Determining the speed of multipartite quantum systems by few local measurements Chao Zhang,1,2 Benjamin Yadin,3 Zhi-Bo Hou,1,2 Huan Cao,1,2 Bi-Heng Liu,1,2 Yun-Feng Huang,1,2, ∗ Reevu Maity,3 Vlatko Vedral,3,4 Chuan-Feng Li,1,2, † Guang-Can Guo,1,2 and Davide Girolami3, ‡
arXiv:1611.02004v2 [quant-ph] 10 Nov 2016
1
Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, China 2 Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, 230026, P.R. China 3 Department of Atomic and Laser Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom 4 Centre for Quantum Technologies, National University of Singapore, 117543 Singapore Measuring the speed of evolution of a quantum system can reveal its key properties and structure. Yet, it usually requires experimental and computational resources which increase exponentially with the system size. Here we show how to evaluate the speed of a multipartite quantum system by measurement networks scaling linearly with the system parts. We employ the scheme to detect fundamental quantum properties including metrologically useful coherence and entanglement in an all-optical experiment. The result paves the way for the investigation of quantum phenomena in large complex systems with limited resources.
Important properties of quantum systems can be disclosed by tracking how fast they change during controlled interactions. In quantum information and metrology protocols [1, 2], a device speed provides information about its computational power [3–5], and the surrounding environment structure [6, 7], helping develop optimal measurement and control strategies [8–11]. Also, singular values of the evolution speed in condensed matter systems detect quantum phase transitions [12, 13]. Despite such strong theoretical motivations, quantifying the speed of a quantum system is experimentally and computationally challenging. Generally, it relies on full reconstruction of the system state at different times, which requires measurement schemes exponentially growing with the system size. Here we show how to estimate the speed of a many-body quantum system by a network of single-site measurements scaling linearly with the system parts. We prove that the speed quantitatively bounds quantum properties, as coherence and entanglement, which yield quantum advantage in metrology protocols. We demonstrate the result in an all-optical experiment. We relate speed to observable quantities by adopting a geometric viewpoint. The dynamics of a finite dimensional quantum system can be visualized as a parametrized path ρt , t ∈ R, in the space of accessible states (Fig.1). The parameter t represents the evolution time. We define the system speed as the average rate of change of the state ρt over an interval t ∈ [0, τ], which is expressed in terms of the mean value of quantum operators h·iρ = Tr(·ρ): sτ (ρ0 ) :=
|ρτ − ρ0 | (hρτ iρτ + hρ0 iρ0 − 2hρτ iρ0 )1/2 = . (1) τ τ
Then, measuring a probe observable on two system copies is sufficient to quantify the speed, hσiρ = hViρ⊗σ , V(|φ1 i ⊗ |φ2 i) = |φ2 i ⊗ |φ1 i, ∀|φ1,2 i, where V is the swap operator. It is possible to evaluate any observable without full state reconstruction by single qubit interferometry [14–16]. In this scheme, the system copies are correlated with an ancillary qubit by a controlled gate. They
τ
ρτ
t
ρ0 Figure 1. A quantum process is modelled as a geometric path ρt , where the coordinate t represents the computation time. The speed in an interval τ is defined as the average rate of change of the system state along the path from ρ0 to ρτ . For periodic motions, ρt = ρ0 , t = kT, k ∈ Z, the quantity is informative if taken over an interval τ < T .
are encoded either by distinct system replicas, or different degrees of freedom of a single copy, e.g. spin and momentum. The operator mean value is extracted by measuring the ancilla polarisation. Unfortunately, the complexity of a controlled gate, i.e. the minimum number of one-qubit and two-qubit operations into which it is decomposable, increases exponentially with the system size [17]. Yet, for systems which exhibit a partition S ≡ {S i }, i = 1, . . . , n, there is an alternative measurement strategy [18, 19], which bypasses the necessity of multipartite interferometers and can be applied to evaluate a system speed. The factorization of the global swap VS = ⊗ni=1 VS i allows one to quantify state overlaps hσS iρS by collectively measuring the n local swaps. This consists of a network of n bipartite gates which correlate each pair of S i subsystem copies, followed by detection of single copy, single-site observables. An important case study is represented by n-qubit systems. The swap mean value is here obtained by readout of n local parity measurements, or projections on the antisymmetric subspace, √ − − VS i = 1 − 2ΠψS i , ΠψS i = |ψ− ihψ− |S i , |ψ− i = 1/ 2(|01i − |10i), a standard routine of quantum information processing, e.g. in optical systems and bosonic lattices [18, 19]. The complexity of the measurement network scales linearly (O(2n)) with the
2 number of subsystems, providing an exponential advantage with respect to the O(22n ) measurements required by state tomography. Measuring speed unveils quantum properties of a system with no further data. Coherence, i.e. P P the possibility to observe coherent superpositions i ci |ii, i |ci |2 = 1, in a given basis {|ii}, is arguably the most fundamental quantum feature. Coherence is a resource which triggers quantum computational speed-up. In particular, the sensitivity to a unitary perturbation Ut ρUt† , Ut = e−iHt , is due to quantum asymmetry, a peculiar property of coherent states in the Hamiltonian eigenbasis [20, 21]. Asymmetry is defined as the system ability to break a symmetry generated by a given Hamiltonian dynamics. It corresponds to coherence in the Hamiltonian function H eigenbasis. Asymmetry underpins the usefulness of coherent superpositions in phase estimation schemes, reference frame alignment, and description of thermodynamic properties of quantum systems. Asymmetry measures are defined as non-increasing functions in dynamics allowed by the symmetry, which are described by operations Φ commuting with the Hamiltonian, [Φ, H] = 0. Experimentally measuring asymmetry is hard [22]. In general, one cannot discriminate P with certainty coherent states from incoherent mixtures i |ci |2 |iihi|, without reconstructing the density matrix off-diagonal entries. We here show how to evaluate the asymmetry of an unknown system P by its speed. To quantify the sensitivity of a probe state ρ = i λi |iihi| to the unitary transformation Ut , we employ the class of quantum Fisher informations: I f (ρ, H) =
1 X (λi − λ j )2 |hi|H| ji|2 , 4 i, j λ j f (λi /λ j )
(2)
being f (x) members of an infinite set of real-valued functions [23]. We prove that each element of the class is an asymmetry measure, being contractive on average under operations commuting with the Hamiltonian: X I f (ρ, H) ≥ pµ I f (Φµ (ρ), H), (3) µ
∀{pµ , Φµ } :
X
pµ = 1, [Φµ , H] = 0, ∀ f.
