Discontinuity of Maximum Entropy Inference and Quantum Phase Transitions

In this paper, we discuss the connection between two genuinely quantum phenomena --- the discontinuity of quantum maximum entropy inference and quantum phase transitions at zero temperature. It is shown that the discontinuity of the maximum entropy inference of local observable measurements signals the non-local type of transitions, where local density matrices of the ground state change smoothly at the transition point. We then propose to use the quantum conditional mutual information of the ground state as an indicator to detect the discontinuity and the non-local type of quantum phase transitions in the thermodynamic limit.


Introduction
Quantum phase transitions happen at zero temperature with no classical counterparts and are believed to be driven by quantum fluctuations [22]. The study of quantum phase transitions has been a central topic in the condensed matter physics community during the past several decades involving the study of exotic phases of matter such as superconductivity [1], fractional quantum Hall systems [12], and recently the topological insulators [2,11,18]. In recent years, it also becomes an intensively studied topic in quantum information science community, mainly because of its intimate connection to the study of local Hamiltonians [6].
In a usual model for quantum phase transitions, one considers a local Hamiltonian H ( ) λ which depends on some parameter vector λ. While H ( ) λ smoothly changes with λ, the change of the ground state ( ) 0 λ ψ | 〉may not be smooth when the system is undergoing a phase transition. Such kind of phenomena is naturally expected to happen at a level-crossing, or at an avoided (but near) level-crossing [22].
Intuitively, the change of ground states can then be measured by some distance between ( ) 0 λ ψ | 〉and For a small change of the parameters λ, such a distance is relatively large near a transition point, while the Hamiltonian changes smoothly from H ( ) λ to H ( ) λ λ δ + . The fidelity approach, using the fidelity of quantum states to measure the change of the global ground states, has demonstrated the idea successfully in many physical models for signaling quantum phase transitions [3,5,25,26]. In addition, the relations between the density functional fidelity susceptibility and the Kullback-Leibler entropy or Rényi entropy have been discussed in [17,21]. While the fidelity approach is believed to provide a signal for many kinds of quantum phase transitions, it does not distinguish between different types of the transition, for instance local or non-local (in a sense that the reduced fidelity of local density matrices may also signal the phase transition, as discussed in [5]). Moreover, one usually needs to compute the fidelity change of a relatively large system in order to clearly signal the transition point.
In this work, we explore an information-theoretic viewpoint to quantum phase transitions. Our approach is based on the structure of the convex set given by all the possible local measurement results, and the corresponding inference of the global quantum states based on these local measurement results. By the principle of maximum entropy, the best such inference compatible with the given local measurement results is the unique quantum state * ρ with the maximum von Neumann entropy [7]. It is known that in the classical case, the maximum entropy inference is continuous [7,9,24]. This means that, for any two sets of local measurement results α and α′ close to each other, the corresponding inference * ( ) α ρ and * ( ) α ρ ′ are also close to each other. Surprisingly, however, the quantum maximum entropy inference can be discontinuous! Namely, a small change of local measurement results may correspond to a dramatic change of the global quantum state.
The main focus of this work is to relate the discontinuity of the quantum maximum entropy inference to quantum phase transitions. We show that the discontinuity of maximum entropy inference signals levelcrossings of the non-local type. That is, at the level-crossing point, a smooth change of the local Hamiltonian H ( ) λ corresponds to a smooth change of the local density matrices of the ground states, while the change of * ρ , the maximal entropy inference of these local density matrices, is discontinuous.
We then move on to discuss the possibility of signaling quantum phase transitions by computing the discontinuity of the maximal entropy inference * ρ . Given the observation on the relation between discontinuity of * ρ and the non-local level-crossings, it is natural to consider signaling quantum phase transitions by directly computing where the discontinuity happens. This approach works well in finite systems, but may fail in the thermodynamic limit of infinite size systems as the places of discontinuity (i.e. where the system 'closes gap') may change when the system size goes to infinity. Hence, computations in finite systems may provide no information of the phase transition point. We propose to solve the problem by using the quantum conditional mutual information of two disconnected parts of the system for the ground states. This idea comes from the relationship between the three-body irreducible correlation and quantum conditional mutual information of gapped systems. As it turns out, the quantum mutual information works magically well to signature the discontinuity point, thereby also signals quantum phase transitions in the thermodynamic limit. In some sense, the quantum conditional mutual information is an analog of the Levin-Wen topological entanglement entropy [13].
We apply the concept of discontinuity of the maximum entropy inference to some well-known quantum phase transitions. In particular, we show that the non-local transition in the ground states of the transverse quantum Ising chain can be detected by the quantum mutual information of two disconnect parts of the system. The scope of the applicability of the quantum conditional mutual information was extended to many other systems, featuring different types of transitions [28,29]. All these studies conclude that the quantum mutual information serves well as a universal indicator of non-trivial phase transitions.
We organize our paper as follows. In section 2, we discuss the concept of the maximum entropy inference and summarize some important relevant facts. In section 3, we analyze several examples of discontinuity of the maximum entropy inference * ρ , ranging from simple examples in dimension 3 to more physically motivated ones. In section 4, we link the discontinuity of * ρ to the concept of the long-range irreducible many-body correlation and propose to detect the non-local type of quantum phase transitions by the quantum conditional mutual information of two disconnect parts of the system. In section 5, further properties of discontinuity of the maximum entropy inference are discussed. We provide both a necessary condition and a sufficient condition for the discontinuity to happen. Finally, section 6 contains a summary of all the main concepts discussed and a discussion of possible future directions.

