(* Author: Anthony Bordg, University of Cambridge, apdb3@cam.ac.uk Yijun He, University of Cambridge, yh403@cam.ac.uk *) section ‹Measurement› theory Measurement imports Quantum begin text ‹ Given an element v such that @{text "state n v"}, its components @{text "v $ i"} (when v is seen as a vector, v being a matrix column) for @{text "0 ≤ i < n"} have to be understood as the coefficients of the representation of v in the basis given by the unit vectors of dimension $2^n$, unless stated otherwise. Such a vector v is a state for a quantum system of n qubits. In the literature on quantum computing, for $n = 1$, i.e. for a quantum system of 1 qubit, the elements of the so-called computational basis are denoted $|0\rangle$,$|1\rangle$, and these last elements might be understood for instance as $(1,0)$,$(0,1)$, i.e. as the zeroth and the first elements of a given basis ; for $n = 2$, i.e. for a quantum system of 2 qubits, the elements of the computational basis are denoted $|00\rangle$, $|01\rangle$, $|10\rangle$,$|11\rangle$, and they might be understood for instance as $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, $(0,0,0,1)$; and so on for higher values of $n$. The idea behind these standard notations is that the labels on the vectors of the computational basis are the binary expressions of the natural numbers indexing the elements in a given ordered basis interpreting the computational basis in a specific context, another point of view is that the order of the basis corresponds to the lexicographic order for the labels. Those labels also represent the possible outcomes of a measurement of the $n$ qubits of the system, while the squared modules of the corresponding coefficients represent the probabilities for those outcomes. The fact that the vector v has to be normalized expresses precisely the fact that the squared modules of the coefficients represent some probabilities and hence their sum should be $1$. Note that in the case of a system with multiple qubits, i.e. $n \geq 2$, one can model the simultaneous measurement of multiple qubits by sequential measurements of single qubits. Indeed, this last process leads to the same probabilities for the various possible outcomes. Given a system with n-qubits and i the index of one qubit among the $n$ qubits of the system, where $0 \leq i \leq n-1$ (i.e. we start the indexing from $0$), we want to find the indices of the states of the computational basis whose labels have a $1$ at the ith spot (counting from $0$). For instance, if $n=3$ and $i=2$ then $1$,$3$,$5$,$7$ are the indices of the elements of the computational basis with a $1$ at the 2nd spot, namely $|001\rangle$,$|011\rangle$,$|101\rangle$,$|111\rangle$. To achieve that we define the predicate @{term "select_index"} below. › definition select_index ::"nat ⇒ nat ⇒ nat ⇒ bool" where "select_index n i j ≡ (i≤n-1) ∧ (j≤2^n - 1) ∧ (j mod 2^(n-i) ≥ 2^(n-1-i))" lemma select_index_union: "{k| k::nat. select_index n i k} ∪ {k| k::nat. (k<2^n) ∧ ¬ select_index n i k} = {0..<2^n::nat}" proof have "{k |k. select_index n i k} ⊆ {0..<2 ^ n}" proof fix x::nat assume "x ∈ {k |k. select_index n i k}" then show "x ∈ {0..<2^n}" using select_index_def by (metis (no_types, lifting) atLeastLessThan_iff diff_diff_cancel diff_is_0_eq' diff_le_mono2 le_less_linear le_numeral_extra(2) mem_Collect_eq one_le_numeral one_le_power select_index_def zero_order(1)) qed moreover have "{k |k. k<2 ^ n ∧ ¬ select_index n i k} ⊆ {0..<2 ^ n}" by auto ultimately show "{k |k. select_index n i k} ∪ {k |k. k<2 ^ n ∧ ¬ select_index n i k} ⊆ {0..<2 ^ n}" by simp next show "{0..<2 ^ n} ⊆ {k |k. select_index n i k} ∪ {k |k. k<2 ^ n ∧ ¬ select_index n i k}" by auto qed lemma select_index_inter: "{k| k::nat. select_index n i k} ∩ {k| k::nat. (k<2^n) ∧ ¬ select_index n i k} = {}" by auto lemma outcomes_sum [simp]: fixes f :: "nat ⇒ real" shows "(∑j∈{k| k::nat. select_index n i k}. (f j)) + (∑j∈{k| k::nat. (k<2^n) ∧ ¬ select_index n i k}. (f j)) = (∑j∈{0..<2^n::nat}. (f j))" proof - have "{k| k::nat. select_index n i k} ⊆ {0..<2^n::nat}" using select_index_union by blast then have "finite {k| k::nat. select_index n i k}" using rev_finite_subset by blast moreover have "{k| k::nat. (k<2^n) ∧ ¬ select_index n i k} ⊆ {0..<2^n::nat}" using select_index_union by blast then have "finite {k| k::nat. (k<2^n) ∧ ¬ select_index n i k}" using rev_finite_subset by blast ultimately have "(∑j∈{k| k::nat. select_index n i k}∪{k| k::nat. (k<2^n) ∧ ¬ select_index n i k}. (f j)) = (∑j∈{k| k::nat. select_index n i k}. (f j)) + (∑j∈{k| k::nat. (k<2^n) ∧ ¬ select_index n i k}. (f j)) - (∑j∈{k| k::nat. select_index n i k}∩{k| k::nat. (k<2^n) ∧ ¬ select_index n i k}. (f j))" using sum_Un by blast then have "(∑j∈{0..