µ
τ→0
Proofs and full details are reported in the Appendix. We extend the analysis to multipartite systems, where the computational advantage provided by quantum correlations can be tested by measuring speed. We consider a phase estimation scenario, which is a building block of several quantum computation and metrology schemes [1, 2, 8]. A phase shift Uτ,i = e−iHi τ is applied in parallel to each site of an n-qubit system. This means that the register undergoes a unitary evolution 1/2-spin HamilP generated by an additive x(y,z) tonian Hadd = H , H = I ⊗ σ ⊗ Ii+1,...,n , where i i 1,...,i−1 i i σix(y,z) are the Pauli matrices. The goal is to estimate the nondirectly observable parameter τ by an estimator τest extracted from measurements on the perturbed system. Assuming the estimation is unbiased, hτest iρ = τ, the quantum CramerRao bound establishes that asymmetry bounds the estimation precision, expressed in terms of the estimator variance, V(ρ, τest ) ≥ (νI f ≡F (ρ, H))−1 , ∀ρ, H, where ν is the number of trials and IF (ρ, H), F(x) = (1 + x)/2 is the Symmetric Logarithmic Derivative quantum Fisher information (SLDF) [24]. Separable states achieve at best IF (ρ, Hadd ) = O(n), while entanglement enables up to a quadratic improvement of scaling precision in the large sample limit (n → ∞), IF (ρ, Hadd ) = O(n2 ). The relation IF (ρ, Hadd ) > n/4 then witnesses enhanced sensitivity due to entanglement [25]. This suggests non-tomographic evaluation of the SLDF as an appealing strategy to verify the advantage due to quantum correlations. For thermal states of condensed matter systems, the quantity can be extracted by measuring the system dynamical susceptibility [26], while lower bounds are obtained by twotime detections of a global observable [27, 28]. By Eq. 5, it is possible to verify entanglement-enhanced precision by monitoring the average speed for an arbitrary phase shift τ: Sτ (ρ, Hadd ) > n/4.
Such quantities are not directly observables. Full knowledge of state and Hamiltonian is required to compute them. Yet, we show they are quantitatively lower bounded by the system speed over a finite interval τ of the time-independent unitary evolution Ut ρUt† : Sτ (ρ, H) ≤ I f (ρ, H), Sτ (ρ, H) := sτ (ρ)2 /2 =
are insensitive to the perturbation, while a non-vanishing speed reliably witnesses coherence in the Hamiltonian eigenbasis, sτ (ρ, H) = 0 ⇐⇒ I f (ρ, H) = 0, ∀ f, τ. The quantum Fisher informations are upper bounded by the Hamiltonian variance up to a constant, 2 f (0)I f (ρ, H) ≤ V(ρ, H) = hH 2 iρ − hHi2ρ , ∀ f, ρ, H. The chain of inequalities is saturated for pure states in the zero time limit, lim Sτ (ρψ , H) = 2 f (0)I f (ρψ , H) = V f (ρψ , H), ρψ = |ψihψ|.
hρiρ −
(4)
hUτ ρUτ† iρ , ∀ f, ρ, τ, H. τ2
It is then possible to experimentally bound asymmetry with respect to an arbitrary Hamiltonian, by estimating the purity hρiρ and the overlap hUτ ρUτ† iρ . Incoherent states
(5)
The speed detection network for n-qubit systems is depicted in Fig. 2. The method has three key advantages. First, it is hardware independent, being applicable to any probe state and experimental setting. Second, only local two-site interactions and single-site detections are needed. This means that quantum speed-up due to entanglement is detectable by classical resources. Distant laboratories {S i } can verify entanglement in a shared system by implementing local operations and classical communication between the sites [1], provided each laboratory S i with two subsystem copies (Fig. 2). Third, the tightness of the bound makes possible to certify the probe optimization, by verifying achieved quadratic scaling:
3 τ
t
ρS
S1
ρS
Uτ,i
UBS
Uτ,i
UBS
Uτ,i
UBS
1
ρ
2
ρ
S2
ρ
Sn
ρ
Sn
Figure 2. The depicted network measures the speed of a n-qubit system along a unitary evolution e−iHadd t ρeiHadd t over an interval τ. Two copies of the system in the state ρ1,2 S enter a two-arm channel (blue and green). For the overlap evaluation, the unitary gates Uτ,i = eiHi τ are applied to the second copy subsystems. The measurement apparatus (orange) interferes each pair of subsystem copies by O(n) correlating beam splitter U BS gates. The swap mean value, and therefore the speed, is extracted by O(2n) single-site detections.