The maximum entropy inference
We start our discussion by introducing the concept of the maximum entropy inference given a set of linear constraints on the state space.

The general case
Let  be the d-dimensional Hilbert space corresponding to the quantum system under discussion and ρ be the state of the system. Let  be the set of all possible quantum states on . Any tuple F F F ( , , , ) The set   can be considered as a projection of  and is a compact convex set in r  . If all the F i ʼs are commuting (i.e. F F [ , ] 0 i j = , corresponding to the classical case), then   is a polytope in r  . The convex set   is mathematically related to the so-called '(joint) numerical range' of the operators F i ʼs. For more mathematical aspects of these joint numeral ranges and the discontinuity of the maximum entropy inference, we refer to [20]. We remark that   is also known as quantum convex support in the literature [23].
As it will be clear in later discussions, the observables F i ʼs usually come from the terms in the local Hamiltonian of interest so that the Hamiltonian is in the span of the observables F i ʼs. We will call H Given any measurement result α ∈   , we are interested in the set of all states in  that can give α as the measurement results. We denote such a set as It is the preimage of α under the mapping in equation (1). In other words, it consists of the states satisfying a set of linear constraints and we call this subset of  a linear family of quantum states.
In general, there will be many quantum states compatible with α and ( ) α  contains more than one state, unless one chooses to measure an informationally complete set of observables (for example, a basis of operators on  as one often does for the case of quantum tomography). Especially, when the dimension of system d is large, it is unlikely that one can really measure an informationally complete set of observables. For instance, for an n-qubit system when n is large, we usually only have access to the expectation values of local measurements, each involving measurements only on a few number of qubits. In this case, quantum states compatible with the local observation data α are usually not unique.
The question is then what would be the best inference of the quantum states compatible with the given measurement results α. The answer to this question is well-known, and is given by the principle of maximum entropy [7,24]. That is, for any given measurement results α, there is a unique state * ( ) α ρ ∈  , given by ρ is the von Neumann entropy of ρ. We call * ( ) α ρ the maximum entropy inference for the given measurement results α. More explicitly, it is the optimal solution of the following optimization problem Maximize: ( ) Subject to: tr( ) , for all 1, 2, , , .
It may seem counter-intuitive that both the maximum entropy inference * ρ and its entropy can be discontinuous [9] as functions of the local measurement data α. When we say * ρ is discontinuous, we mean the state itself, not its entropy, is discontinuous. Indeed there could be examples where these two concepts are not the same (e.g. the energy gap of the system closes but the ground-state degeneracy does not change). For all examples considered in this paper, however, the entropy is also discontinuous when the state is. We note that the discontinuity of the maximum entropy inference is a genuinely quantum effect as the classical maximum entropy inference is always continuous [7,24].