<2^n::nat}. (f j)) = (∑j∈{k| k::nat. select_index n i k}. (f j)) + (∑j∈{k| k::nat. (k<2^n) ∧ ¬ select_index n i k}. (f j)) - (∑j∈{}. (f j))" using select_index_union select_index_inter by simp thus ?thesis by simp qed text ‹ Given a state v of a n-qbit system, we compute the probability that a measure of qubit i has the outcome 1. › definition prob1 ::"nat ⇒ complex mat ⇒ nat ⇒ real" where "prob1 n v i ≡ ∑j∈{k| k::nat. select_index n i k}. (cmod(v $$ (j,0)))⇧^{2}" definition prob0 ::"nat ⇒ complex mat ⇒ nat ⇒ real" where "prob0 n v i ≡ ∑j∈{k| k::nat. (k<2^n) ∧ ¬ select_index n i k}. (cmod(v $$ (j,0)))⇧^{2}" lemma shows prob1_geq_zero:"prob1 n v i ≥ 0" and prob0_geq_zero:"prob0 n v i ≥ 0" proof - have "(∑j∈{k| k::nat. select_index n i k}. (cmod(v $$ (j,0)))⇧^{2}) ≥ (∑j∈{k| k::nat. select_index n i k}. (0::real))" by (simp add: sum_nonneg) then have "(∑j∈{k| k::nat. select_index n i k}. (cmod(v $$ (j,0)))⇧^{2}) ≥ 0" by simp thus "prob1 n v i ≥ 0" using prob1_def by simp next have "(∑j∈{k| k::nat. (k < 2 ^ n) ∧ ¬ select_index n i k}. (cmod(v $$ (j,0)))⇧^{2}) ≥ (∑j∈{k| k::nat. (k < 2 ^ n) ∧ ¬ select_index n i k}. (0::real))" by (simp add: sum_nonneg) then have "(∑j∈{k| k::nat. (k < 2 ^ n) ∧ ¬ select_index n i k}. (cmod(v $$ (j,0)))⇧^{2}) ≥ 0" by simp thus "prob0 n v i ≥ 0" using prob0_def by simp qed lemma prob_sum_is_one [simp]: assumes "state n v" shows "prob1 n v i + prob0 n v i = 1" proof- have "prob1 n v i + prob0 n v i = (∑j∈{0..<2^n::nat}. (cmod(v $$ (j,0)))⇧^{2})" using prob1_def prob0_def outcomes_sum by simp also have "… = ∥col v 0∥⇧^{2}" using cpx_vec_length_def assms state_def atLeast0LessThan by fastforce finally show ?thesis using assms state_def by simp qed lemma assumes "state n v" shows prob1_leq_one:"prob1 n v i ≤ 1" and prob0_leq_one:"prob0 n v i ≤ 1" apply(metis assms le_add_same_cancel1 prob0_geq_zero prob_sum_is_one) apply(metis assms le_add_same_cancel2 prob1_geq_zero prob_sum_is_one) done lemma prob0_is_prob: assumes "state n v" shows "prob0 n v i ≥ 0 ∧ prob0 n v i ≤ 1" by (simp add: assms prob0_geq_zero prob0_leq_one) lemma prob1_is_prob: assumes "state n v" shows "prob1 n v i ≥ 0 ∧ prob1 n v i ≤ 1" by (simp add: assms prob1_geq_zero prob1_leq_one) text ‹Below we give the new state of a n-qubits system after a measurement of the ith qubit gave 0.› definition post_meas0 ::"nat ⇒ complex mat ⇒ nat ⇒ complex mat" where "post_meas0 n v i ≡ of_real(1/sqrt(prob0 n v i)) ⋅⇩_{m}|vec (2^n) (λj. if ¬ select_index n i j then v $$ (j,0) else 0)⟩" text ‹ Note that a division by 0 never occurs. Indeed, if @{text "sqrt(prob0 n v i)"} would be 0 then @{text "prob0 n v i"} would be 0 and it would mean that the measurement of the ith qubit gave 1. › lemma post_meas0_is_state [simp]: assumes "state n v" and "prob0 n v i ≠ 0" shows "state n (post_meas0 n v i)" proof - have "(∑j∈{0..<2^n::nat}. (cmod (if ¬ select_index n i j then v $$ (j,0) else 0))⇧^{2}) = (∑j∈{k| k::nat. select_index n i k}. (cmod (if ¬ select_index n i j then v $$ (j,0) else 0))⇧^{2}) + (∑j∈{k| k::nat. (k<2^n) ∧ ¬ select_index n i k}. (cmod (if ¬ select_index n i j then v $$ (j,0) else 0))⇧^{2})" using outcomes_sum[of "λj. (cmod (if ¬ select_index n i j then v $$ (j,0) else 0))⇧^{2}" n i] by simp moreover have "(∑j∈{k| k::nat. (k<2^n) ∧ ¬ select_index n i k}. (cmod (if ¬ select_index n i j then v $$ (j,0) else 0))⇧^{2}) = prob0 n v i" by(simp add: prob0_def) ultimately have "∥vec (2 ^ n) (λj. if ¬ select_index n i j then v $$ (j,0) else 0)∥ = sqrt(prob0 n v i)" using lessThan_atLeast0 by (simp add: cpx_vec_length_def) moreover have "∥col (complex_of_real (1/sqrt (prob0 n v i)) ⋅⇩_{m}|vec (2^n) (λj. if ¬ select_index n i j then v $$ (j,0) else 0)⟩) 0∥ = (1/sqrt (prob0 n v i)) * ∥vec (2^n) (λj. if ¬ select_index n i j then v $$ (j,0) else 0)∥" using prob0_geq_zero smult_vec_length_bis by(metis (no_types, lifting) real_sqrt_ge_0_iff zero_le_divide_1_iff) ultimately show ?thesis using state_def post_meas0_def by (simp add: ket_vec_def post_meas0_def assms(2)) qed text ‹Below we give the new state of a n-qubits system after a measurement of the ith qubit gave 1.› definition post_meas1 ::"nat ⇒ complex mat ⇒ nat ⇒ complex mat" where "post_meas1 n v i ≡ of_real(1/sqrt(prob1 n v i)) ⋅⇩_{m}|vec (2^n) (λj. if select_index n i j then v $$ (j,0) else 0)⟩" text ‹ Note that a division by 0 never occurs. Indeed, if @{text "sqrt(prob1 n v i)"} would be 0 then @{text "prob1 n v i"} would be 0 and it would mean that the measurement of the ith qubit gave 0. › lemma post_meas_1_is_state [simp]: assumes "state n v" and "prob1 n v i ≠ 0" shows "state n (post_meas1 n v i)" proof - have "(∑j∈{0..