The entanglement, one of the central mysteries of the quantum mechanics, plays a significant role in a variety of applications of the quantum information theory. A natural question in theoretical and experimental importance is whether it is possible to detect a universal entanglement without full diagnostics. The diagnostics relies on a set of quantum trajectories and their records from measurements. This model reflects the probability that each of the measurements may be damaged from interference and decoherence, and may also be associated with recording of continuous signals for an end-time period. The goal is then to retrieve the quantum state such as it had been in the beginning of this measurement process. The proposed solution relies on explicit expression of the probability function through effective matrices contained in the quantum approximation and solutions of ad-joint quantum filters. In this article, we prove а no-go theorem, which outlines this possibility for non-adaptive schemes, which use only single-copy measurements. We also examine in detail а previously conducted experiment, for which it is claimed that detects the entanglement of two-qubit states through adaptive single-copy measurements without full diagnostics. With the conduct of the experiment and the analysis of the data, we demonstrate that the information collected is really sufficient to reconstruct the state. These results reveal a fundamental limit of the single-copy measurements upon the entanglement detection, and provide a common framework for learning the detection of other interesting properties of the quantum states, such as the positivity of partial transposition and the k-symmetric-extendibility.
Extraction of the state from a dynamic and non-ideal prepared system is of fundamental importance for the quantum computational physics. Contrary to the classical physics, the determination of the quantum state
In this article, we offer a method for measurement of the parameters of a partially unknown or partially coherent superposition of system quantum states. Such a circuit is created, when connecting the magnetic sublevels m = −1, m = 0, and m = 1 of J = 1 level with J = 0 level with laser pulses with linear (π), right circular (σ+), and left circular (σ−) polarizations. There are two ways to select the signal-through optical pumping of the population density of the upper state or through fast pulse excitation. The population density ρee (t) depends from the unknown elements of the superposition, as well as from the laser parameters (intensities and phases). By measuring the signal at different laser parameters, we receive a system of algebraic equations for the unknown elements of the superposition, the latter of which we define. The unknown factor F can be eliminated, taking relations of signals, because it does not depend from the laser parameters. The method of optical pumping has priority over the fast pulse excitation, because it is not compulsory at each new measurement of the signal, the Raby frequencies and the frequency differences to be one and the same.
Upon different laser phases and ellipticities are necessary 16% - 20% more independent measurements of the signal for the determination of the elements of the matrix density, compared to the cases when the unknown state is clear.
By default, the measurement process is assumed to be instantaneous and without defects from the decoherence. In specific experimental situations it is not difficult to be taken into account certain measuring imperfections. As a whole, the obtaining of a probability function in the presence of imperfections and decoherence during the measurement has not been examined sufficiently in depth. This is one of the goals of this document-to determine how the deficiencies and decoherence for quantum filtering affect the measurement
More specifically, the bipartite quantum state рАВ of the systems А and В, is separable, if it can be written as a combination of states of the products рАВ = ∑iрiriA⊗riB as pi ³ 0 and ∑iрi = 1, for certain states riA of the system A and riB of the system В; otherwise рАВ is entangled [
In practice, an entanglement can be detected by measurement of the “entanglement witnesses”, physical observables with known values, which prove the existence of quantum entanglement in a given state рАВ [
These widely known limitations on the Bell’s inequalities, the entanglement witnesses and the entanglement measurement raise a fundamental question: How do we universally detect the entanglement by physical observables? The traditional method for solving this problem is to fully characterize the quantum state by diagnostic of the state [
A natural solution is to find a way to obtain the value of the entanglement measurement without full diagnostics of the state (FDS). In fact, many efforts were made in this aspect in the last decade [
In this article, we study the possibility for detecting entanglement without full diagnostics of the state by measuring only the single-copy observables. Surprisingly, despite the previous signs, we find that this attractive task, unfortunately is impossible, if only single-copy observables are measured. This means that there is no way to determine with certainty each entanglement measurement or to determine even whether the value is zero or not, without full diagnostics of the state. For greater accuracy, this means that for any set of informationally-in- complete measurements, there are always two different states, an entangled рАВ, and a separable sAB, which give the same results of the study under this measurement. This sounds a bit counter-intuitive at first glance, but as the entanglement is only a single value, while the diagnostics of the quantum state requires measurement of a set of observables, which are informationally-incomplete, there is a need to determine the scale through square of the Hilbert space of the system.