The case of local measurements
The discussions in the above subsection specialize to the important case of many-body physics with local measurements.
Consider an n-particle system where each particle has dimension d. The Hilbert space  of the systems is ( ) We know that, for an n-particle state ρ, we usually only have access to the measurement results of a set of local measurements on the system, where each F i acts on at most k particles for k n ⩽ . The most interesting case is where n is large and k is small (usually a constant independent of n). In this sense, we will just call such a measurement setting k-local.
Notice that each measurement result F tr( ) i ρ now depends only on the k-particle reduced density matrix (k-RDM) of the particles that F i is acting non-trivially on. It is convenient to write the set of all the k-RDMs of ρ (in some fixed order) as a vector The k-RDMs k ( ) ρ will play the role of expectation values α as in the general case. Along this line, the set of results of all k-local measurements can be defined in terms of k-RDMs, and we write the set k Similarly, the linear family can also be defined in terms of k-RDMs The maximum entropy inference given the k-RDMs k We remark that, in practice, one may not be interested in all the m n k k-RDMs, but rather only those k-RDMs that are geometrically local. For instance, for a lattice spin model, one may only be interested in the two-RDMs of the nearest-neighbour spins. Our discussion can also be generalized to these cases, as in the discussion in [29] for one-dimensional spin chains. There could also be cases that the system has certain symmetry (for instance a bosonic system or fermionic system where all the k-RDMs are the same), and our theory can be naturally adapted to these cases. The maximum entropy inference * ρ given local density matrices has a more concrete physical meaning. For , then ρ is uniquely determined by its k-RDMs using the maximum entropy principle. One can argue, in this case, that all the information (including all correlations among particles) contained in ρ are already contained in its k-RDMs. In other words, ρ does not contain any irreducible correlation [30] of order higher than k. On the other hand, if * ρ ρ ≠ , then ρ can not be determined by its k-RDMs and there are more information/correlations in ρ than those in its k-RDMs. Therefore, ρ contains nonlocal irreducible correlation that can not be obtained from its local RDMs.

Discontinuity of * ρ
In this section, we explore the discontinuity of * ρ based on several simple examples. The first three of them involve only two different measurement observables, but they do demonstrate almost all the key ideas in the general case.

The examples of two observables
We will choose d = 3 for the Hilbert space dimension as it is enough to demonstrate most of the phenomena we need to see. Fix an arbitrary orthonormal basis of 3  , say, Example 1.  consists of the following two observables First, notice that F F , 1 2 do not commute. The set of all possible measurement results   is a convex set in 2  . We plot this convex set in figure 1(a). To obtain this figure, we let ρ vary for all the density matrices on 3  , and let the corresponding F tr( ) 1 ρ be the horizontal coordinate and F tr( ) 2 ρ the vertical coordinate. The resulting picture is nothing but the numerical range of the matrix F F i 1 2 + .
As discussed in section 2, the Hamiltonian H related to  has the form for some parameters , 1 2 θ θ ∈ . Notice that the Hamiltonian corresponds to supporting hyperplanes of   , as the inner product has the form H tr( ) ( , ) · ( , ) T We demonstrate these supporting hyperplanes of   in figure 1 It is straightforward to see that the ground state of H is non-degenerate except for the case 0, 0 We now show that the maximum entropy inference * ( ) α ρ is indeed discontinuous at the point To see this, first notice that the corresponding * ( 0 0 1 1 ) While for any small ϵ, the corresponding ground state space of F F Therefore, for any sequence of α on the boundary of   approachingα 0 , and the discontinuity of * ( ) α ρ follows. This example seems to indicate that the discontinuity simply comes from degeneracy: as in general degeneracy is rare, whenever such a point of degeneracy exists, we have a singularity on the boundary of   so discontinuity happens. However, it is important to point out that this is not quite true. For example, degeneracy also happens in classical systems where there can have no discontinuity of * ρ . We further explain this point in the following example. By simple calculations, now the maximum entropy inference * ( ) α ρ is in fact continuous at the point For any small perturbation ϵ, the corresponding ground state space of F sudden with respect to the small change of ϵ, which, however, is accompanied by a sudden change also in the measurement results (from a point near (1, 1) to (1, 0)). As we are considering the discontinuity of * ρ with respect to the measurement dataα, not the parameter ϵ in the Hamiltonian, * ρ is in fact continuous. This example demonstrates that when Hamiltonian changes smoothly, ground states have sudden changes accompanied with the sudden change of measurement results. In other words, the change of ground states can be described already by the change of local measurement results. This is somewhat a classical feature, as discussed in the next example. The convex set   in given in figure 3(a). It is a triangle for this example, and a polytope in the general classical case.
Consider the related Hamiltonian H F F 1 1 2 2 θ θ = + for some , 1 2 θ θ ∈ , as illustrated as supporting hyperplanes in figure 3(b). Similarly, the ground state of H is two-fold degenerate for 0, 0 , the corresponding ground-state space is nondegenerate: it is 1 | 〉for 0 ϵ < and 2 | 〉for 0 ϵ > . So from 0 ϵ < to 0 ϵ > , we also see sudden changes of both the measurement results and the ground states.
In the above three examples, the first one is the most interesting and exhibits smooth change in measurement results and discontinuity of the maximum entropy inference * ρ . The second and third behave in a similar classical way where a small change in the Hamiltonian will induce a sudden change of measurement results and there is no discontinuity of * ρ . We summarize our observations from the three examples in this subsection as below. Although the examples involve two observables only, we state the observation in the more general setting of arbitrarily many observables. • Type I (local type): level-crossing that can be detected by a sudden change of the measurement results.
• Type II (non-local type): level-crossing that can not be detected by a sudden change of the measurement results.
More importantly, only Type II corresponds to discontinuity of the maximum inference * ( ) α ρ .

