be done if we prove the relation (n, m) refers to n electrons on the left dot and m electrons on the right dot). This blockade physics Yk Xk ðiÞ provides an interesting initialisation procedure of the ½1 r~ii ð1 expðgtiÞÞ ¼ 1 ½1 expðgtiÞr~ii; i¼1 i¼1 quantum register – when the single-electron current ð63Þ stops flowing, we are confident that the two-electron state is in a (1,1)T state, because in the absence of (notice that the right-hand side of this equation is spin flip processes, the tunnelling transition to the (0, equal to the numerator in Equation (61) with 2) state is forbidden. From this configuration, it is substitution k ! i). Equation (63) can be provenQ by possible to manipulate the system by applying ðiÞ i1 induction using the relation r~ii ¼ r~ii= j¼1 ½1 electron spin resonance pulses [42], transitioning the ðjÞ state to have overlap with the singlet state. Thus, the ½1 exp ðgtjÞr~jj , which can be easily derived recur- electron on the left dot may tunnel (with rate )to sively, r~ðjÞ ! r~ðjþ1Þ, starting from r~ð1Þ ¼ r~ . In this way ii ii ii ii the right dot and exit the system, giving rise to a we show consistency between the null-result probabil- small electrical current at the drain when this process ities given by Equations (61) and (62), permitting the is repeated many times. Of course, this will happen calculation of P in two independent ways. S with some probability controlled by the overlap of the Let us mention again that the uncollapsing proce- state with the singlet. dure considered in this section reaches the upper Drawing on our experience with the phase qubit bound (28) for the success probability P , that can be S (see Section 3), it is clear how to devise a weak seen both by construction and explicitly. measurement experiment and an uncollapsing experi- ment: the allowed transition can be permitted for 8. Recent developments in wavefunction uncollapse a time of one’s choosing and then forbidden by Before concluding, we wish to give a summary of detuning the energy levels with a voltage pulse to one Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 some interesting recent developments in this area of of the quantum dot’s gates. In this way one can research. We will briefly discuss two theory proposals weakly probe the two-electron state, and in the null- and one experiment. result case (no single electron tunnelling) partially collapse it to the triplet subspace. In order to propose the uncollapse part of the experiment, it is easiest to 8.1. Spin qubit consider the case when the surrounding nuclear spins The examples given above mainly concern quantum [42] quickly admix the singlet state with a triplet state, dot charge qubits. It is a natural question if a similar permitting the two-qubit state to encode one effective kind of partial collapse/uncollapse can be carried over qubit: parallel or anti-parallel spins. The weak to spin qubits. An analysis of this situation was carried measurement technique described above will then out by Trauzettel, Burkard, and one of the authors partially collapse the state toward the parallel state [41]. There it was shown how an uncollapse measure- under a null-measurement (no single electron tunnel- ment can be realised using a scheme similar to the ling). If now a p-pulse is applied to one of the spins recent experiments by Koppens et al. [42]. The essential with electron spin resonance, followed by a second idea of the spin-qubit experiments [42–45] is to null-measurement, this was shown to uncollapse the manipulate and measure the spin of a single electron state of the effective qubit [41]. 142 A.N. Jordan and A.N. Korotkov