Sτ (ρ, Hadd ) = O(n2 ) ⇒ IF (ρ, Hadd ) = O(n2 ). For mixtures of arbitrary pure states with white noise, ρ = (1−)ρψ + dI , false negatives (linearly scaling lower bound and quadratic scaling SDLF) are ruled out, as the speed fully determines the scaling of the SLDF (proof in Appendix). All the asymmetry measures are connected by the inequality chain 2 f (0)I f (ρ, H) ≤ IF (ρ, H) ≤ I f (ρ, H), ∀ f [29]. Thus, super-linear speed scaling of any contractive metric is sufficient to certify an optimal probe, I f (ρ, H) = O(n2 ) ⇔ IF (ρ, H) = O(n2 ), ∀ f . In fact, for separable states, it holds 2 f (0)I f (ρ, Hadd ) ≤ n/4, ∀ f . We experimentally demonstrate the speed detection protocol in an optical set-up. We bound the metrologically useful coherence (i.e. asymmetry) and entanglement of a twoqubit system AB, where the qubits are encoded in the photon polarisations, by measuring its speed in a controlled unitary evolution. The system is prepared in a mixture of Belldiagonal states, ρ p,AB = p|φ+ ihφ+ | + (1 − p)|φ− ihφ− |, |φ± i = √ 1/ 2(|00i ± |11i), p ∈ [0, 1]. We run a series of experiments for equally stepped values of the mixing parameter, p = 0, 0.1, 0.2, . . . , 0.9, 1. We measure the speed function Sτ (ρ pP , H) in dynamics generated by the spin Hamiltonians H = i=A,B Hi , Hi = σ x,y,z , for a fixed shift τ = π/6. Each run of the experiment implements the scheme illustrated in Fig. 3. First, we extract two copies of a maximally entangled two-qubit state |φ+ i = √12 (HH + VV), where H, V label horizontal and vertical polarizations, from three spontaneous parametric down-conversion (SPDC) sources generated by an ultrafast pump pulse from a mode-locked Ti:Sapphire laser with a central wavelength of 780 nm, a pulse duration of 140 fs, and a repetition rate of 76 MHz. The pump power for each source is 90 mW. Then, we simulate the preparation φ+ φ+ of the two copy mixture ρ p,A1 B1 ⊗ ρ p,A2 B2 = p2 Π12 + p(1 −
Figure 3. Experimental set-up. We prepare two copies (Copy 1,2) of a Bell state |φ+ i by a laser-emitted ultraviolet pulse split into three beams pumping SPDC sources (Source 1,2,3). The scheme guarantees that the two copies are always emitted by a different source. Conversely, in a two source setting, fourfold coincidences in the BSMs could be generated by two photon pairs emitted from only one source, invalidating the experiment. Copy 1 (photons 1,2) is directly generated from Source 1 by employing sandwich-like BBO crystal [30]. Copy 2 is prepared from Source 2,3. First, two product states (photons 3-6) are obtained via single BBO crystals (beamlike type-II phase matching). The polarisations of photons 3,4 are rotated via HWPs. They are then interfered by a PBS for parity check measurements. The Bell state is triggered by post-selection. All the photons pass through single mode fibers for spatial mode selection. For spectral mode selection, photons 1-4 (5,6) pass through 3-nm (8-nm) bandwidth filters. By tuning QWP1,2, the four terms of the mixture are created. A 45◦ HWP is placed in one port (A1 , B1 ) of each BSM to deterministically project into the Bell singlet. A multi-channel unit counts the sixfold coincidences (one detector fire in each output mode). + −
− +
φ φ φ φ φ φ p)Π12 + p(1− p)Πφ12 φ +(1− p)2 Π12 , Π12 = |φ± ihφ± |A1 B1 ⊗ ± ± |φ ihφ |A2 B2 . Classical mixing is obtained by applying a quarter-wave plate to each system copy (QWP1,QWP2). A 90◦ rotated QWP swaps the Bell states, |φ± i → |φ∓ i, by implementing a π phase shift between H, V polarisations. The four terms of the mixture are generated in separate runs by engineering the rotation sequences (QWP1,QWP2) = {(0◦ , 0◦ ), (0◦ , 90◦ ), (90◦ , 0◦ ), (90◦ , 90◦ )}, with a duration proportional to {p2 , p(1 − p), p(1 − p), (1 − p)2 } respectively. The collected data from the four cases are then identical to the ones obtained from direct preparation of the mixture. By reminding Eq. 4, we quantify speed by measuring the purity hV12 iρ p,A1 B1 ⊗ρ p,A2 B2 and the overlap hV12 iρ p,A B ⊗Uπ/6 ρ p,A B U † . 1 1 2 2 π/6 The unitary gate Uπ/6 = Uπ/6,A2 ⊗ Uπ/6,B2 , Uπ/6,A2 (B2 ) = e−iHA2 (B2 ) π/6 , is applied to the second system copy by a sequence of one half-wave plate (HWP) sandwiched by two QWPs. The rotation angles which implement each Hamiltonian choice are given in Appendix. The mean value of the swap operator is extracted by local and bi-local projections on − −
± ±
4 Hi σx σy σz IF (ρ p , H) p (1 − p) (1 − 2p)2 Sτ (ρ p , H) (p sin τ/4τ)2 ((1 − p) sin τ/4τ)2 ((1 − 2p) sin τ/4τ)2 IF (ρ p , H) > 0.5 p > 0.5 p < 0.5 p < 0.147, p > 0.853 Oπ/6 (ρ p , H) > 0.5 p > 0.741 p < 0.259 p < 0.129, p > 0.870 (a) Speed X 1 0.5
◆
◆
0 0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
p
(b) Speed Y 1 0.5
◆
◆
0 0
0.2
0.4
p
(c) Speed Z 1 0.5
◆◆
◆◆
0 0
0.2
0.4
0.6
0.8
1
p
(d)
Figure 4. Experimental results. We evaluate the speed function Sτ (ρ, H) of a two-photon probe in the state ρ p = p|φ+ ihφ+ | + (1 − p)|φ− ihφ− |, p = 0, 0.1, 0.2, . . . 0.9, 1, for unitary evolutions Uτ ρ p Uτ† , Uτ = e−iHτ , H = σ x,y,z A ⊗ IB + IA ⊗ σ x,y,z B , over a time shift τ = π/6. The Table (a) reports the theoretical values of the SLDF, which lower bounds any quantum Fisher information, the speed function extracted from purity and overlap measurements, and the related entanglement witness conditions (Eq. 5). The experimental results are plotted in Figs. (b)-(d). The green dashed line depicts the SLDF, while the blue continuous line is the lower bound Sπ/6 (ρ p , H). Superlinear scaling due to entanglement is witnessed for values above the black dotted line. The black diamonds mark the point where the quantities surpass such threshold. The error bars are determined by Monte Carlo simulation with Poisson-distributed error (1000 samples for each state).
