aa r X i v : . [ qu a n t - ph ] J un Priors in quantum Bayesian inference
Christopher A. Fuchs ∗ and Rüdiger Schack † ∗ Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, CanadaN2L 2Y5 † Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX,UK
Abstract.
In quantum Bayesian inference problems, any conclusions drawn from a finite numberof measurements depend not only on the outcomes of the measurements but also on a prior. Herewe show that, in general, the prior remains important even in the limit of an infinite number ofmeasurements. We illustrate this point with several examples where two priors lead to very differentconclusions given the same measurement data.
Keywords:
Bayesian probability, quantum inference, quantum measurement
PACS:
INTRODUCTION
This paper addresses the problem of inference in quantum mechanics in a very generalsetting. Consider a sequence of quantum systems, each with Hilbert space H which, forsimplicity, is assumed to have finite dimension. Now some of the systems are measured.What conclusions can be drawn from the measurement outcomes?It is often useful to consider less general situations. One of the most studied quantuminference problems is quantum state estimation, which is frequently described as follows[1]. Each system is assumed to have the same unknown state s , where s is a densityoperator on H . A sequence of appropriately chosen measurements is then used to deter-mine s . The most general single-system measurement is described by a POVM, whichis a collection of positive operators, { E , . . . , E r } , acting on H , such that (cid:229) k E k is theidentity operator. The index k ∈ { , . . . , r } labels the possible measurement outcomes. Ifthe state of the system is r , the probability of obtaining outcome k is given by tr ( r E k ) .The Bayesian solution to the quantum state estimation problem is straightforward. Itsstarting point is a prior probability distribution p prior ( r ) on the set of density operatorson H . Assume that the POVM { E k } is measured for a single system. If the outcome ofthe measurement is k , the distribution p prior ( r ) is updated using a quantum version ofBayes’s rule [2, 3], resulting in the posterior distribution p posterior ( r ) = p prior ( r ) tr ( r E k ) R d r p prior ( r ) tr ( r E k ) . (1)This process is iterated, each time using a fresh copy of the system and possibly a dif-ferent measurement, and each time setting p prior ( r ) equal to the previously obtained p posterior ( r ) . Given some mild conditions on the original prior distribution, and assum-ing that the measurements are appropriately chosen, it can be shown that the iterated dis-tribution approaches a delta distribution centered at the unknown state s . For instance,his convergence holds if the original p prior ( r ) is nonzero for all r and the measuredPOVM is informationally complete [4].In practice, the iteration will have to stop after a finite number of measurements.The posterior distribution will then generally depend on the initial prior distribution.From a Bayesian perspective, this dependence on the prior distribution is unavoidable.Finding the appropriate mathematical form of the prior distribution is a central part ofthe Bayesian approach to quantum state estimation.There are a number of (non-Bayesian) state estimation methods that attempt to cir-cumvent the dependence of the estimate on the prior distribution [1]. An important ex-ample is maximum likelihood estimation. We are not going to discuss non-Bayesianmethods any further in this paper.In the next section, we will show that quantum state estimation as described above is aspecial case of a far more general quantum inference scenario in which priors retain theirimportance even in the limit of an infinite number of measurements. This is followed bya section illustrating this point with some examples. The paper concludes with a briefdiscussion. GENERAL QUANTUM INFERENCE PROBLEM
There is something peculiar about the quantum Bayes rule (1). Since a density operator r encodes the probabilities for outcomes of quantum measurements, the rule (1) mixestwo kinds of probabilities, namely the “classical” probabilities p prior and p posterior on theone hand, and the “quantum” probabilities encoded in r on the other hand. We will nowshow that the rule (1) is a special case of a more general rule phrased entirely in terms ofquantum states. It is therefore not necessary to make the distinction between two kindsof probability.There is a general updating rule built into quantum theory. To describe a quantummeasurement fully, one needs to provide, in addition to the POVM { E , . . . , E r } , a set of Kraus operators A