The example of local measurements
We now give a simple example showing the discontinuity of * ρ in a three-qubit system with two-local interactions.
To see this, consider a family of perturbations H X i i 1 3 ∑ ϵ + = of the Hamiltonian H. For any 0 ϵ ≠ , the ground space is non-degenerate and the unique ground state converges to GHZ 3 | 〉when 0 ϵ → − and to ( 000 111 ) 2 | 〉 − | 〉 when 0 ϵ → +. As the ground state is unique when 0 ϵ ≠ and the Hamiltonian is two-local, the two-RDMs of the ground state determines the state. This means that * ρ is pure and coincide with the ground state for all 0 ϵ ≠ . However, at 0 ϵ = , * ρ is (2) ρ ρ = | 〉〈 | + | 〉〈 | and the discontinuity of * ρ follows. It is worth pointing out the similarity in the structure of the above example and example 1, despite their totally different specific form. First notice that 000 111 + and X i i 1 3 ∑ = has a rather similar block structure as F 1 and F 2 in example 1.
We generalize the observation 1 in terms of local measurements as follows.
Observation 1′. For an n-particle system, consider the set of all k-local measurements  , which then corresponds to a local Hamiltonian H c F j j j ∑ = with F j ∈  acting nontrivially on at most k particles. There are two kinds of ground state level crossing: • Type I: level-crossing that can be detected by a sudden change of the k-RDMs k ( ) ρ .
• Type II: level-crossing that can not be detected by a sudden change of the local k-RDMs k ( ) ρ .
Only Type II corresponds to discontinuity of the maximum entropy inference * ( )

The example of transverse quantum Ising model
Our next example is an n-qubit generalization of example 4 and is known as the transverse quantum Ising model. for J 0 > . For any finite n the discontinuity of * ρ determined by the two-RDMs happen at 0 λ = . For infinite n, the discontinuity of * ρ happen at 1 λ = .

Example 5. The Ising Hamiltonian is given by
The Hamiltonian H ( ) λ has a 2  symmetry, which is given by X n ⊗ , i.e.
In the thermodynamic limit of n → ∞, it is well-known that when λ increases from 0 to , ∞ quantum phase transition happens at the point 1 λ = [19]. For 1 λ → + , 1 λ = is exactly the point where the ground space of H ( ) λ undergoes the transition from non-degenerate to degenerate. A discontinuity of * ( ) ρ λ happens at 1 λ = when 1 λ → + , which is a sudden jump of rank from 1 to 2, while the local RDMs of * ( ) ρ λ change smoothly. For 0 1 λ < ⩽ , the two-fold degenerate ground states, although not exactly the same as those two at 0 λ = , are qualitatively similar. For the range of 0 1 λ ⩽ ⩽ , the ground states are all two-fold degenerate. For finite n, however, in the region of 0 1 λ < ⩽ , an (exponentially) small gap exists between two near degenerate states, and the true ground state does not break the 2  symmetry of the Hamiltonian H ( ) λ . This example demonstrates the dramatic difference between the case of finite n and the case of the thermodynamic limit of infinite n. It also foretells the difficulty of signaling phase transitions by computing the discontinuity of * ρ of finite systems directly. We will propose a solution to this problem in section 4.