incidences with a 50/hour rate for about 6 hours in each experiment run. The results are summarised in Fig. 4. For each Hamiltonian, we reconstruct the speed function Sπ/6 (ρ p , H), and compare it against the theoretical value of the smallest quantum Fisher information, the SLDF. We detect entanglement by witnessing supra-linear speed scaling Sπ/6 (ρ p , H) ≥ 1/2. We stress that for each Hamiltonian, the speed certifies entanglement yielding non-classical metrological efficiency, rather than just non-separability of the density matrix (the state ρ p is entangled for p , 1/2). We provide a full charaterization of the experimental setting. The three Hang-Ou-Mandel interferometers (one for the PBS and each BSM) have a visibility of 0.91. We performed tomographies of the input states and of the BSMs. The complete data are reported in the Appendix. The fidelity of the input Bell states are respectively 0.9889(φ+1 ), 0.9901 (φ−1 ), 0.9279 (φ+2 ), 0.9319 (φ−2 ). The average fidelities of BSM1 and BSM2 are 0.9389 ± 0.0030 and 0.9360 ± 0.0034, being the error bars calculated from the standard deviation of 100 run results by assuming Poisson statistics. We here showed how to directly measure the speed of a quantum system. For n-partite systems, the speed is evaluated by implementing O(n) bipartite interactions and singlesite detections, exponentially outperforming state tomography. We demonstrated the method in an all-optical experiment, extracting quantitative bounds to metrologically useful coherence and entanglement from speed measurements. These are considered key resources to develop quantum information processors, and valuable tools to study correlated quantum systems. Yet, their experimental verification is a serious challenge in networks of growing size and complexity. The scalability of the speed detection scheme overcomes the problem, making possible to certify quantum computations operated in large scale registers [8, 21], and to study phase transitions of matter and critical phenomena in manybody quantum systems [12, 13, 26], by limited laboratory resources. A further interesting development would be to test macroscopic quantum effects, recently linked to quadratic (IF (ρ, H) = O(n2 )) precision scaling in phase estimation protocols [18, 27, 33], by speed measurements of continuous variable systems.
ACKNOWLEDGEMENTS
the Bell singlet: V12 = I12 − 2Πψ1 ⊗ I2 − 2I1 ⊗ Πψ2 + 4Πψ12 ψ . −
−
− −
(6)
We measure the projectors by Bell state measurement (BSM) schemes applied to each subsystem pair. They consist of polarizing beam splitters (PBSs), 45◦ HWPs, and single photon detectors. We insert a 45◦ HWP in the input ports of the PBS corresponding to the A1 , B1 susbsystems. This is crucial to deterministically project into the singlet, as a single BSM run discriminates only two Bell states [31, 32]. A multi-channel coincidence unit counted the sixfold co-
We thank Paolo Gibilisco for fruitful discussions. This work was supported by: the Oxford Martin School and the Wolfson College, University of Oxford; the EPSRC (UK) Grant EP/L01405X/1; the Leverhulme Trust (UK); the John Templeton Foundation; the EU Collaborative Project TherMiQ (Grant Agreement 618074); the COST Action MP1209; the NRF, Prime Ministers Office, Singapore; the Ministry of Manpower (Singapore) under its Competitive Research Programme (CRP Award No. NRF- CRP14-2014-02), administered by the Centre for Quantum Technologies (NUS);
5 the Key Research Program of Frontier Sciences, CAS (No. QYZDY-SSW-SLH003); the National Natural Science Foundation of China (Grant Nos. 11274289, 11325419, 11374288, 11474268, 61327901, 61225025 and 61490711); the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB01030300); the Fundamental Research Funds for the Central Universities, China (Grant Nos. WK2470000018 and WK2470000022); the National Youth Top Talent Support Program of National High-level Personnel of Special Support Program (No. BB2470000005).
∗
[email protected] [email protected] ‡
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1
Appendix
Chentsov-Morozova operator monotones Fop consists of the real-valued functions f : R+ → R+ such that i) For any hermitian operators A, B such that 0 ≤ A ≤ B, we have 0 ≤ f (A) ≤ f (B)
THEORY Quantum Fisher informations as measures of state sensitivity.