k j such that E k = (cid:229) j A k j A † k j for k = , . . . , r [5]. If the measurementgives outcome k for a system in state r , the state of the system after the measurementwill be r k = tr ( r E k ) − (cid:229) j A k j r A † k j . (2)This rule has an interpretation very similar to the Bayes rule. If we call r the prior stateand r k the posterior state, we see that the prior state is changed into the posterior stateupon acquisition of the data k .To see that the quantum Bayes rule (1) is a special case of the Kraus rule (2) [2],consider a sequence of quantum systems each with Hilbert space H as before. Wedefine a prior on our sequence of systems as a sequence of n -system states r ( n ) prior , n = , , . . . , where each r ( n ) prior is a density operator on the n -fold tensor-product Hilbertspace H ⊗ n = H ⊗ · · · ⊗ H , and where r ( n ) prior = tr n + r ( n + ) prior for all n ≥
1. By tr n + wedenote the trace over the ( n + ) -th system. In words, each member of the sequence isobtained from the next by tracing over the additional system.ow assume that the first system is measured and an outcome k is obtained. Applyingthe rule (2) we find, for any n ≥
1, that the state of the first n systems after themeasurement is r ( n ) k = tr ( r ( n ) prior E k ) − (cid:229) j A k j r ( n ) prior A † k j , (3)where it is understood that the operators A k j and E k act on the first system only. Bytracing over the first system, we obtain what we call the posterior on our sequence ofsystems, r ( n ) posterior = tr r ( n + ) k ( n = , , . . . ) . (4)The posterior is again a sequence of states; its n -th member is obtained from the ( n + ) -th member of the prior by measuring and then discarding the first system. The posteriorhas the property r ( n ) posterior = tr n + r ( n + ) posterior for all n ≥ r ( ) posterior is the marginal state for the first unmeasured system. Thisstate is sometimes called the Bayesian mean estimator.To recover the familiar rule (1), only one simple additional assumption has to made,namely that for any n ≥ r ( n ) prior is symmetric under permutations of the n systems. In this case we say that the prior is exchangeable [6]. Given this extraassumption, it is the content of the quantum de Finetti theorem [4, 6] that the priorcan be written as r ( n ) prior = Z d r p prior ( r ) r ⊗ n ( n = , , . . . ) , (5)where p prior ( r ) is a probability distribution on the space of single-system density oper-ators, and r ⊗ n is the n -fold tensor product r ⊗ · · · ⊗ r . It is not difficult to establish [2]that a measurement on the first system with outcome k will lead to the posterior r ( n ) posterior = Z d r p posterior ( r ) r ⊗ n ( n = , , . . . ) , (6)with p posterior ( r ) given by the rule (1).As was pointed out in the introduction, in the limit of an infinite number of iterations, p posterior ( r ) typically approaches a delta function which is independent of the detailedfunctional form of p prior ( r ) . This conclusion, however, depends crucially on the assump-tion that the prior is of the form (5), i.e., that the prior is exchangeable. In other words,even in the limit of an infinite number of iterations, conclusions depend on the prior. Allwe can say is that some details of the prior become irrelevant in this limit.Exchangeable priors are an important class of priors that are used so frequently that itis sometimes overlooked that exchangeability is an assumption. Making this assumptionis equivalent to choosing a prior from a restricted set. What we have therefore shown inthis section is that conclusions drawn in Bayesian quantum inference situations generallyepend on the prior as well as measurement data, even in the limit of an infinite numberof measurements. In the next section, we illustrate this point with some examples. EXAMPLE PRIORS
The first example is a sequence of qubits, i.e., two-dimensional quantum systems. Wedenote by | i and | i two orthogonal basis states and consider three priors.Our first prior is exchangeable and given by r ( n ) a = Z d | y i p a ( | y i ) ( | y ih y | ) ⊗ n ( n = , , . . . ) , (7)where d | y i p a ( | y i ) is the Haar measure on the space of pure one-qubit states.Our second prior is also exchangeable, but consists of a sequence of pure productstates. It could be called a Rosenkrantz and Guildenstern prior [7]. This is the stateone might assign to a quantum random number generator manufactured by a trustedcompany. It is given by r ( n ) b = (cid:16) ( | i + | i )( h | + h | ) (cid:17) ⊗ n ( n = , , . . . ) . (8)Our third prior is not exchangeable. We call it a counter-inductive prior [8] becauseit leads one to predict outcomes that are the opposite of what