Our observation is that the universal detection of any characteristic without full diagnostics of the state imposes strong geometrical structural conditions for the set of states that have that characteristic. The set of separable states does not fulfill these conditions due to their non-linear nature, and hence the universal detection of entanglement without full diagnostics of the state through the use of single-copy measurements is impossible. There is a good geometric picture of this fact: unless the form of these separable states is not “cylindrical”, it is not possible to find a projection of the space of the state on a lower measurable hyperplane with non-overlap- ping image for the set of separable states or entangled states.
If adaptive measurements are allowed (the observable that must be measured may depend on previous results of measurement), a protocol is applied in [
We design an experiment with which to apply this adaptive protocol as proposed in [
In addition, it is worthwhile to underline that to the best of our knowledge that this is the first experimental realization of the quantum filters (or equivalently of the channel for reducing the amplitude) through an auxiliary method. In comparison with the optical platform that does not require additional ancilla qubits for realization of the channel for reduction of the amplitude [
We demonstrate also that if joint measurements of r-copies (i.e. rАБ⊗r ) are allowed even for r = 2 can be found protocols that detect the entanglement of рАВ, without FDS. Therefore, our constant result reveals a fundamental limit of the single-copy measurements, and provides a common framework for studying other interesting quantities for a bipartite quantum state, such as the positivity of the partial transposition and k-symmetric- extendibility [
We discuss the constant result, claiming that it is impossible to determine universally whether a given state is entangled or not without FDS, with single-copy measurements. We first prove a no-go theorem for the non- adaptive measurements, and then examine protocol with adaptive measurements as proposed in detail in [
Non-adaptive measurement. For any given bipartite state рАВ, is allowed only measurement of the physical observables of one copy of this given state. I.e. we can only measure the Hilbert’s operators SK, which act on A⊗B. For simplicity, we consider the case in which both А, В, are qubits. Our method naturally includes also the general case of any bipartite system (see “Additional information” for details).
Now we are looking at a two-qubit state рАВ. To obtain information on р, we measure a range of S physical observables
Let’s suppose that we can solve universally whether a random рАВ is entangled or not, without a measurement of informationally-complete range of observables. I.e. there is a set S of almost к = 14 physical observables such that, as we measure S, we can say whether рАВ is entangled or not. For this purpose, it is sufficient to assume that к = 14.
The set of all two-qubit states рАВ, designated with А, is characterized with 15 real parameters, which form a convex set in R15. The separable two-qubit states S form a convex subset A. It is common knowledge that S has non-vanishing volume [
The set of measurements S with k = 14 may be rendered as a definition of the projections of A (and hence of S) on a 14-dimensional hyperplane. If the measurement of the observables in S can demonstrate with certainty whether рАВ is entangled or not, the images on the hyperplane of the separable states S and the entangled state Е should not overlap. We illustrate this geometric idea in
In fact, the only possibility to separate any set from the other states without FDS is the set of the intersection of the set of all states (i.e. set A as in
So to show that the detection of entanglement without diagnostics is impossible, it is sufficient to prove that S is not “cylindrical” (at R15). To do this, we show that for each projection on the hyperplane with 14 dimensions with normal direction R, there always is a two-qubit state р, which is on the border of the set S, as R + tR is entangled for some t (see “Additional information” for details). i.e. р and р + tR have the same image of the hyperplane with 14 dimensions.
To be useful, the quantum computers ultimately must include a large number of qubits. At the same time the scientists studying mainly the quantum physics aim at examining larger and larger objects. For example, an entanglement of up to fourteen ions or eight photons has been generated and checked. However, strictly analyzing the quantum properties of these entangled states in these expanded systems with instruments, which are available is cumbersome, mostly because of difficulties like time consuming and difficult data processing. To overcome these restrictions, we need innovative techniques for the analysis of the quantum state. A possible approach is the testing of an entanglement in a range of previously unexplored quantum systems.
Because of their non-local structure, the entanglement witnesses may not directly be provided from local measurements, but it is still easy to obtain the expectation value: Several types of correlations between the quantum systems can simply be measured and then by a linear combination are retrieved the mean values for calculating the value of non-local observables. These ratios may take the form of experimental measurements of polarization on spatially separated photons, where the ratios may be such that two photons will always have opposite polarization. Geometrically, the set of all separable states is convex, which means that for any two points in it, the line for connection is also in the set. This means that can be found such entanglement witnesses W for any entangled state, since there is always a possibility for separation of entangled from the separable states.