Quantum Information geometry studies quantum states and channels as geometric objects. The Hilbert space of a finite ddimensional quantum system admits a Rimenannian structure, thus it is possible to apply differential geometry concepts and tools to characterize quantum processes. For an introduction to the subject, see Refs. [A1, A2]. The information about a d-dimensional physical system is encoded in states represented by d × d complex hermitian matrices ρ ≥ 0, Tr(ρ) = 1, ρ = ρ† , in the system Hilbert space H. Each subset of rank k states is a smooth manifold Mk (H) of dimension 2dk − k2 − 1 [A3]. The set of all states M(H) = ∪dk=1 Mk (H) forms a stratified manifold, where the stratification is induced by the rank k. The boundary of the manifold is given by the density matrices satisfying the condition det ρ = 0. State transformations are represented on M(H) as piecewise smooth curves ρ : t → ρt , where ρt represents the quantum state of the system at time t ⊆ R. By employing differential geometry techniques, it is possible to study the space of quantum states M(H) as a Riemannian structure. The length of a path ρt , t ∈ [0, τ], on the manifold is given by the integral of the line element Z τ Z τ lρt = ds = ||∂t ρt || dt, (A.1) 0
0
where the norm is induced by equipping M(H) with a symmetric, semi-positive definite metric. The path length is invariant under monotone reparametrizations of the coordinate t. The definition of a metric function yields the notion of distance d(ρ, σ) between two quantum states ρ, σ. The choice of the metric is arbitrary. However, Morozova, Chentsov and Petz identified a class of functions, the quantum Fisher informations, which extend the contractivity of the classical Fisher-Rao metric under noisy operations to quantum manifolds. This means that they have the appealing feature to be the unique class of contractive Riemannian metrics under CPTP maps Φ: d(Φ(ρ), Φ(σ)) ≤ d(ρ, σ), ∀ρ, σ, Φ [A4, A5]. For such a class of P metrics, given the spectral decomposition of an input ρ = i λi |iihi|, the line element associated to an infinitesimal displacement ρ → ρ + dρ takes the form sX X (dλi )2 /4λi + c f (λi , λ j )/2|hi|dρ| ji|2 . ds f = i
i< j
(A.2) The terms c f (i, j) = ( j f (i/ j)) , where the f s are the Chentsov-Morozova functions [A5], identify the elements of the class. We here describe their main properties, by focusing the analysis on the subclass of function identified by the regularity condition f (0) > 0. The set of symmetric, normalised −1
ii) f (x) = x f (x−1 ) iii) f (1) = 1. Thus, the following properties are satisfied: i) 1/c f (x, 1) : R+ → R+ ii) c f (x, y) = c f (y, x), c f (zx, zy) = z−1 c f (x, y) iii) x c f (x, 1) = c f (1/x, 1) iv) x ≤ y ⇒ c f (y, 1) ≤ c f (x, 1) v) c f (1, 1) = 1. By extending the domain of these functions to positive square matrices, they enjoy a one-to-one correspondence with the set Mm op of matrix means m(A, B); see Ref. [A6] for a list of defining properties. The link between the two sets is 1
1
1
1
m f (A, B) := A 2 f (A− 2 BA− 2 )A 2 ,
(A.3)
which reduces to m f (A, B) = A f (BA−1 ) for commuting A, B. Thus, matrix means also have a bijection with the set of monotone Riemannian metrics which give rise to norms ||A||ρ, f defined by ||A||2ρ, f := Tr A m f (Lρ , Rρ )−1 (A) , (A.4) where Rρ and Lρ are the right- and left-multiplication superoperators: Rρ (A) = Aρ, Lρ (A) = ρA. The monotonicity property of these metrics is that, for any CPTP map Φ, one has ||Φ(A)||Φ(ρ), f ≤ ||A||ρ, f .
(A.5)
When applied to the stratified manifold of quantum states, such norms correspond to the quantum Fisher informations. Indeed, any metric defined onP the manifold induces a metric on a parametrized curve ρt = i λi (t)|i(t)ihi(t)|. The squared rate of change at time t is then given by the tangent vector length X |hi(t)|∂t ρt | j(t)i|2 λ j (t) f (λi (t)/λ j (t)) i, j X = (dt λi (t))2 /4λi (t)
||∂t ρt ||2f =
i
+
X
c f (λi (t), λ j (t))/2|hi(t)|∂t ρt | j(t)i|2 . (A.6)
i< j
The dynamics of the quantum Fisher informations for closed and open quantum systems has been studied in Ref. [A7].
2 All P such metrics reduce to the classical Fisher-Rao metric i (dt λi (t))2 /(λi (t)) for stochastic dynamics of probability distributions {λi (t)}, represented at any time by a diagonal density matrix. On the other hand, unitary transformations ρt = Ut ρUt† are genuinely quantum, as only the eigenbasis elements evolve. We focus on the latter case. Let us consider the unitary transformation Ut ρUt† , Ut = e−iHt . The quantum Fisher informations associated with f ∈ Fop read f (0) 2 2 ||i[ρ, H]|| f . We here absorb the constant factor and recast the quantity in the more compact form
Proof. This follows from Lemma 1 by tracing out the ancillary system, as I f must be monotonically decreasing under partial trace: ! X X pµ I f (ρµ , H) = I f pµ ρµ ⊗ |µihµ|
I f (ρ, H) := 1/4||i[ρ, H]||2f .
We are now ready to prove determinsitic monotonicity (note that monotonicity under unitary transformations is shown already in Ref. [A6]). Recall that a U(1)-covariant channel, i.e. a symmetric operation, Φ is defined to be such that [Φ, Ut ] = 0, where Ut (ρ) := e−iHt ρeiHt . Noting that −i[H, ρ] = dt Ut (ρ)|t=0 , we have from Eq. A.7 that
(A.7)
For pure states, one has 2 f (0)I f (|ψihψ|, H) = V(|ψihψ|, P H) = hH 2 iψ − hHi2ψ , ∀ f . For an arbitrary initial state ρ = i λi |iihi|, it can be shown that I f (ρ, H) =
1 X (λi − λ j )2 |hi|H| ji|2 , 4 i, j λ j f (λi /λ j )
(A.8)
where each term in the sum is taken to be zero whenever λi = λ j [A2].
µ
µ
! X
≥ If
pµ ρµ , H
.