an argument by inductionwould suggest. In particular, we will see that updating this prior after a string of m identical measurement outcomes, the probability for obtaining the opposite outcome inthe next measurement approaches 1 as m increases. The counter-inductive prior is givenby r ( n ) c = N h n − (cid:229) k = − k (cid:16) | ih | ⊗ k ⊗ | ih | ⊗ ( n − k ) + | ih | ⊗ k ⊗ | ih | ⊗ ( n − k ) (cid:17) + ¥ (cid:229) k = n − k (cid:16) | ih | ⊗ n + | ih | ⊗ n (cid:17) i ( n = , , . . . ) , (9)where the normalization constant N is determined by the equation1 = N ¥ (cid:229) k = − k . (10)It is not difficult to check that this sequence of states satisfies the defining condition of aprior, r ( n ) c = tr n + r ( n + ) c for n ≥ {| i , | i} basis is carried out and that each measurement produces the same outcome, 0. Themeasurement data thus consist of a string of zeros. These data are used to updateiteratively each of our three priors above. For each prior, we compute the marginal one-system posteriors in the limit of an infinite number of iterations.or the two exchangeable priors, we obtain easily the limits r ( ) a → | ih | (number of iterations → ¥ ) (11)and r ( ) b → ( | i + | i )( h | + h | ) (number of iterations → ¥ ) . (12)To compute the limit for the counter-inductive prior r ( n ) c , we first compute the posteriorafter m iterations, r ( n ) c , m . We find, for n = , , . . . , r ( n ) c , m = N m (cid:16) | ih | ⊗ n + n − (cid:229) k = − k ( m + k ) | ih | ⊗ k ⊗ | ih | ⊗ ( n − k ) + ¥ (cid:229) k = n − k ( m + k ) | ih | ⊗ n (cid:17) , (13)where N m is determined by the condition1 = N m ¥ (cid:229) k = − k ( m + k ) , (14)implying that lim m → ¥ N m =
1. Hence r ( ) c → | ih | (number of iterations → ¥ ) . (15)Clearly, the three priors lead to radically different conclusions for the same infinitesequence of data.We now move on to our second example, where we compare two priors, again for asequence of qubits. For our first prior, we choose a generic exchangeable prior, r ( n ) d = Z d r p d ( r ) r ⊗ n ( n = , , . . . ) , (16)where d r p d ( r ) is a measure on one-qubit density operators, i.e., density operators on atwo-dimensional Hilbert space, H . We assume that p d ( r ) is nonzero for all r on H .This prior entails that there is no entanglement between the qubits.By contrast, our second prior, though identical with our first on the single systemmarginals, does not rule out entanglement between pairs of qubits. For even numbers ofsystems, it is defined by r ( n ) e = Z d s p e ( s ) s ⊗ n ( n = , , . . . ) , (17)where d s p e ( s ) is a measure on the space of two-qubit density operators, i.e., densityoperators on the four-dimensional Hilbert space H ⊗ H . For odd numbers of systems,the prior is defined by r ( n − ) e = tr n r ( n ) e ( n = , , . . . ) . (18)ssume now that a sequence of informationally complete measurements is performedon the sequence of qubits. As before, the measurement data are used to update both ourpriors iteratively. We assume that the data are such that for the second prior, the marginaltwo-system posterior converges to an entangled two-qubit state, e.g., the maximallyentangled state r ME = ( | i + | i )( h | + h | ) . We thus assume that r ( ) e → r ME (number of iterations → ¥ ) . (19)Since for the maximally entangled state, both marginal states are equal to the totallymixed state r M = ( | ih | + | ih | ) , it follows that given the same data, the marginalone-system posterior for our first prior converges to r M , r ( ) d → r M (number of iterations → ¥ ) , (20)and the marginal two-system posterior converges to r ( ) d → r M ⊗ r M (number of iterations → ¥ ) , (21)which is equal to the maximally mixed state of two qubits and, of course, not entangled.Once more, we see that the same infinite sequence of data leads to radically differentconclusions for the two priors. CONCLUSION
The most general way in quantum mechanics for obtaining a quantum state from datais via the Kraus rule (2). It is clear that the quantum state obtained in this way gener-ally depends on some prior state in addition to the data. In an earlier paper [9] we haveestablished the general principle that a quantum state is never determined by measure-ment data alone. This is true even in state preparation, because the prepared state alwaysdepends on the prior quantum state of the preparation device [9].What we have illustrated here it that this general principle continues to hold in sit-uations where measurements are repeated many times. A quantum state is never deter-mined by measurement data alone, even in the limit of infinitely many measurements.
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