measurement can differ from pt. The calculations show 8.2. Optical polarisation qubit [23] that this procedure can significantly increase the Another interesting development is the experimental quantum memory fidelity, and theoretically it can be implementation of wavefunction uncollapse for optical made arbitrarily close to 100% even for a significant qubits using the polarisation degree of freedom of energy relaxation during the storage period. This single photons by Kim et al. [22] The weak measure- is because the uncollapsing procedure preferentially ment was implemented by passing the photon through selects the cases without energy relaxation. For the a glass plate oriented at the Brewster angle. Only the initial state j0i the energy relaxation is absent by itself, vertical polarisation is reflected off of the glass plate while for the initial state j1i the operating principle (with some probability). By placing a single-photon is the following: the first measurement keeps it as j1i, detector where the photon would have gone had it but if the state jumps down to j0i during the storage reflected, a null-click measurement partially collapses period, then most likely there will be tunnelling during the polarisation state to the horizontal polarisation. the second measurement, and therefore such events The strength of the measurement can be increased by will be eliminated by the selection of only null-result placing a series of plates in a row, effectively increasing cases. This is surely a simplified explanation, but the the net probability of a vertically-polarised photon same idea works in the rigorous calculations as well. reflecting at some point. Notice though that stronger decoherence suppression The wavefunction uncollapse is done by inserting a requires a stronger measurement and therefore a half-wave plate (exchanging the amplitude of horizon- smaller probability of selection. Estimates show that tal with vertical polarisation), and having the same a significant decoherence suppression by uncollapsing number of plates traversed by the photon again. If can be demonstrated with present-day phase qubits. none of the single-photon detectors click, the polarisa- tion state is uncollapsed. This has been verified [22] with quantum state tomography (with polariser and 9. Conclusion single-photon counter placed after all of the reflecting We have reviewed and extended recent developments plates) on the photon, conditioned on none of the in the theory (and experiment) of wavefunction other photon detectors firing. The experiment showed uncollapse by undoing quantum measurements. We an uncollapsing fidelity of above 94% for measure- have formulated the problem of wavefunction uncol- ment strengths up to 0.9. It was also pointed out that lapse in terms of a contest between the uncollapse the information from the first weak measurement proponent, Plato, and the uncollapse skeptic, Socrates, can be used for developing guessing strategies about monitored by the arbiter, Aristotle. Plato claims to the unknown initial state. Two such strategies have the ability to uncollapse wavefunctions, and this were presented, and one was shown to be optimal. ability can be tested under the rules of the contest set Of course, in the case where the measurement was forth. subsequently undone, these strategies did no better We have discussed several general features of the than random guessing. uncollapse process in the abstract case, such as the upper bound on the success probability and quantum information aspects of the problem. In order to 8.3. Decoherence suppression by uncollapsing probabilistically undo the measurement, it is necessary Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 It has been recently shown [23] that uncollapsing can to erase the information extracted about the state in be used to suppress qubit decoherence due to energy the first measurement. This is a necessary condition relaxation at low temperature (this is a nearly to uncollapse the wavefunction, because otherwise dominant decoherence process in superconducting various paradoxes arise. However, the information phase qubits [24] and the dominant process in super- erasure is surely not a sufficient condition: the unitary conducting ‘transmon’ qubits [47]). The proposed evolution should also be properly reversed and, as the procedure is very close to the existing experiment [21] most experimentally challenging condition, the process with the phase qubit. To protect the qubit state from should not bring decoherence, which requires a very energy relaxation, it is first partially measured with the good (ideal) detector. strength pt ¼ 1 7 exp (7t) (see Section 3.1), so that In addition to discussing the theory of wavefunc- in the null-result case the qubit state moves towards tion uncollapse in the abstract case, we have also the ground state j0i. The qubit is kept there during the considered a variety of solid-state implementations storage/protection period, and then it is uncollapsed in and specific practical strategies for wavefunction the usual way (using a p-pulse, another null-result uncollapse. The cases of the phase qubit, the charge partial measurement and one more p-pulse – see qubit (with and without Hamiltonian dynamics), and Section 3.2), though the strength pt~ of the second many entangled charge qubits have been examined Contemporary Physics 143