(A.11)
µ
f (0) ||dt Ut (ρ)|0 ||2f . (A.12) 2 The linearity of Φ and the monotonicity property (A.5) then give I f (ρ, H) =
||dt Ut (Φ(ρ))|| f = ||dt Φ(Ut (ρ))|| f = ||Φ(dt Ut (ρ))|| f
Proof that any quantum Fisher information is an ensemble asymmetry monotone, Eq. 3
≤ ||dt Ut (ρ)|| f ,
We prove two preliminary results upon which the monotonicity of the speed measures will be demonstrated. Lemma 1. For any set of states ρµ and normalised probabilities pµ , and an orthonormal set {|µi}, ! X X If pµ ρµ ⊗ |µihµ|, H ⊗ I = pµ I f (ρµ , H), ∀ f. µ
µ
(A.9) Proof. Let each ρµ have a spectral decomposition ρµ = P i λi|µ |ψµ,i ihψµ,i |. Using (A.8) and defining λµ,i := pµ λi|µ , we then have ! X If pµ ρµ ⊗ |µihµ|, H ⊗ I = µ
1 X (λµ,i − λν, j )2 = |hψµ,i |hµ|(H ⊗ I)|ψν, j i|νi|2 4 µ,ν,i, j λν, j f (λµ,i /λν, j ) =
so that I f (Φ(ρ), A) ≤ I f (ρ, A), ∀ f . To prove the ensemble monotonicity, we introduce a quantum instrument as a set of covariant maps {ΦµP } which are not necessarily trace preserving, while the sum µ Φµ is. For every quantum instrument, one can construct a channel (i.e., so that trace is preserved) by including in the output an ancilla that records which outcome was obtained via a set of orthonormal states {|µi}: X Φ0 (ρ) := Φµ (ρ) ⊗ |µihµ|. (A.14) µ
P Tracing out the ancilla results in the channel µ Φµ . It is 0 clear thatP Φ is covariant whenever each of the Φµ is. Writing Φ0 (ρ) = µ pµ ρµ ⊗ |µihµ|, we can apply Lemma 1 followed by deterministic monotonicity to find ! X X pµ I f (ρµ , H) = I f pµ ρµ ⊗ |µihµ|, H ⊗ I µ
µ
= I f (Φ0 (ρ), H ⊗ I)
1 X (λµ,i − λµ, j )2 |hψµ,i |H|ψµ, j i|2 4 µ,i, j λµ, j f (λµ,i /λµ, j )
1 X p2µ (λi|µ − λ j|µ )2 |hψµ,i |H|ψµ, j i|2 4 µ,i, j pµ λ j|µ f (λi|µ /λ j|µ ) X = pµ I f (ρµ , H).
≤ I f (ρ, H).
(A.15)
Proof that the speed bounds any quantum Fisher information, Eq.4
=
(A.10)
µ
Corollary 1. I f (ρ, H) is convex in ρ.
(A.13)
We here show that Sτ (ρ, H) ≤ I f (ρ, H), Sτ (ρ, H) := sτ (ρ)2 /2 =
hρiρ − hUτ ρUτ† iρ . τ2
(A.16)
3 Proof. It is possible to write the system speed in terms of the Hilbert-Schmidt distance DHS (ρ, σ) = Tr((ρ − σ)2 ) and the related norm,
IF (ρ , H) has been proven previously. Also, the bound is saturated for pure states in the zero shift limit, S0 (ρ , H) = lim Sτ (ρ , H) = −1/2Tr([ρ , H]2 ), thus we would expect the
Sτ (ρ, H) = D2HS (ρ, Uτ ρUτ† )/(2τ2 )
two quantities to be close for sufficiently small . For an upper bound, by convexity, we have
=
||Uτ ρUτ†
−
ρ||2HS /(2τ2 ).
(A.17)
τ→0
IF (ρ , H) ≤ (1 − )IF (|ψihψ|, H)
The zero shift limit is given by S0 (ρ, H) := lim Sτ (ρ, H) = −1/2Tr([ρ, H]2 ). (A.18) τ−>0
By expanding the quantity in terms P of the state spectrum and eigenbasis, one has S0 (ρ, H) = i, j (λi − λ j )2 /2|hi|H| ji|2 . We recall the norm inequality chain f (0)/2 ||A|| f ≤ 1/4||A||F ≤ 1/4||A|| f , ∀ f, A, which, for unitary transformations e−iHt ρeiHt , implies the topological equivalence of the quantum Fisher information measures:
≤ (1 − )S0 (|ψihψ|, H), ∀H,
since I/d is an incoherent state in any basis. Few algebra steps give S0 (ρ , H) = −2Tr([H, (1 − )|ψihψ|]2 ) = (1 − )2 S0 (|ψihψ|, H) = (1 − )2 IF (|ψihψ|, H).
IF (ρ , H) ≤
(A.19) where F labels the Symmetric Logarithmic Derivative quantum Fisher information (SLDF) [A6]. We note that P its expansion for unitary transformations reads IF (ρ, H) = i, j (λi − λ j )2 /(2(λi + λ j ))|hi|H| ji|2 . Since λi + λ j ≤ 1, ∀i, j, it follows that (A.20)
Any distance between two states is defined as the length of the shortest path between them. By recalling the von Neumann equation ∂t ρt = i[ρt , H], and integrating over the unitary evolution Ut , one then obtains Z ρt ≡Uτ ρUτ† † DHS (ρ, Uτ ρUτ ) ≤ ||∂t ρt ||HS dt ρt ≡ρ
Z
ρt ≡Uτ ρUτ†
≤
1/2 −Tr([ρ, H]2 ) dt
ρt ≡ρ
=
Z
ρt ≡Uτ ρUτ†
ρt ≡ρ
(2S0 (ρ, H))1/2 dt
= (2S0 (ρ, H))1/2 τ ≤ (2IF (ρ, H))1/2 τ 1/2 ≤ 2I f (ρ, H) τ, ∀ f.
(A.24)
It follows that
2 f (0)I f (ρ, H) ≤ IF (ρ, H) ≤ I f (ρ, H), ∀ f, ρ, H,
S0 (ρ, H) ≤ IF (ρ, H), ∀ρ, H.
(A.23)
(A.21)
Hence, the bound is proven. The inequality is saturated for pure states in the limit τ → 0.