Signaling discontinuity by quantum conditional mutual information 4.1. Irreducible correlation and quantum conditional mutual information
We have mentioned the relation between the maximum entropy inference and the theory of irreducible manybody correlations [30]. For an n-particle quantum state ρ, denote its k-RDMs by k ( ) ρ . Then its k-particle irreducible correlation is given by [15,30] Consider a partition A B C , , of the n particles so that A and C are far apart. Define Then the three-body irreducible correlation of ABC ρ is given by in the definition of ABC * ρ . The reason for this is that the region of A and C are chosen to be far apart and, therefore, there will be no k-local terms in the Hamiltonian that act non-trivially on both A and C.
In the discussion on the example of quantum Ising chain, we have observed the difficulty of signaling the discontinuity of * ρ in the thermodynamic limit by computations of finite systems. In the following, we propose a quantity that can reveal the physics in the thermodynamic limit by investigating relatively small finite systems.
The quantity we will use is the quantum conditional mutual information We will also omit the subscript ρ when there is no ambiguity. Usually, the state ρ will be chosen to be a reduced state of the ground state of the Hamiltonian. It is known that the quantum conditional mutual information is an upper bound of C ABC [13,16]. Namely, we have which is equivalent to the strong subadditivity [14] for the state ABC * ρ . The equality holds when the state ABC * ρ satisfies I A C B ( : ) 0 | = , or is a so-called quantum Markovian state. We will use the quantum conditional mutual information I A C B ( : ) | of the ground state, instead of threebody irreducible correlation C ABC , to signal the discontinuity and phase transitions in the system. We do this for two reasons. First, it is conjectured that the equality in equation (21) always holds in the thermodynamic limit for gapped systems. In other words, the corresponding ABC * ρ of the ground state is always a quantum Markovian state (there are reasons to believe this, see e.g. [8,16]). Assuming this conjecture, I A C B ( : ) | is indeed a good quantity to signal the discontinuity and phase transition in the thermodynamic limit. Second, as it turns out, quantum conditional mutual information performs much better as in indicator when we do computations in systems of small system sizes. Most importantly, it doesn't seem to suffer from the problem C ABC has in finite systems. For more discussion on the physical aspects of I A C B ( : ) | , we refer to [28].

The transverse Ising model
We now illustrate the mutual information approach in one-dimensional systems. First, consider a onedimensional system with periodic boundary conditions. As we need A and C to be large regions far away from each other, the partition A B C , , can be chosen as in figure 4. ( : ) | ʼs clearly indicate a phase transition at 1 λ = (where the curves intersect). This is consistent with our discussions for the quantum Ising chain with transverse field in section 3.3.
However, the phase transition of the Hamiltonian with a Z direction magnetic field, given by ( ) ( : ) | for this model, we will not see any transition in the system. We emphasize that the above approach employs calculations of extremely small systems yet still precisely signals the transition point of the corresponding system in the thermodynamic limit. ( : ) | as in figure 5, which is given in figure 7. This clearly shows a discontinuity of * ρ and a quantum phase transition at 1 λ = . However, if the partition in figure 6 is used for the Ising model with periodical boundary condition, as given in figure 8, the behaviour of I A C B ( : ) | will be very different. In fact, in this case I A C B ( : ) | reflects nothing but the 1D area law of entanglement, which will diverge at the critical point 1 λ = in the thermodynamic limit. For a   figure 4. A similar result is presented in [28], from a different viewpoint. finite system as illustrated in figure 9, I A C B ( : ) | does no clearly signal the two different quantum phases and the phase transition.