in detail. Additionally, we have also discussed two Plato himself will not know whether the strategy experimental realisations of this physics, based on the succeeded or not, and therefore cannot claim to have an uncollapse procedure. phase qubit and the polarisation qubit, both of which 7. It is easy to generalise Equations (44)–(47) to include the have clearly demonstrated wavefunction uncollapse detector non-ideality and cross-correlation between the with high fidelity. output and back-action noises (as in [13]). Non-ideality leads to the extra dephasing term 7g s in Equation The ideality of the detector is necessary for perfect : d 12 uncollapsing, and we have only dealt with these kinds (47) for s 12, while the cross-correlation brings the term iK[I(t) 7 I0]s12 into the same equation. The formulation of detectors in the theory section of this paper (by this of Equations (44)–(47) corresponds to the approach we mean the detector adds no extra decoherence to the developed in [39]. system). If a detector is slightly non-ideal, then even a perfectly executed uncollapse strategy will result in a Notes on contributors slight infidelity in the final state. This is indeed the case Andrew Jordan received his Ph.D. in in the experiments mentioned above although the Physics from University of California, fidelity was quite high. In such a situation there are Santa Barbara in 2002. He subsequently two characteristics to contend with: the fidelity of worked at the University of Geneva, uncollapsing as well as the probability of claimed Switzerland and Texas A&M University, success. It is an open topic for future research how and in 2006 he joined the faculty of the University of Rochester (NY), where he these characteristics are related. remains professor of physics. Interests include mesoscopic physics and founda- Acknowledgements tional topics in quantum mechanics, The work was supported by NSA and IARPA under ARO measurement theory in particular. He has recently been Grant W911NF-08-1-0336, the National Science Foundation working to develop applications of weak values to precision under Grant No. DMR-0844899, and the University of measurements in optics. Rochester. Alexander Korotkov obtained his M.S. Notes (1986) and Ph.D. (1991) degrees in Physics from the Moscow State Univer- 1. One of the possible procedures is the following. If the sity. He worked at the Moscow State post-measured state is not pure, we can apply more University, the State University of New measurements to make it pure, and then probabilistically York at Stony Brook, and since 2000 he apply a unitary operation from an easily calculable set, joined the faculty of the University of which creates a mixture of pure states identical to the California, Riverside, where at present initial state. he is Professor of Electrical Engineering. 2. An exception is when the initial state belongs to a very In the 1990s he was mainly working limited set, in which the measurement result corresponds on problems in Coulomb blockade and single-electron to only one state. devices. He was always interested in the controversy of the 3.P The classical Bayes rule PðkjmÞ¼PðmjkÞPk= ~ wavefunction collapse, and since 1998 has been developing k~ PðmjkÞPk~ relates the posterior probability PðkjmÞ the theory of non-projective quantum measurement of solid- of a hypothesis k (after observing an event m) to the prior state qubits. probability Pk of this hypothesis and the conditional probability PðkjmÞ of the event m. The hypotheses k should form a complete and mutually exclusive set. References 4. Strictly speaking, the QPT language may not be [1] J. von Neumann, Mathematical Foundations of Quantum Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 applicable to the uncollapsing experiment, because it Mechanics, Princeton University Press, Princeton, 1955. requires linearity of the quantum operation, while after [2] J.A. Wheeler, Law without law,in Quantum Theory and selection of particular realisations and state normal- Measurement, J.A. 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We use normalisation of the shot noise, in which and Quantum Information, Cambridge University Press, SI ¼ 2eIð1 TÞ, where T is the QPC transparency. Cambridge, 2000. See Secs. 2.2.3 and 2.4.1 for quantum 6. In this example, Plato could change his uncollapsing measurement. J. Preskill, Lecture Notes for Physics 229: strategy to simply apply a tailored unitary to shift the Quantum Information and Computation. Available at disturbed state jc1,mi back to its original state jc1i. http://theory.caltech.edu/people/preskill, see Ch. 3 for Under this modified strategy, the success probability will quantum measurement. be P, which may or may not exceed PS, depending on the [5] M.B. Mensky, Decoherence and the theory of continuous strength of the measurement. However, in this case, quantum measurements, Phys. Usp. 41 (1998), p. 923. 144 A.N. Jordan and A.N. Korotkov

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Nazarov, Statistics of measurement 1;2 2pt 2t of noncommuting quantum variables: Monitoring and purification of a qubit, Phys. Rev. B 78 (2008), after changing variables from I to r. These are the solutions 045308. of two different classical random walks with dimensionless 146 A.N. Jordan and A.N. Korotkov

drift velocity v˜1,2 ¼ +1 and dimensionless diffusion coefficient D˜ ¼ 1/2 described by the Fokker–Planck equations,

~ 2 @tPiðr; tÞ¼v~i@rPi þ D@r Pi: ð65Þ In order to solve the first passage problem, we solve first for the Green functions G(r,t) of the above equations starting from the initial condition r ¼ r0. The solutions from the different drift velocities will be weighted with probabilities p~1,2. These Green function equations are supplemented with an absorbing boundary condition at the origin (r ¼ 0),

Giðr ¼ 0; tÞ¼0; ð66Þ

in order to account for the statistics of random walks that cross this point at least once. Let us again start by assuming r0 4 0 and consider the other case later. The solution of Figure 4. Probability distribution of the time required to Equation (65) subject to the condition (66) is most easily undo the measurement. Different plots are for different found by guessing: values of r0 "# 2 1 ðr r0 v~itÞ Giðr; tÞ¼ exp ð4pD~tÞ1=2 4D~t because the probability current at infinity vanishes. Applied "#!to our problem, we find 2 ~ ðr þ r0 v~itÞ exp v~ir0=D exp : ð67Þ hi 4D~t ðiÞ r0 2 ~ PfptðtÞ¼ exp ðr0 þ v~itÞ =ð4DtÞ : ð71Þ ð4pD~t3Þ1=2 In the form written above, it is obvious that the solution

obeys the equation of motion (65) and has the correct initial The probability PC that the point r ¼ 0 is ever crossed is condition r0 at t ¼ 0 (because the absorbing boundary found by integrating (71) over all positive time to obtain condition only permits r 0 solutions). Further inspection of the solution is facilitated by factoring out the free 2 ˜ 1/2 Green function, Gfree(r,t) ¼ exp [7(r7r07v˜it) /(4Dt)]/4p exp ðv~ r =D~Þ¼exp ð2r Þ; i ¼ 1; P ¼ 1 0 0 ð72Þ to write the solution as C 1; i ¼ 2:

This result may be understood intuitively because if the state rr0 is in i ¼ 2 then the drift v˜2 ¼ 71 causes r(t) to evolve from r0 Giðr; tÞ¼Gfreeðr; tÞ 1 exp : ð68Þ D~t to 7? and therefore must cross 0 at some time, while if the system is in state i ¼ 1 then the drift v˜1 ¼ 1 causes r(t)to One can now explicitly see that the absorbing boundary evolve from r0 to þ?. Therefore, in order to cross r ¼ 0, the condition (66) is satisfied, and the solution is completely noise term must fight against the drift, causing a successful positive (as it must be to represent a probability density). crossing only occasionally. To calculate the first passage time distribution, we first In order to obtain the normalised first passage dis- note that the total survival probability that the random tribution (conditioned on crossing), we divide (71) by the Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010 walker willR be in the interval r e (0, ?) at time t is given by probabilities (72) to obtain 1 PsurðtÞ¼ 0 drGðr; tÞ. However, the only place for the particle to be lost from the system is at the origin. ðiÞ hi ðiÞ r0 2 Therefore, the first passage time distribution Pfpt is given by ~ PfptðtjCÞ¼ exp ðr0 jv~ijtÞ =ð4DtÞ : ð73Þ ð4pD~t3Þ1=2 Z 1 ðiÞ Pfpt ¼@tPsur ¼ dr@tGiðr; tÞ: ð69Þ 0 The mean first passage time may also be calculated from (73) to obtain tc,i ¼ r0/jv˜ij¼r0. The next step is to note that the Fokker–Planck equation (65) Obtaining analogous results for r0 5 0 is straightforward may be rewritten as a continuity equation, @tGi þ @r Ji ¼ 0. because the Green function for the Fokker–Planck equation This simply means that locally, probability is conserved. (65) is invariant under the transformation {r ! 7r, r0 ! 7r0, The probability current in the continuity equation is Ji ¼ v˜ ! 7v˜ (or 1 $ 2)} which is also reflection symmetry ˜ i i 7D@rGi þ v˜iGi from (65). Substituting this into (69) we find about the origin. Therefore, results (71), (72) and (73) can be the general result extended using this symmetry. Combining results, we can now calculate the total uncollapsing probability, Ps ¼ p~1PC,1 þ Z p~ P 1 2 C,2 to obtain the result (15) in this new, more powerful way. ðiÞ In addition to the probability of success, the complete PfptðtÞ¼ dr@rJi ¼ Jið1Þ Jið0Þ¼Jið0Þ; ð70Þ 0 solution of the first-passage problem given above now allows Contemporary Physics 147

us to specify further information about the uncollapsing Using the distribution (74), we can find the mean waiting process. In particular, an important question for an time to uncollapse experimental implementation of this idea is how long it is necessary to wait. Twait ¼ TM jr0j; ð75Þ Since the distribution of the first passage time (73) does not depend on the bit state, it directly gives the distribution the standard deviation of the waiting time to uncollapse any qubit state. Therefore, rescaling back the time axis in Equation (73), we find the 1=2 waiting time distribution is DTwait ¼ TMðjr0jÞ ; ð76Þ

"#and the most likely waiting time (which maximises Pwait) jr j ðjr jt=T Þ2 P ðtÞ¼ 0 exp 0 M : ð74Þ wait 1=2 ð2pt3=T Þ 2t=TM M 2 1=2 Tl ¼ TM ðr0 þ 9=4Þ 3=2 : ð77Þ This distribution is normalised, since we consider only successful attempts of uncollapsing. The fact that the distribution is independent of the initial qubit state is not The distribution (74) of the waiting time is plotted in surprising, since otherwise a successful uncollapsing instance Figure 4 for several values of r0. Note that it has a long tail, would give us an information about the qubit state (see which makes the average value Twait to be longer than the discussion in Section 5.3). most likely value Tl. Downloaded By: [University of California, Riverside] At: 20:00 18 February 2010