S0 (ρ , H) . 1−
(A.25)
We now express the result in terms of the purity of ρ . One has I 2 Tr(I) + 2(1 − )hψ| |ψi d2 d 2 1 = (1 − )2 + + 2(1 − ) d d 1 1 2 =1−2 1− + 1− . (A.26) d d
Tr(ρ2 ) = (1 − )2 +
Solving for gives r =1−
dTr(ρ2 ) − 1 . d−1
(A.27)
Finally, combined with the lower bound, we have s d−1 S0 (ρ , H). (A.28) S0 (ρ , H) ≤ IF (ρ , H) ≤ dTr(ρ2 ) − 1 By Taylor expansion about τ = 0, one has Sτ (ρ, H) = S0 (ρ, H) + O(τ2 ), ∀ρ, H. Thus, measuring the speed function Sτ (ρ ) and the state purity determines both upper and lower bounds to the SDLF, and consequently of any quantum Fisher information, up to an experimentally controllable error due to the selected time shift. False negatives are then avoided. EXPERIMENTAL SETTING
Determining the scaling of the quantum Fisher informations from speed measurements: the case of pure states mixed with white noise
Suppose we are given the state I ρ = (1 − )ρψ + , d
(A.22)
where I is the identity of dimension d, while ρψ is an arbitrary pure state and is unknown. The lower bound Sτ (ρ , H) ≤
Implementation of single qubit unitary gates
Single qubit unitary gates implement SU(2) group transformations. We parametrize the rotations by the Euler angles (ξ, η, ζ): 1 1 1 u(ξ, η, ζ) := exp −i ξσy exp −i ησ x exp −i ζσy , 2 2 2 (A.29)
4 where σ x,y,z are the Pauli matrices. According to Ref [A8], one can engineer arbitrary single qubit gates designing a θ-rotated HWP implementing the transformation Hθ , sandwiched by two rotated QWPs (transformations Qθ ): u(ξ, η, ζ) = Qθ3 Hθ2 Qθ1 ,
Tomography of the input Bell state copies
(A.30)
where θ1,2,3 are the rotation angles to apply to each plate. In particular, any unitary transformation is prepared by a gate sequence of the form θ1 = π/4 − ζ/2 mod π,
(A.31)
θ2 = −π/4 + (ξ + η − ζ)/4 mod π, θ3 = π/4 + ξ/2 mod π. The phase shift angles to design each spin Hamiltonian are shown in Table I. Angles I U X UY UZ π π π π θ1 4 2 4 4 π π 5π 5π θ2 − 24 24 24 4 π π π π θ3 4 2 6 4 Table I. Angles of the wave plates for the single qubit gates.
0.5146 + 0.0000i −0.0158 − 0.0031i = 0.0058 − 0.0029i 0.4923 − 0.0071i 0.5072 + 0.0000i −0.0065 − 0.0008i = −0.0052 − 0.0028i −0.4931 + 0.0090i 0.4881 + 0.0000i −0.0108 − 0.0041i = 0.0063 − 0.0091i 0.4486 − 0.0509i 0.4911 + 0.0000i 0.0041 + 0.0184i = 0.0058 − 0.0075i −0.4502 + 0.0462i
φ+A1 B1
φ−A1 B1
φ+A2 B2
φ−A2 B2
We perform full state reconstruction of the two copies (Copy 1,2) of the Bell states φ± obtained by SPDC sources. We remind that Copy 1 (subsystems A1 B1 ) is generated by the sandwich-like Source 1 (photons 1,2), while Copy 2 (A2 B2 ) is triggered by Sources 2,3 via parity check gate and postselection applied to two product states (photons 3-6). The counting rate for the Copy 1 photon pair is 32000/s, while for the four photons of Copy 2 is 110/s. We use the maximum likelihood estimation method to reconstruct the related density matrices, which read:
−0.0158 + 0.0031i 0.0039 + 0.0000i −0.0003 + 0.0026i −0.0173 + 0.0021i
0.4923 + 0.0071i −0.0173 − 0.0021i , 0.0029 − 0.0043i 0.4787 + 0.0000i −0.0065 + 0.0008i −0.0052 + 0.0028i −0.4931 − 0.0090i 0.0030 + 0.0000i 0.0007 + 0.0021i 0.0065 + 0.0016i , 0.0007 − 0.0021i 0.0029 + 0.0000i 0.0056 + 0.0034i 0.0065 − 0.0016i 0.0056 − 0.0034i 0.4869 + 0.0000i −0.0108 + 0.0041i 0.0063 + 0.0091i 0.4486 + 0.0509i 0.0216 + 0.0000i −0.0029 − 0.0066i −0.0140 − 0.0068i , −0.0029 + 0.0066i 0.0198 + 0.0000i 0.0044 − 0.0073i −0.0140 + 0.0068i 0.0044 + 0.0073i 0.4706 + 0.0000i 0.0041 − 0.0184i 0.0058 + 0.0075i −0.4502 − 0.0462i 0.0155 + 0.0000i 0.0005 + 0.0080i 0.0041 − 0.0089i . 0.0005 − 0.0080i 0.0209 + 0.0000i −0.0085 + 0.0182i 0.0041 + 0.0089i −0.0085 − 0.0182i 0.4724 + 0.0000i
The fidelity for each state, also reported in the main text, is respectively: 0.9889(1), 0.9901(1), 0.9279(17), 0.9319(16).