I A C B
( : ) | as a universal indicator From our previous discussions, we observe that to use I A C B ( : ) | to detect quantum phase and phase transitions, it is crucial to choose the areas A C , that are far from each other. Here 'far' is determined by the locality of the system. For instance, on an 1D chain, the areas A C , in figure 6 are far from each other, but in figure 8 are not.   If such an areas A C , are chosen, then for a gapped system, a nonzero I A C B ( : ) | of a ground state will then indicates non-trial quantum order. We have already demonstrated it using the transverse Ising model, where for 0 1 λ < < , the system exhibits the 'symmetry-breaking' order. In fact, we can also use I A C B ( : ) | to detect other kind of non-trivial quantum orders.
For instance, I A C B ( : ) | was recently applied to study the quantum phase transitions related to the so-called 'symmetry-protected topological (SPT) order', which also has a 'nonlocal' nature despite that the corresponding ground states are only short-range entangled (in the usual sense as discussed in this paper) [29].
It was shown that for a 1D gapped system on an open chain, a non-zero I A C B ( : ) | for the choice of the regions A B C , , as in figure 6 also detects non-trivial SPT order. However, it does not distinguish SPT order from the symmetry-breaking order. In stead, one can use a cutting as given in figure 10, where the whole system is divided into four parts A B C D , , , , and I A C B ( : ) | the detects the non-trivial correlation in the reduced density matrix of the state of ABC. Under this cutting, I A C B ( : ) | is zero for a symmetry-breaking ground state, but has non-zero value for an SPT ground state.
A similar idea also applies to 2D systems. For instance, for a 2D system on a disk with boundary, we can consider three different kinds of cuttings [13,28,29], as shown in figure 11. For each of these cuttings, a nontrivial I A C B ( : ) | detects different orders of the system. For figure 11(a), I A C B ( : ) | detects both symmetrybreaking order and SPT order and topological phase transitions [16]. Figure 11(b) is nothing but the choices of A B C , , to define the topological entanglement entropy by Levin and Wen [13], which detects topological order. And similarly as the 1D case, figure 11(c) detects SPT order, which distinguishes it from symmetry-breaking order (in this case I A C B ( : ) 0 | = for symmetry-breaking order) [29]. In this sense, by choosing proper areas A B C , , with A C , far from each other, a non-zero I A C B ( : ) | universally indicates a non-trivial quantum order in the system. Furthermore, by analyzing the choices of A B C , , , it also tells which order the system exhibits (symmetry-breaking, SPT, topological, or a mixture of them).
We remark that, for a pure state, the cuttings of figures 4 and 6 give that I A C B I A C ( : ) ( : ) | = . However, this is not the case for a mixed state. Therefore, although one may be able to detect nontrivial quantum order simply using I A C

Further properties of the discontinuity
In this section, we further explore the structure associated with the discontinuity of the maximum entropy inference.

Path dependence of discontinuity
We continue our discussion of examples 1-3 in dimension 3, but with more than two observables. The following example illustrates that one may need to choose the right path in order to see the discontinuity of * ρ . It is an example that combines examples 1 and 2 together. Example 6. We consider the tuple  of 3 operators, with F F , 1 2 the same as given in example 1 and However, if α is approached from 0 ϵ → in a Hamiltonian F F 1 2 ϵ − + , then there is discontinuity of * ( ) α ρ , similar as the discussion in example 1.
This example shows that, in general for k measurements, whether there is discontinuity of * ( ) α ρ at the point α ∈   depends on the direction on the boundary of   along which α is approached. If there is a sequence α s approaching α but  . The yellow line corresponds to x (1, 1, ), where the discontinuity happens.
The same situation can happen in example 4. If we approach the two-RDM (2) ρ of the GHZ state using the has a similar structure as that in figure 12, as given in figure 1 (c) of [27]. Now consider the situation of the thermodynamic limit, corresponding to the transverse Ising model with also a magnetic field in the Z direction, i.e. the Hamiltonian ( ) is quite different, as the line of discontinuity (similar as the line x (1, 1, ) in figure 12) will expand to become a 'ruled surface' (see figure 1(b) of [27]), which is nothing but the symmetry-breaking phase [27] (this corresponds to the phase transition at 1 λ = ). Another interesting thing of example 6 is that the discontinuities of * ( ) α ρ do not only happen at the point (1, 1, 0.5) α = . In fact they can happen at any point s (1, 1, ) with s (0 1) < < . This can be done by engineering the Hamiltonian This condition guarantees that the state ψ | 〉 and the maximally mixed state * ρ have the same local density matrices. The discontinuity of maximum entropy inference therefore follows when considering the sequence of reduced density matrices of ( ) ϵ ψ | 〉. Notice that equation (31) is the quantum error-detecting condition for the error set  but without the coherence condition of F c a j a j ψ ψ 〈 | | 〉 = for a = b [10], where c j is a constant that is independent of a. We will refer to this condition as the partial error-detecting condition.