Suppose we have an ensemble {pX (x), ρx} of density operators and a POVMwith elements {Λx} that should identify the states ρx with high probability, i.e., we would likeTr {Λxρx} to be as high as possible. The expected success probability of the POVM is then∑xpX (x) Tr {Λxρx} . (4.99)Suppose that there exists some operator τ such thatτ ≥ pX (x)ρx , (4.100)where the condition τ ≥ pX (x)ρx is the same as τ − pX (x)ρx ≥ 0 (i.e., that the operatorτ − pX (x)ρx is a positive semi-definite operator). Show that Tr {τ } is an upper bound onthe expected success probability of the POVM. After doing so, consider the case of encodingn bits into a d-dimensional subspace. By choosing states uniformly at random (in the caseof the ensemble {2−n, ρi}i∈{0,1}n ), show that the expected success probability is bounded aboveby d 2−n. Thus, it is not possible to store more than n classical bits in n qubits and have aperfect success probability of retrieval.

Show that performing a measurement with measurement operators {ΛjA}on system A is the same as performing a measurement of the ensemble in (4.173). That is,show that Tr{ρAΛjA} = Tr{ρXA(IX ⊗ ΛjA)}, where ρA is defined in (4.175).

Consider the following sentence: a Markov model tags easilyAssume that based on a  HMM, we have the following probabilities: Emission:P1(a|DET) = 0.1,    P1(easily|ADV) = 0.1,   P1(Markov|N) = 0.1,    P1(model|N)   = 0.095,  P1(model|V)  =0.005, P1(tags|N) = 0.080,  P1(tags|V) = 0.020,Transition probabilities: P(Y|X) Y DET   N   V   ADJ   ADV X DET 0 0.55 0 0.02 0.03   N 0.01 0.1 0.08 0.01 0.02   V 0.16 0.11 0.06 0.08 0.08   ADJ 0.01 0.65 0 0.05 0   ADV 0.08 0.02 0.09 0.04 0.04 Initial probabilitiesP3(DET) = 0.20, P3(N) = 0.06, P3(V) = 0.08, P3(ADV) = 0.07, P3(ADJ) = 0.02.What are the possible tag(s) of the sentence?Assume you want to use the Viterbi algorithm to decode the sentence, write down the expression to be evaluated at the initial step.Write down the expression for the second iteration if the first tag is DET, for a second tag of ADJ

When a probabilistic bit is in (10)(10) , we apply the operator (1/21/201)(1/201/21)  three times. What is the probability of being in state  ⌈1⌋⌈1⌋   at the end?

A decision-maker under conditions of risk knows a. With certainty the outcome of any decision or alternative b. The probability of occurrence of the outcomes or consequences for each choice c. The optimistic outcome and the pessimistic outcomes of any decision or alternative d. The maximin, maximax and Laplace outcomes of any decision or alternative e. Nothing about the outcome of a decision or alternative

When a probabilistic bit is in (10)(10) , we apply the operator (1/21/201)(1/201/21)  three times. What is the probability of being in state  ⌈1⌋⌈1⌋   at the end?Group of answer choices3/41/87/81/21/4