<2^n::nat}. (cmod (if select_index n i j then v $$ (j,0) else 0))⇧^{2}) = (∑j∈{k| k::nat. select_index n i k}. (cmod (if select_index n i j then v $$ (j,0) else 0))⇧^{2}) + (∑j∈{k| k::nat. (k<2^n) ∧ ¬ select_index n i k}. (cmod (if select_index n i j then v $$ (j,0) else 0))⇧^{2})" using outcomes_sum[of "λj. (cmod (if select_index n i j then v $$ (j,0) else 0))⇧^{2}" n i] by simp then have "∥vec (2^n) (λj. if select_index n i j then v $$ (j,0) else 0)∥ = sqrt(prob1 n v i)" using lessThan_atLeast0 by (simp add: cpx_vec_length_def prob1_def) moreover have "∥col(complex_of_real (1/sqrt (prob1 n v i)) ⋅⇩_{m}|vec (2^n) (λj. if select_index n i j then v $$ (j,0) else 0)⟩) 0∥ = (1/sqrt(prob1 n v i)) * ∥vec (2^n) (λj. if select_index n i j then v $$ (j,0) else 0)∥" using prob1_geq_zero smult_vec_length_bis by (metis (no_types, lifting) real_sqrt_ge_0_iff zero_le_divide_1_iff) ultimately have "∥col(complex_of_real (1/sqrt (prob1 n v i)) ⋅⇩_{m}|vec (2^n) (λj. if select_index n i j then v $$ (j,0) else 0)⟩) 0∥ = (1/sqrt(prob1 n v i)) * sqrt(prob1 n v i)" by simp thus ?thesis using state_def post_meas1_def by (simp add: ket_vec_def post_meas1_def assms(2)) qed text ‹ The measurement operator below takes a number of qubits n, a state v of a n-qubits system, a number i corresponding to the index (starting from 0) of one qubit among the n-qubits, and it computes a list whose first (resp. second) element is the pair made of the probability that the outcome of the measurement of the ith qubit is 0 (resp. 1) and the corresponding post-measurement state of the system. Of course, note that i should be strictly less than n and v should be a state of dimension n, i.e. state n v should hold". › definition meas ::"nat ⇒ complex mat ⇒ nat ⇒ _list" where "meas n v i ≡ [(prob0 n v i, post_meas0 n v i), (prob1 n v i, post_meas1 n v i)]" text ‹ We want to determine the probability that the first n qubits of an n+1 qubit system are 0. For this we need to find the indices of the states of the computational basis whose labels do not have a 1 at spot $i=0,...,n$. › definition prob0_fst_qubits:: "nat ⇒ complex Matrix.mat ⇒ real" where "prob0_fst_qubits n v ≡ ∑j∈{k| k::nat. (k<2^(n+1)) ∧ (∀i∈{0..<n}. ¬ select_index (n+1) i k)}. (cmod(v $$ (j,0)))⇧^{2}" lemma select_index_div_2: fixes n i j::"nat" assumes "i < 2^(n+1)" and "j<n" shows "select_index n j (i div 2) = select_index (n+1) j i" proof- have "2^(n-Suc j) ≤ i div 2 mod 2^(n-j) ⟹ 2^(n-j) ≤ i mod 2^(n+1-j)" proof- define a::nat where a0:"a = i div 2 mod 2^(n-j)" assume "2^(n-Suc j) ≤ a" then have "2*a + i mod 2 ≥ 2^(n-(Suc j)+1)" by simp then have f0:"2*a + i mod 2 ≥ 2^(n-j)" by (metis Suc_diff_Suc Suc_eq_plus1 assms(2)) have "a < 2^(n-j)" using a0 by simp then have "2*a + i mod 2 < 2*2^(n-j)" by linarith then have "2*a + i mod 2 < 2^(n-j+1)" by simp then have f1:"2*a + i mod 2 < 2^(n+1-j)" by (metis Nat.add_diff_assoc2 Suc_leD Suc_leI assms(2)) have "i = 2*(a + 2^(n-j)*(i div 2 div 2^(n-j))) + i mod 2" using a0 by simp then have "i = 2*a + i mod 2 + 2^(n-j+1)*(i div 2 div 2^(n-j))" by simp then have "i = 2*a + i mod 2 + 2^(n+1-j)*(i div 2 div 2^(n-j))" by (metis Nat.add_diff_assoc2 Suc_leD Suc_leI assms(2)) then have "i mod 2^(n+1-j) = 2*a + i mod 2" using f1 by (metis mod_if mod_mult_self2) then show "2^(n-j) ≤ i mod 2^(n+1-j)" using f0 by simp qed moreover have "2^(n-j) ≤ i mod 2^(n+1-j) ⟹ 2^(n-Suc j) ≤ i div 2 mod 2^(n-j)" proof- define a::nat where a0:"a = i div 2 mod 2^(n-j)" assume a1:"2^(n-j) ≤ i mod 2^(n+1-j)" have f0:"2^(n-j) = 2^(n-Suc j+1)" by (metis Suc_diff_Suc Suc_eq_plus1 assms(2)) have "a < 2^(n-j)" using a0 by simp then have "2*a + i mod 2 < 2*2^(n-j)" by linarith then have "2*a + i mod 2 < 2^(n-j+1)" by simp then have f1:"2*a + i mod 2 < 2^(n+1-j)" by (metis Nat.add_diff_assoc2 Suc_leD Suc_leI assms(2)) have "i = 2*(a + 2^(n-j)*(i div 2 div 2^(n-j))) + i mod 2" using a0 by simp then have "i = 2*a + i mod 2 + 2^(n-j+1)*(i div 2 div 2^(n-j))" by simp then have "i = 2*a + i mod 2 + 2^(n+1-j)*(i div 2 div 2^(n-j))" by (metis Nat.add_diff_assoc2 Suc_leD Suc_leI assms(2)) then have "i mod 2^(n+1-j) = 2*a + i mod 2" using f1 by (metis mod_if mod_mult_self2) then have "2*a + i mod 2 ≥ 2^(n-j)" using a1 by simp then have "(2*a + i mod 2) div 2 ≥ (2^(n-j)) div 2" using div_le_mono by blast then show "2^(n-Suc j) ≤ a" by (simp add: f0) qed ultimately show ?thesis using select_index_def assms by auto qed lemma select_index_suc_even: fixes n k i:: nat assumes "k < 2^n" and "select_index n i k" shows "select_index (Suc n) i (2*k)" proof- have "select_index n i k = select_index n i (2*k div 2)" by simp moreover have "… = select_index (Suc n) i (2*k)" proof- have "i < n" using assms(2) select_index_def by (metis (no_types, opaque_lifting) Suc_eq_plus1 assms(1) calculation diff_diff_left diff_le_self diff_self_eq_0 div_by_1 le_0_eq le_eq_less_or_eq less_imp_diff_less mod_div_trivial mult.left_neutral mult_eq_0_iff mult_le_mono1 not_less plus_1_eq_Suc power_0 semiring_normalization_rules(7)) thus ?thesis using select_index_div_2 assms(1) select_index_def by(metis Suc_1 Suc_eq_plus1 Suc_mult_less_cancel1 power_Suc) qed ultimately show "select_index (Suc n) i (2*k)" using assms(2) by simp qed lemma select_index_suc_odd: fixes n k i:: nat assumes "k ≤ 2^n -1" and "select_index n i k" shows "select_index (Suc n) i (2*k+1)" proof- have "((2*k+1) mod 2^(Suc n - i) ≥ 2^(n - i)) = (((2*k+1) div 2) mod 2^(n - i) ≥ 2^(n-1-i))" proof- have "2*k+1 < 2^(n + 1)" using assms(1) by (smt Suc_1 Suc_eq_plus1 Suc_le_lessD Suc_le_mono add_Suc_right distrib_left_numeral le_add_diff_inverse mult_le_mono2 nat_mult_1_right one_le_numeral one_le_power plus_1_eq_Suc power_add power_one_right) moreover have "i < n" using assms(2) select_index_def by (metis (no_types, opaque_lifting) add_cancel_left_left add_diff_inverse_nat diff_le_self div_by_1 le_antisym less_le_trans less_one mod_div_trivial not_le power_0) ultimately show ?thesis using select_index_div_2[of "2*k+1" "n" i] select_index_def by (metis Nat.le_diff_conv2 Suc_eq_plus1 Suc_leI assms(2) diff_Suc_1 less_imp_le less_power_add_imp_div_less one_le_numeral one_le_power power_one_right) qed moreover have "… = (k mod 2^(n - i) ≥ 2^(n-1-i))" by simp ultimately show ?thesis proof- have "i ≤ Suc n -1" using assms(2) select_index_def by auto moreover have "2*k+1 ≤ 2^(Suc n)-1" using assms(1) by (smt Suc_diff_1 Suc_eq_plus1 add_diff_cancel_right' diff_Suc_diff_eq2 diff_diff_left diff_is_0_eq diff_mult_distrib2 le_add2 mult_2 mult_Suc_right plus_1_eq_Suc pos2 power_Suc zero_less_power) ultimately show ?thesis using select_index_def by (metis ‹(2 ^ (n - 1 - i) ≤ (2 * k + 1) div 2 mod 2 ^ (n - i)) = (2 ^ (n - 1 - i) ≤ k mod 2 ^ (n - i))› ‹(2 ^ (n - i) ≤ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) ≤ (2 * k + 1) div 2 mod 2 ^ (n - i))› assms(2) diff_Suc_1) qed qed lemma aux_range: fixes k:: nat assumes "k < 2^(Suc n + 1)" and "k ≥ 2" shows "k = 2 ∨ k = 3 ∨ (∃l. l≥2 ∧ l≤2^(n+1)-1 ∧ (k = 2*l ∨ k = 2*l + 1))" proof(rule disjCI) assume "¬ (k = 3 ∨ (∃l≥2. l ≤ 2^(n + 1) - 1 ∧ (k = 2 * l ∨ k = 2 * l + 1)))" have "k > 3 ⟶ (∃l≥2. l ≤ 2^(n + 1) - 1 ∧ (k = 2 * l ∨ k = 2 * l + 1))" proof assume asm:"k > 3" have "even k ∨ odd k" by simp then obtain l where "k = 2*l ∨ k = 2*l+1" by (meson evenE oddE) moreover have "l ≥ 2" using asm calculation by linarith moreover have "l ≤ 2^(n+1) - 1" using assms(1) by (metis Suc_diff_1 Suc_eq_plus1 calculation(1) dvd_triv_left even_Suc_div_two less_Suc_eq_le less_power_add_imp_div_less nonzero_mult_div_cancel_left pos2 power_one_right zero_less_power zero_neq_numeral) ultimately show "∃l≥2. l ≤ 2^(n + 1) - 1 ∧ (k = 2 * l ∨ k = 2 * l + 1)" by auto qed then have "k ≤ 2" using ‹¬ (k = 3 ∨ (∃l≥2. l ≤ 2 ^ (n + 1) - 1 ∧ (k = 2 * l ∨ k = 2 * l + 1)))› less_Suc_eq_le by auto thus "k = 2" using assms(2) by simp qed lemma select_index_with_1: fixes n:: nat assumes "n ≥ 1" shows "∀k. k < 2^(n+1) ⟶ k ≥ 2 ⟶ (∃i<n. select_index (n+1) i k)" using assms proof(rule nat_induct_at_least) show "∀k< 2^(1+1). 2 ≤ k ⟶ (∃i<1. select_index (1+1) i k)" proof- have "select_index 2 0 2 = True" using select_index_def by simp moreover have "select_index 2 0 3" using select_index_def by simp ultimately show ?thesis by (metis Suc_leI add_Suc_shift le_eq_less_or_eq mult_2 not_less one_add_one one_plus_numeral plus_1_eq_Suc power.simps(2) power_one_right semiring_norm(3) zero_less_one_class.zero_less_one) qed next show "⋀n. 1 ≤ n ⟹ ∀k < 2^(n+1). 2 ≤ k ⟶ (∃i<n. select_index (n+1) i k) ⟹ ∀k < 2^(Suc n + 1). 2 ≤ k ⟶ (∃i<Suc n. select_index (Suc n +1) i k)" proof- fix n:: nat assume asm:"n ≥ 1" and IH:"∀k < 2^(n+1). 2 ≤ k ⟶ (∃i<n. select_index (n+1) i k)" have "select_index (Suc n + 1) n 2" proof- have "select_index (Suc n) n 1" using select_index_def by(smt Suc_1 Suc_diff_Suc Suc_lessI add_diff_cancel_right' diff_Suc_1 diff_commute diff_zero le_eq_less_or_eq less_Suc_eq_le nat.simps(3) nat_power_eq_Suc_0_iff one_mod_two_eq_one plus_1_eq_Suc power_one_right zero_less_power) thus ?thesis using select_index_suc_even by (metis Suc_eq_plus1 less_numeral_extra(4) mult_2 not_less_less_Suc_eq one_add_one one_less_power zero_less_Suc) qed moreover have "select_index (Suc n + 1) n 3" proof- have "select_index (Suc n) n 1" using select_index_def by(smt Suc_1 Suc_diff_Suc Suc_lessI add_diff_cancel_right' diff_Suc_1 diff_commute diff_zero le_eq_less_or_eq less_Suc_eq_le nat.simps(3) nat_power_eq_Suc_0_iff one_mod_two_eq_one plus_1_eq_Suc power_one_right zero_less_power) thus ?thesis using select_index_suc_odd by (metis One_nat_def Suc_eq_plus1 mult_2 numeral_3_eq_3 select_index_def) qed moreover have "∃i<Suc n. select_index (Suc n +1) i (2*k)" if "k ≥ 2" and "k ≤ 2^(n + 1)-1" for k:: nat proof- obtain i where "i<n" and "select_index (n+1) i k" using IH by(metis One_nat_def Suc_diff_Suc ‹2 ≤ k› ‹k ≤ 2 ^ (n + 1) - 1› diff_zero le_imp_less_Suc pos2 zero_less_power) then have "select_index (Suc n +1) i (2*k)" using select_index_suc_even by (metis One_nat_def Suc_diff_Suc add.commute diff_zero le_imp_less_Suc plus_1_eq_Suc pos2 that(2) zero_less_power) thus ?thesis using ‹i < n› less_SucI by blast qed moreover have "∃i<Suc n. select_index (Suc n +1) i (2*k +1)" if "k ≥ 2" and "k ≤ 2^(n + 1)-1" for k:: nat proof- obtain i where "i<n" and "select_index (n+1) i k" using IH by(metis One_nat_def Suc_diff_Suc ‹2 ≤ k› ‹k ≤ 2 ^ (n + 1) - 1› diff_zero le_imp_less_Suc pos2 zero_less_power) then have "select_index (Suc n +1) i (2*k+1)" using select_index_suc_odd that(2) by simp thus ?thesis using ‹i < n› less_SucI by blast qed ultimately show "∀k< 2^(Suc n + 1). 