This geometric picture leads to a common framework for studying the detection of other interesting quantities for the bipartite quantum state with single-copy measurements. Indeed, our proof also showed that the states sets with positive partial transposition (PPT) are not “cylinder-like”, and it follows herefrom that they cannot be
detected by single-copy measurements without full diagnostics of the state. With a similar method, we can show that the states sets allowing k-symmetric-extendibility, are also not “cylinder-like”, even for a two-qubit system. This shows the fundament limit of the single-copy measurements, i.e., a full diagnostics of the state is required in order to universally reveal the non-trivial properties of the quantum states (i.e. separability, PPT, k-symm- etric-extendibility, see “Additional information” for details).
Adaptive measurement. In case of adaptive protocols, the observable, which must be measured at each step may depend on the previous study results. The type of protocol for measurement may be formulated as follows. First, the observable Н1 is selected, and tr(H1p) is measured. Let’s suppose that the result of the study is α1. On the basis of α1, the observable Н2α1 is selected and tr(Н2α1 p) is measured. Let’s assume that the outcome of the survey is α2. On the basis of α1, α2, the observable Н3α1α2, is selected tr( Н3α1α2 р), is measured and so on.
The protocol in [
Let’s first briefly introduce the idea for distillation of the entanglement by guiding filtering procedure (33). For unknown two-qubit state
In case that
At first glance, the above procedure seems acceptable to determine the entanglement value without FDS because only the single-qubit matrices of the density
To demonstrate the relationship between the local filters and FDS, we simulate a procedure with a local filter as we choose another number of applied filters. It appears that in many cases k = 4 (five filter) is sufficient in order to uniquely determine
As an example, we illustrate the simulation of the input state as equation (5) with l = 0.2. Initially, we have 15 real parameters (i.e. degrees of freedom, DOF for short), which to determine
Experimental protocol in the setup of nuclear magnetic resonance. In order to apply experimentally the protocol, we first must discuss how to realize the local filters at a NMR system. Without loss of generality, we can look at the local filter FA applied on qubit A for example. For any FA, it may always be decomposed to the form UAΛAVA through decomposition of the only value where UA and VA are inseparable values and ΛA is the diagonal operator of Kraus:
the quantum channel of the system qubit A, will be ΛA. By selecting later the subspace, where the ancilla qubit
The operation U1A in this way is a controlled rotation: when the qubit of the system А is
The schematic circuit for the application of a proposal for a filter-based distillation of the entanglement for an unknown two-qubit state
Application of NMR. In order to implement the abovementioned protocol based filter of the distillation of the entanglement in NMR, we need a 4-qubit quantum processor, which shall consist of two system qubits A and B and two ancilla qubits 1 and 2. Our 4-qubit sample is 13С with trans-crotonic acid, dissolved in d6-acetone. The Methyl-group М, Н1 and Н2 are distinguished through all experiments. The internal operator of Hamilton of this system may be described as
where υj is the chemical change of the j-th rotation and Jjk is the J-th strength of the connection between rotations j and k. We denoted C3 and С2 as the system qubits A and B, and C4 and С1 as ancilla qubits 1 and 2 which can help at in imitating the filters, respectively. All experiments were carried out on a Bruker DRX 700MHz spectrometer at room temperature.
Our target input state was selected as a combined state, including one proportion of the state of the Bell and two proportions of the state of the product, as the state of the Bell can be adjusted. This state is written as:
where
have matches 1, 0 and 0 respectively. The parameter λ in (0.1) is thus proportional to the entanglement value
A sequence of NMR, which to realize the proposal based on filter, for the distillation of the entanglement. More precisely, this sequence shows how to realize the first two filters
The two-qubit diagnostics of the state is applied on the system qubits after the creation of
Now we should demonstrate how to realize the local operations for filtering in NMR. By measuring the local reduced matrix of the density
where
Here we have assumed that t2 > 0 (
Every time after executing one local filter, we implement diagnostics with single qubit on the other qubit, rather than the working qubit at which was applied the filter. The reason is that the working qubit has developed to identity due to the properties of this filter. The tomographic result was used for development of the next filter on the other qubit. In principle, before the application of any filters, it is necessary to be set the two ancilla qubits to
From the above discussions, we showed that the experiments with NMR contain only evolution of free J-couplings and unitaries with single qubit. For the evolutions of the J-couplings, we imposed in the system to be applied free equation of Hamilton (4) for the same time. For local unitaries, we use techniques for Gradient Ascent Pulse Engineering (GRAPE), in order to optimize them (46), (47). The method GRAPE provides 1 ms width of the pulse and above 99.8% fidelity for each local unitary, and furthermore all pulses are corrected by setup of the control of the feedback in NMR spectrometer in order to reduce the contradictions between the ideal and the applied pulses (48)-(50).