Tomography of Bell state measurements
We analyse the efficiency of the measurement apparata. A BSM consists of Hang-Ou-Mandel (HOM) interferometers and coincidence counts. The BSM is only partially deterministic, discriminating two of the four Bell states (|φ± i, or |ψ± i if a 90◦ QWP rotates one of the photons before en-
0.0058 + 0.0029i −0.0003 − 0.0026i 0.0029 + 0.0000i 0.0029 + 0.0043i
(A.32)
tering in the HOM interferometer) at a time. The interferometry visibility in our setting is 0.91. Two BSM (1,2) are required to evaluate purity and overlap by measurements on two system copies. This requires the indistinguishability of the four interfering photons 1-4, including their arriving time, spatial mode and frequency. As explained in the main text, our three source scheme ensures that, post-selecting sixfold coincidences, each detected photon pair is emitted by a different source. We test our measurement hardware by performing BSM tomography. N The probe states are chosen of the form |{H, V, D, A, R, L}i |{H, V, D, A, R, L}i, where the
5 labels identify the following photon polarisations: horizon√ 2), anti-diagonal tal (H), vertical (V), diagonal ((H + V)/ √ √ ((H − V)/ √2), right circular ((H + iV)/ 2), and left circular ((H − iV)/ 2). The measurement results for all the possible outcomes are recorded accordingly. An iterative maximum
likelihood estimation algorithm yields the estimation of what projection is performed in each run [A9]. The estimated Bell state projections Π1(2) = |xihx|A1 (B1 )A2 (B2 ) , x = φ± , ψ± , reconx structed from BSM1 (detecting on A1 A2 ) and BSM2 (detecting on B1 B2 ), read
+
Πφ1
Πφ1
−
+
Πψ1
Πψ1
−
+
Πφ2
Πφ2
−
Πψ2
+
Πψ2
−
0.5142 0.0096 − 0.0102i 0.0043 − 0.0055i 0.4443 − 0.0088i 0.0096 + 0.0102i 0.0024 −0.0005 + 0.0007i −0.0037 + 0.0018i , = 0.0043 + 0.0055i −0.0005 − 0.0007i 0.0052 0.0003 + 0.0110i 0.4443 + 0.0088i −0.0037 − 0.0018i 0.0003 − 0.0110i 0.4863 0.4816 −0.0088 + 0.0057i −0.0081 + 0.0039i −0.4481 + 0.0048i −0.0088 − 0.0057i 0.0031 0.0013 + 0.0019i 0.0136 − 0.0096i , = −0.0081 − 0.0039i 0.0013 − 0.0019i 0.0018 −0.0001 − 0.0055i −0.4481 − 0.0048i 0.0136 + 0.0096i −0.0001 + 0.0055i 0.5033 0.0014 −0.0000 − 0.0083i 0.0100 − 0.0010i 0.0006 + 0.0006i −0.0000 + 0.0083i 0.4954 0.4382 − 0.0059i −0.0136 + 0.0147i , = 0.0100 + 0.0010i 0.4382 + 0.0059i 0.5059 −0.0057 + 0.0143i 0.0006 − 0.0006i −0.0136 − 0.0147i −0.0057 − 0.0143i 0.0014 0.0027 −0.0008 + 0.0128i −0.0062 + 0.0026i 0.0032 + 0.0033i −0.0008 − 0.0128i 0.4991 −0.4390 + 0.0033i 0.0038 − 0.0068i , = −0.0062 − 0.0026i −0.4390 − 0.0033i 0.4871 0.0054 − 0.0198i 0.0032 − 0.0033i 0.0038 + 0.0068i 0.0054 + 0.0198i 0.0090 0.4893 0.0043 − 0.0223i 0.0064 − 0.0182i 0.4397 − 0.0667i 0.0043 + 0.0223i 0.0017 0.0008 − 0.0004i 0.0003 + 0.0159i , = 0.0064 + 0.0182i 0.0008 + 0.0004i 0.0012 0.0123 + 0.0107i 0.4397 + 0.0667i 0.0003 − 0.0159i 0.0123 − 0.0107i 0.4942 0.5036 0.0050 − 0.0021i −0.0015 + 0.0040i −0.4413 + 0.0636i 0.0050 + 0.0021i 0.0023 −0.0011 + 0.0008i 0.0091 − 0.0072i , = −0.0015 − 0.0040i −0.0011 − 0.0008i 0.0011 −0.0069 + 0.0007i −0.4413 − 0.0636i 0.0091 + 0.0072i −0.0069 − 0.0007i 0.4987 0.0032 −0.0098 + 0.0070i −0.0140 + 0.0192i 0.0018 + 0.0016i −0.0098 − 0.0070i 0.4919 0.4375 + 0.0446i −0.0101 − 0.0085i , = −0.0140 − 0.0192i 0.4375 − 0.0446i 0.5012 −0.0059 − 0.0061i 0.0018 − 0.0016i −0.0101 + 0.0085i −0.0059 + 0.0061i 0.0050 0.0039 0.0005 + 0.0173i 0.0091 − 0.0049i −0.0001 + 0.0014i 0.0005 − 0.0173i 0.5041 −0.4371 − 0.0451i 0.0007 − 0.0002i . = 0.0091 + 0.0049i −0.4371 + 0.0451i 0.4965 0.0004 − 0.0052i −0.0001 − 0.0014i 0.0007 + 0.0002i 0.0004 + 0.0052i 0.0021
The average fidelities of BSM1 and BSM2 are, as reported in the main text as well, 0.9389±0.0030 and 0.9360±0.0034 respectively.
∗
[email protected] [email protected] ‡
[email protected] [A1] S. Amari, Differential-Geometrical Methods of Statistics (Springer, Berlin, 1985). †
(A.33)
[A2] I. Bengtsson and K. Zyczkowski, Geometry of Quantum States (Cambridge University Press, Cambridge, 2007). [A3] J. Grabowski, M. K´us, and G. Marmo, Geometry of quantum systems: density states and entanglement, J. Phys. A 38, 10217 (2005). [A4] D. Petz and C. Ghinea, Introduction to quantum Fisher information, QP–PQ: Quantum Probab. White Noise Anal. 27, 261 (2011). [A5] E. A. Morozova and N. N. Chentsov, Markov invariant geometry on state manifolds (in Russian), Itogi Nauki i Tehniki 36, 69 (1990). [A6] P. Gibilisco, D. Imparato, and T. Isola, Inequalities for Quantum Fisher Information, Proc. of the Am. Math. Soc. 137, 317
6 (2008). [A7] D. Paiva Pires, M. Cianciaruso, L. C. C´eleri, G. Adesso, and D. O. Soares-Pinto, Generalized Geometric Quantum Speed Limits, Phys. Rev. X 6, 021031 (2016).
[A8] B. N. Simon, C. M. Chandrashekar, and S. Simon, Hamilton’s turns as a visual tool kit for designing single-qubit unitary gates, Phys. Rev. A 85, 022323 (2012). [A9] J. Fiur´asˇek, Maximum-likelihood estimation of quantum measurement, Phys. Rev. A 64, 024102 (2001).