Experimental results and analysis of errors. We prepared six input states by varying λ from 0.2 to 0.7 with 0.1 size of the step in the form of equation. After the preparation, we carried out two-qubit full diagnostics of the state for each state, and reconstructed them as
After the initial preparation of the state and each filter we received the reduced matrix of the density of qubit A and/or B through diagnostics of single qubit in the subspace, where the ancillary qubits are
The Fidelities between
With seven states with single qubit, we can recreate the initial two-qubit state
Then we calculated the concurrence for each one of the cases with different input two-qubit state. The concurrence is monotone entanglement defined for combined state p of two qubits
where l1, l2, l3 и l4 are the own values of
in descending order. Obviously, the concurrence is proportional to the λ, as the λ is the weight of the state of the Bell, which is the only thing, which contributes to the entanglement. The blue squares show the concurrence of
For convenience, we assume that the errors come from three main aspects and are additive. One error is caused by the inaccuracy of the procedure of the diagnostics with single qubit. As we used the least-square fitting algorithm to analyze the spectrum of the outcome, and we have converted the data into quantum states, the concurrence caused around 3.00% uncertainty of the final result with single qubit. The second is the error from the application of imperfect filters in the experiment. It originates mainly from errors in the accumulation of GRAPE pulses, which is about 1.59% for each operation of the filter. The third error, to a lesser extent, was around 1.20%, caused by decoherence. Therefore, in total we have concluded that at most 5.79% error may appear in the whole process. We identified them as artificial noise and we’ve incorporated them in the theoretical input state
We proved the no-go theorem, and that there is no way to detect the entanglement of an arbitrary bipartite state рАВ without FDS if only nonadaptive measurements with one copy are allowed. Our conclusion follows from a good geometric picture: It is not possible to detect a projection of the space of the state on a lower hyperplane of the dimension with non-overlapping image for a set of separable states and entangled states. Our method provides a common framework for studying the detection of other interesting quantities for the bipartite quantum state, such as positive partial transposition and k-symmetric-extendibility. We also have analyzed the case of the adaptive measurements. It is assumed in (33) that the concurrence in the measurement of the entanglement for two-qubit states can be determined without FDS, through measurements with single copy. In order to apply this protocol, we have developed a method with maintaining of ancillaries to finalize the filters. Practically, the technique can be extended to cover other quantum systems, except the optics in order to apply a channel reducing the amplitude that is of enormous importance in the quantum information. Through the application of this protocol, we show that when the experimental data are collected for a given state рАВ, this protocol leads in fact to FDS of рАВ. Therefore, this protocol does not lead to universal detection of рАВ without FDS.
Concurrence of
This justifies our unchanging results, which indicates that even when at the adaptive measurements, the universal entanglement detection by measurements with single-copy is impossible without FDS.
It is important to underline that the large identity does not arise at any unitary propagator and it is not observed in NMR. So there is only a need for focusing on the part for the deviation |0000>, as the entire system is functioning exactly as it.
In consequence, each component undergoes the entire procedures for filtering and measuring respectively, as the final result is obtained by summarizing all three experiments.
Diagnosis with single qubit after each filter. The procedure for the distillation of the entanglement described to in Ref. (33), includes local filtering operations, which means that each filter depends on the previous result from the measurement with single qubit. In one experiment, we performed diagnostics with single qubit on the system qubits C2 and С3, respectively. It requires measurement of the expected values of sx, sy and sz, respectively. In our 4-qubit system, this diagnostics with single qubit (as we assume the measurement of is equal to the measurement
So our survey shows the fundamental ratio between the detection of the entanglement and the diagnostics of the quantum state. Namely that universal entanglement detection without FDS is not possible only with single-copy measurements. Naturally occurs also the question of whether the joint measurements on r copies of the state
In fact there are also cases in which this is possible even for r = 2. For example, we found such a scheme, which reveals the entanglement of an arbitrary two-qubit state рАВ without FDS, if we allow joint measurements of two copies. The idea is that рАВ is entangled only and only when (23) Det (
Furthermore, if only measurements with single-copy are permitted, the value of Det (
Nikolay Raychev, (2016) Fundamental Limit for Universal Entanglement Detection. Journal of Applied Mathematics and Physics,04,1567-1577. doi: 10.4236/jamp.2016.48166