2 ≤ k ⟶ (∃i<Suc n. select_index (Suc n +1) i k)" using aux_range by (metis lessI) qed qed lemma prob0_fst_qubits_index: fixes n:: nat and v:: "complex Matrix.mat" shows "{k| k::nat. (k<2^(n+1)) ∧ (∀i∈{0..<n}. ¬ select_index (n+1) i k)} = {0,1}" proof(induct n) case 0 show "{k |k. k < 2^(0+1) ∧ (∀i∈{0..<0}. ¬ select_index (0+1) i k)} = {0,1}" by auto next case (Suc n) show "⋀n. {k |k. k < 2^(n+1) ∧ (∀i∈{0..<n}. ¬ select_index (n+1) i k)} = {0,1} ⟹ {k |k. k < 2^(Suc n + 1) ∧ (∀i∈{0..<Suc n}. ¬ select_index (Suc n + 1) i k)} = {0, 1}" proof- fix n assume IH: "{k |k. k < 2^(n+1) ∧ (∀i∈{0..<n}. ¬ select_index (n+1) i k)} = {0,1}" then have "{0,1} ⊆ {k |k. k < 2^(Suc n + 1) ∧ (∀i∈{0..<Suc n}. ¬ select_index (Suc n + 1) i k)}" proof- have "k < 2^(n+1) ⟶ k < 2^(Suc n + 1)" for k::nat by simp moreover have "(∀i∈{0..<n}. ¬ select_index (n+1) i 0) ∧ (∀i∈{0..<n}. ¬ select_index (n+1) i 1)" using IH by auto then have "(∀i∈{0..<n}. ¬ select_index (Suc n +1) i 0) ∧ (∀i∈{0..<n}. ¬ select_index (Suc n +1) i 1)" using select_index_suc_odd[of 0 "n+1"] Suc_eq_plus1 by (smt One_nat_def Suc_1 add_Suc_shift add_diff_cancel_right' atLeastLessThan_iff diff_diff_cancel le_eq_less_or_eq less_Suc_eq linorder_not_le mod_less nat_power_eq_Suc_0_iff select_index_def zero_less_power) moreover have "select_index (Suc n + 1) n 0 = False" using select_index_def by simp moreover have "select_index (Suc n + 1) n 1 = False" using select_index_def by simp ultimately show ?thesis by (smt One_nat_def Suc_1 Suc_eq_plus1 Suc_lessI atLeast0_lessThan_Suc empty_iff insertE mem_Collect_eq nat.simps(1) nat_power_eq_Suc_0_iff pos2 subsetI zero_less_power) qed moreover have "{k |k. k < 2^(Suc n + 1) ∧ (∀i∈{0..<Suc n}. ¬ select_index (Suc n + 1) i k)} ⊆ {0,1}" proof- have "∀k<2^(Suc n +1). k ≥ 2 ⟶ (∃i<Suc n. ¬ select_index (Suc n +1) i k = False)" using select_index_with_1[of "Suc n"] by (metis Suc_eq_plus1 add.commute le_add1) thus ?thesis by auto qed ultimately show "{k |k. k<2^(Suc n + 1) ∧ (∀i∈{0..<Suc n}. ¬ select_index (Suc n +1) i k)} = {0,1}" by auto qed qed lemma prob0_fst_qubits_eq: fixes n:: nat shows "prob0_fst_qubits n v = (cmod(v $$ (0,0)))⇧^{2}+ (cmod(v $$ (1,0)))⇧^{2}" proof- have "prob0_fst_qubits n v = (∑j∈{k| k::nat. (k<2^(n+1)) ∧ (∀i∈{0..<n}. ¬ select_index (n+1) i k)}. (cmod(v $$ (j,0)))⇧^{2})" using prob0_fst_qubits_def by simp moreover have "… = (∑j∈{0,1}. (cmod(v $$ (j,0)))⇧^{2})" using prob0_fst_qubits_index by simp finally show ?thesis by simp qed (* Below in iter_post_meas0, the first argument n corresponds to the number of qubits of the system , and the second argument m corresponds to the number of qubits that have been measured. *) primrec iter_post_meas0:: "nat ⇒ nat ⇒ complex Matrix.mat ⇒ complex Matrix.mat" where "iter_post_meas0 n 0 v = v" | "iter_post_meas0 n (Suc m) v = post_meas0 n (iter_post_meas0 n m v) m" (* iter_prob0 outputs the probability that successive measurements of the first m qubits (out of n qubits in the system) give m zeros. *) definition iter_prob0:: "nat ⇒ nat ⇒ complex Matrix.mat ⇒ real" where "iter_prob0 n m v = (∏i<m. prob0 n (iter_post_meas0 n i v) i)" (* To do: lemma iter_prob0_eq: fixes n:: nat and v:: "complex Matrix.mat" assumes "n ≥ 1" shows "iter_prob0 (Suc n) n v = prob0_fst_qubits n v" *) subsection ‹Measurements with Bell States› text ‹ A Bell state is a remarkable state. Indeed, if one makes one measure, either of the first or the second qubit, then one gets either $0$ with probability $1/2$ or $1$ with probability $1/2$. Moreover, in the case of two successive measurements of the first and second qubit, the outcomes are correlated. Indeed, in the case of @{text "|β⇩_{0}⇩_{0}⟩"} or @{text "|β⇩_{1}⇩_{0}⟩"} (resp. @{text "|β⇩_{0}⇩_{1}⟩"} or @{text "|β⇩_{1}⇩_{1}⟩"}) if one measures the second qubit after a measurement of the first qubit (or the other way around) then one gets the same outcomes (resp. opposite outcomes), i.e. for instance the probability of measuring $0$ for the second qubit after a measure with outcome $0$ for the first qubit is $1$ (resp. $0$). › lemma prob0_bell_fst [simp]: assumes "v = |β⇩_{0}⇩_{0}⟩ ∨ v = |β⇩_{0}⇩_{1}⟩ ∨ v = |β⇩_{1}⇩_{0}⟩ ∨ v = |β⇩_{1}⇩_{1}⟩" shows "prob0 2 v 0 = 1/2" proof - have set_0 [simp]:"{k| k::nat. (k<4) ∧ ¬ select_index 2 0 k} = {0,1}" using select_index_def by auto have "v = |β⇩_{0}⇩_{0}⟩ ⟹ prob0 2 v 0 = 1/2" proof - fix v assume asm:"v = |β⇩_{0}⇩_{0}⟩" show "prob0 2 v 0 = 1/2" proof - have "prob0 2 v 0 = (∑j∈{k| k::nat. (k<4) ∧ ¬ select_index 2 0 k}. (cmod(bell00 $$ (j,0)))⇧^{2})" by (auto simp: prob0_def asm) also have "… = (∑j∈{0,1}. (cmod(bell00 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(1/sqrt(2)))⇧^{2}+ (cmod(0))⇧^{2}" by (auto simp: bell00_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed moreover have "v = |β⇩_{0}⇩_{1}⟩ ⟹ prob0 2 v 0 = 1/2" proof - fix v assume asm:"v = |β⇩_{0}⇩_{1}⟩" show "prob0 2 v 0 = 1/2" proof - have "prob0 2 v 0 = (∑j∈{k| k::nat. (k<4) ∧ ¬ select_index 2 0 k}. (cmod(bell01 $$ (j,0)))⇧^{2})" by (auto simp: prob0_def asm) also have "… = (∑j∈{0,1}. (cmod(bell01 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(0))⇧^{2}+ (cmod(1/sqrt(2)))⇧^{2}" by (auto simp: bell01_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed moreover have "v = |β⇩_{1}⇩_{0}⟩ ⟹ prob0 2 v 0 = 1/2" proof - fix v assume asm:"v = |β⇩_{1}⇩_{0}⟩" show "prob0 2 v 0 = 1/2" proof - have "prob0 2 v 0 = (∑j∈{k| k::nat. (k<4) ∧ ¬ select_index 2 0 k}. (cmod(bell10 $$ (j,0)))⇧^{2})" by (auto simp: prob0_def asm) also have "… = (∑j∈{0,1}. (cmod(bell10 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(1/sqrt(2)))⇧^{2}+ (cmod(0))⇧^{2}" by (auto simp: bell10_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed moreover have "v = |β⇩_{1}⇩_{1}⟩ ⟹ prob0 2 v 0 = 1/2" proof - fix v assume asm:"v = |β⇩_{1}⇩_{1}⟩" show "prob0 2 v 0 = 1/2" proof - have "prob0 2 v 0 = (∑j∈{k| k::nat. (k<4) ∧ ¬ select_index 2 0 k}. (cmod(bell11 $$ (j,0)))⇧^{2})" by (auto simp: prob0_def asm) also have "… = (∑j∈{0,1}. (cmod(bell11 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(0))⇧^{2}+ (cmod(1/sqrt(2)))⇧^{2}" by (auto simp: bell11_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed ultimately show ?thesis using assms by auto qed lemma prob_1_bell_fst [simp]: assumes "v = |β⇩_{0}⇩_{0}⟩ ∨ v = |β⇩_{0}⇩_{1}⟩ ∨ v = |β⇩_{1}⇩_{0}⟩ ∨ v = |β⇩_{1}⇩_{1}⟩" shows "prob1 2 v 0 = 1/2" proof - have set_0 [simp]:"{k| k::nat. select_index 2 0 k} = {2,3}" using select_index_def by auto have "v = |β⇩_{0}⇩_{0}⟩ ⟹ prob1 2 v 0 = 1/2" proof - fix v assume asm:"v = |β⇩_{0}⇩_{0}⟩" show "prob1 2 v 0 = 1/2" proof - have "prob1 2 v 0 = (∑j∈{k| k::nat. select_index 2 0 k}. (cmod(bell00 $$ (j,0)))⇧^{2})" by (auto simp: prob1_def asm) also have "… = (∑j∈{2,3}. (cmod(bell00 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(1/sqrt(2)))⇧^{2}+ (cmod(0))⇧^{2}" by (auto simp: bell00_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed moreover have "v = |β⇩_{0}⇩_{1}⟩ ⟹ prob1 2 v 0 = 1/2" proof - fix v assume asm:"v = |β⇩_{0}⇩_{1}⟩" show "prob1 2 v 0 = 1/2" proof - have "prob1 2 v 0 = (∑j∈{k| k::nat. select_index 2 0 k}. (cmod(bell01 $$ (j,0)))⇧^{2})" by (auto simp: prob1_def asm) also have "… = (∑j∈{2,3}. (cmod(bell01 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(0))⇧^{2}+ (cmod(1/sqrt(2)))⇧^{2}" by (auto simp: bell01_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed moreover have "v = |β⇩_{1}⇩_{0}⟩ ⟹ prob1 2 v 0 = 1/2" proof - fix v assume asm:"v = |β⇩_{1}⇩_{0}⟩" show "prob1 2 v 0 = 1/2" proof - have "prob1 2 v 0 = (∑j∈{k| k::nat. select_index 2 0 k}. (cmod(bell10 $$ (j,0)))⇧^{2})" by (auto simp: prob1_def asm) also have "… = (∑j∈{2,3}. (cmod(bell10 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(1/sqrt(2)))⇧^{2}+ (cmod(0))⇧^{2}" by (auto simp: bell10_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed moreover have "v = |β⇩_{1}⇩_{1}⟩ ⟹ prob1 2 v 0 = 1/2" proof - fix v assume asm:"v = |β⇩_{1}⇩_{1}⟩" show "prob1 2 v 0 = 1/2" proof - have "prob1 2 v 0 = (∑j∈{k| k::nat. select_index 2 0 k}. (cmod(bell11 $$ (j,0)))⇧^{2})" by (auto simp: prob1_def asm) also have "… = (∑j∈{2,3}. (cmod(bell11 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(0))⇧^{2}+ (cmod(1/sqrt(2)))⇧^{2}" by (auto simp: bell11_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed ultimately show ?thesis using assms by auto qed lemma prob0_bell_snd [simp]: assumes "v = |β⇩_{0}⇩_{0}⟩ ∨ v = |β⇩_{0}⇩_{1}⟩ ∨ v = |β⇩_{1}⇩_{0}⟩ ∨ v = |β⇩_{1}⇩_{1}⟩" shows "prob0 2 v 1 = 1/2" proof - have set_0 [simp]:"{k| k::nat. (k<4) ∧ ¬ select_index 2 1 k} = {0,2}" by (auto simp: select_index_def) (metis Suc_le_mono add_Suc add_Suc_right le_numeral_extra(3) less_antisym mod_Suc_eq mod_less neq0_conv not_mod2_eq_Suc_0_eq_0 numeral_2_eq_2 numeral_Bit0 one_add_one one_mod_two_eq_one one_neq_zero) have "v = |β⇩_{0}⇩_{0}⟩ ⟹ prob0 2 v 1 = 1/2" proof - fix v assume asm:"v = |β⇩_{0}⇩_{0}⟩" show "prob0 2 v 1 = 1/2" proof - have "prob0 2 v 1 = (∑j∈{k| k::nat. (k<4) ∧ ¬ select_index 2 1 k}. (cmod(bell00 $$ (j,0)))⇧^{2})" by (auto simp: prob0_def asm) also have "… = (∑j∈{0,2}. (cmod(bell00 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(1/sqrt(2)))⇧^{2}+ (cmod(0))⇧^{2}" by (auto simp: bell00_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed moreover have "v = |β⇩_{0}⇩_{1}⟩ ⟹ prob0 2 v 1 = 1/2" proof - fix v assume asm:"v = |β⇩_{0}⇩_{1}⟩" show "prob0 2 v 1 = 1/2" proof - have "prob0 2 v 1 = (∑j∈{k| k::nat. (k<4) ∧ ¬ select_index 2 1 k}. (cmod(bell01 $$ (j,0)))⇧^{2})" by (auto simp: prob0_def asm) also have "… = (∑j∈{0,2}. (cmod(bell01 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(0))⇧^{2}+ (cmod(1/sqrt(2)))⇧^{2}" by (auto simp: bell01_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed moreover have "v = |β⇩_{1}⇩_{0}⟩ ⟹ prob0 2 v 1 = 1/2" proof - fix v assume asm:"v = |β⇩_{1}⇩_{0}⟩" show "prob0 2 v 1 = 1/2" proof - have "prob0 2 v 1 = (∑j∈{k| k::nat. (k<4) ∧ ¬ select_index 2 1 k}. (cmod(bell10 $$ (j,0)))⇧^{2})" by (auto simp: prob0_def asm) also have "… = (∑j∈{0,2}. (cmod(bell10 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(1/sqrt(2)))⇧^{2}+ (cmod(0))⇧^{2}" by (auto simp: bell10_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed moreover have "v = |β⇩_{1}⇩_{1}⟩ ⟹ prob0 2 v 1 = 1/2" proof - fix v assume asm:"v = |β⇩_{1}⇩_{1}⟩" show "prob0 2 v 1 = 1/2" proof - have "prob0 2 v 1 = (∑j∈{k| k::nat. (k<4) ∧ ¬ select_index 2 1 k}. (cmod(bell11 $$ (j,0)))⇧^{2})" by (auto simp: prob0_def asm) also have "… = (∑j∈{0,2}. (cmod(bell11 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(0))⇧^{2}+ (cmod(1/sqrt(2)))⇧^{2}" by (auto simp: bell11_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed ultimately show ?thesis using assms by auto qed lemma prob_1_bell_snd [simp]: assumes "v = |β⇩_{0}⇩_{0}⟩ ∨ v = |β⇩_{0}⇩_{1}⟩ ∨ v = |β⇩_{1}⇩_{0}⟩ ∨ v = |β⇩_{1}⇩_{1}⟩" shows "prob1 2 v 1 = 1/2" proof - have set_0:"{k| k::nat. select_index 2 1 k} = {1,3}" by (auto simp: select_index_def) (metis Suc_le_lessD le_SucE le_less mod2_gr_0 mod_less mod_self numeral_2_eq_2 numeral_3_eq_3) have "v = |β⇩_{0}⇩_{0}⟩ ⟹ prob1 2 v 1 = 1/2" proof - fix v assume asm:"v = |β⇩_{0}⇩_{0}⟩" show "prob1 2 v 1 = 1/2" proof - have "prob1 2 v 1 = (∑j∈{k| k::nat. select_index 2 1 k}. (cmod(bell00 $$ (j,0)))⇧^{2})" by (auto simp: prob1_def asm) also have "… = (∑j∈{1,3}. (cmod(bell00 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(1/sqrt(2)))⇧^{2}+ (cmod(0))⇧^{2}" by (auto simp: bell00_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed moreover have "v = |β⇩_{0}⇩_{1}⟩ ⟹ prob1 2 v 1 = 1/2" proof - fix v assume asm:"v = |β⇩_{0}⇩_{1}⟩" show "prob1 2 v 1 = 1/2" proof - have "prob1 2 v 1 = (∑j∈{k| k::nat. select_index 2 1 k}. (cmod(bell01 $$ (j,0)))⇧^{2})" by (auto simp: prob1_def asm) also have "… = (∑j∈{1,3}. (cmod(bell01 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(0))⇧^{2}+ (cmod(1/sqrt(2)))⇧^{2}" by (auto simp: bell01_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed moreover have "v = |β⇩_{1}⇩_{0}⟩ ⟹ prob1 2 v 1 = 1/2" proof - fix v assume asm:"v = |β⇩_{1}⇩_{0}⟩" show "prob1 2 v 1 = 1/2" proof - have "prob1 2 v 1 = (∑j∈{k| k::nat. select_index 2 1 k}. (cmod(bell10 $$ (j,0)))⇧^{2})" by (auto simp: prob1_def asm) also have "… = (∑j∈{1,3}. (cmod(bell10 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(1/sqrt(2)))⇧^{2}+ (cmod(0))⇧^{2}" by (auto simp: bell10_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed moreover have "v = |β⇩_{1}⇩_{1}⟩ ⟹ prob1 2 v 1 = 1/2" proof - fix v assume asm:"v = |β⇩_{1}⇩_{1}⟩" show "prob1 2 v 1 = 1/2" proof - have "prob1 2 v 1 = (∑j∈{k| k::nat. select_index 2 1 k}. (cmod(bell11 $$ (j,0)))⇧^{2})" by (auto simp: prob1_def asm) also have "… = (∑j∈{1,3}. (cmod(bell11 $$ (j,0)))⇧^{2})" using set_0 by simp also have "… = (cmod(0))⇧^{2}+ (cmod(1/sqrt(2)))⇧^{2}" by (auto simp: bell11_def ket_vec_def) finally show ?thesis by(simp add: cmod_def power_divide) qed qed ultimately show ?thesis using assms by auto qed lemma post_meas0_bell00_fst [simp]: "post_meas0 2 |β⇩_{0}⇩_{0}⟩ 0 = |unit_vec 4 0⟩" proof fix i j::nat assume "i < dim_row |unit_vec 4 0⟩" and "j < dim_col |unit_vec 4 0⟩" then have "i ∈ {0,1,2,3}" and "j = 0" by(auto simp add: ket_vec_def) moreover have "post_meas0 2 |β⇩_{0}⇩_{0}⟩ 0 $$ (0,0) = |unit_vec 4 0⟩ $$ (0,0)" by(simp add: post_meas0_def unit_vec_def select_index_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{0}⇩_{0}⟩ 0 $$ (1,0) = |unit_vec 4 0⟩ $$ (1,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{0}⇩_{0}⟩ 0 $$ (2,0) = |unit_vec 4 0⟩ $$ (2,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{0}⇩_{0}⟩ 0 $$ (3,0) = |unit_vec 4 0⟩ $$ (3,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide) ultimately show "post_meas0 2 |β⇩_{0}⇩_{0}⟩ 0 $$ (i,j) = |unit_vec 4 0⟩ $$ (i,j)" by auto next show "dim_row (post_meas0 2 |β⇩_{0}⇩_{0}⟩ 0) = dim_row |unit_vec 4 0⟩" by(auto simp add: post_meas0_def ket_vec_def) show "dim_col (post_meas0 2 |β⇩_{0}⇩_{0}⟩ 0) = dim_col |unit_vec 4 0⟩" by(auto simp add: post_meas0_def ket_vec_def) qed lemma post_meas0_bell00_snd [simp]: "post_meas0 2 |β⇩_{0}⇩_{0}⟩ 1 = |unit_vec 4 0⟩" proof fix i j::nat assume "i < dim_row |unit_vec 4 0⟩" and "j < dim_col |unit_vec 4 0⟩" then have "i ∈ {0,1,2,3}" and "j = 0" by(auto simp add: ket_vec_def) moreover have "post_meas0 2 |β⇩_{0}⇩_{0}⟩ 1 $$ (0,0) = |unit_vec 4 0⟩ $$ (0,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide del:One_nat_def) moreover have "post_meas0 2 |β⇩_{0}⇩_{0}⟩ 1 $$ (1,0) = |unit_vec 4 0⟩ $$ (1,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{0}⇩_{0}⟩ 1 $$ (2,0) = |unit_vec 4 0⟩ $$ (2,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{0}⇩_{0}⟩ 1 $$ (3,0) = |unit_vec 4 0⟩ $$ (3,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide) ultimately show "post_meas0 2 |β⇩_{0}⇩_{0}⟩ 1 $$ (i,j) = |unit_vec 4 0⟩ $$ (i,j)" by auto next show "dim_row (post_meas0 2 |β⇩_{0}⇩_{0}⟩ 1) = dim_row |unit_vec 4 0⟩" by(auto simp add: post_meas0_def ket_vec_def) show "dim_col (post_meas0 2 |β⇩_{0}⇩_{0}⟩ 1) = dim_col |unit_vec 4 0⟩" by(auto simp add: post_meas0_def ket_vec_def) qed lemma post_meas0_bell01_fst [simp]: "post_meas0 2 |β⇩_{0}⇩_{1}⟩ 0 = |unit_vec 4 1⟩" proof fix i j::nat assume "i < dim_row |unit_vec 4 1⟩" and "j < dim_col |unit_vec 4 1⟩" then have "i ∈ {0,1,2,3}" and "j = 0" by(auto simp add: ket_vec_def) moreover have "post_meas0 2 |β⇩_{0}⇩_{1}⟩ 0 $$ (0,0) = |unit_vec 4 1⟩ $$ (0,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell01_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{0}⇩_{1}⟩ 0 $$ (1,0) = |unit_vec 4 1⟩ $$ (1,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell01_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{0}⇩_{1}⟩ 0 $$ (2,0) = |unit_vec 4 1⟩ $$ (2,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell01_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{0}⇩_{1}⟩ 0 $$ (3,0) = |unit_vec 4 1⟩ $$ (3,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell01_def ket_vec_def real_sqrt_divide) ultimately show "post_meas0 2 |β⇩_{0}⇩_{1}⟩ 0 $$ (i,j) = |unit_vec 4 1⟩ $$ (i,j)" by auto next show "dim_row (post_meas0 2 |β⇩_{0}⇩_{1}⟩ 0) = dim_row |unit_vec 4 1⟩" by(auto simp add: post_meas0_def ket_vec_def) show "dim_col (post_meas0 2 |β⇩_{0}⇩_{1}⟩ 0) = dim_col |unit_vec 4 1⟩" by(auto simp add: post_meas0_def ket_vec_def) qed lemma post_meas0_bell01_snd [simp]: "post_meas0 2 |β⇩_{0}⇩_{1}⟩ 1 = |unit_vec 4 2⟩" proof fix i j::nat assume "i < dim_row |unit_vec 4 2⟩" and "j < dim_col |unit_vec 4 2⟩" then have "i ∈ {0,1,2,3}" and "j = 0" by(auto simp add: ket_vec_def) moreover have "post_meas0 2 |β⇩_{0}⇩_{1}⟩ 1 $$ (0,0) = |unit_vec 4 2⟩ $$ (0,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell01_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{0}⇩_{1}⟩ 1 $$ (1,0) = |unit_vec 4 2⟩ $$ (1,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell01_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{0}⇩_{1}⟩ 1 $$ (2,0) = |unit_vec 4 2⟩ $$ (2,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell01_def ket_vec_def real_sqrt_divide del:One_nat_def) moreover have "post_meas0 2 |β⇩_{0}⇩_{1}⟩ 1 $$ (3,0) = |unit_vec 4 2⟩ $$ (3,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell01_def ket_vec_def real_sqrt_divide) ultimately show "post_meas0 2 |β⇩_{0}⇩_{1}⟩ 1 $$ (i,j) = |unit_vec 4 2⟩ $$ (i,j)" by auto next show "dim_row (post_meas0 2 |β⇩_{0}⇩_{1}⟩ 1) = dim_row |unit_vec 4 2⟩" by(auto simp add: post_meas0_def ket_vec_def) show "dim_col (post_meas0 2 |β⇩_{0}⇩_{1}⟩ 1) = dim_col |unit_vec 4 2⟩" by(auto simp add: post_meas0_def ket_vec_def) qed lemma post_meas0_bell10_fst [simp]: "post_meas0 2 |β⇩_{1}⇩_{0}⟩ 0 = |unit_vec 4 0⟩" proof fix i j::nat assume "i < dim_row |unit_vec 4 0⟩" and "j < dim_col |unit_vec 4 0⟩" then have "i ∈ {0,1,2,3}" and "j = 0" by(auto simp add: ket_vec_def) moreover have "post_meas0 2 |β⇩_{1}⇩_{0}⟩ 0 $$ (0,0) = |unit_vec 4 0⟩ $$ (0,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell10_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{1}⇩_{0}⟩ 0 $$ (1,0) = |unit_vec 4 0⟩ $$ (1,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell10_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{1}⇩_{0}⟩ 0 $$ (2,0) = |unit_vec 4 0⟩ $$ (2,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell10_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{1}⇩_{0}⟩ 0 $$ (3,0) = |unit_vec 4 0⟩ $$ (3,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell10_def ket_vec_def real_sqrt_divide) ultimately show "post_meas0 2 |β⇩_{1}⇩_{0}⟩ 0 $$ (i,j) = |unit_vec 4 0⟩ $$ (i,j)" by auto next show "dim_row (post_meas0 2 |β⇩_{1}⇩_{0}⟩ 0) = dim_row |unit_vec 4 0⟩" by(auto simp add: post_meas0_def ket_vec_def) show "dim_col (post_meas0 2 |β⇩_{1}⇩_{0}⟩ 0) = dim_col |unit_vec 4 0⟩" by(auto simp add: post_meas0_def ket_vec_def) qed lemma post_meas0_bell10_snd [simp]: "post_meas0 2 |β⇩_{1}⇩_{0}⟩ 1 = |unit_vec 4 0⟩" proof fix i j::nat assume "i < dim_row |unit_vec 4 0⟩" and "j < dim_col |unit_vec 4 0⟩" then have "i ∈ {0,1,2,3}" and "j = 0" by(auto simp add: ket_vec_def) moreover have "post_meas0 2 |β⇩_{1}⇩_{0}⟩ 1 $$ (0,0) = |unit_vec 4 0⟩ $$ (0,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell10_def ket_vec_def real_sqrt_divide del:One_nat_def) moreover have "post_meas0 2 |β⇩_{1}⇩_{0}⟩ 1 $$ (1,0) = |unit_vec 4 0⟩ $$ (1,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell10_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{1}⇩_{0}⟩ 1 $$ (2,0) = |unit_vec 4 0⟩ $$ (2,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell10_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{1}⇩_{0}⟩ 1 $$ (3,0) = |unit_vec 4 0⟩ $$ (3,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell10_def ket_vec_def real_sqrt_divide) ultimately show "post_meas0 2 |β⇩_{1}⇩_{0}⟩ 1 $$ (i,j) = |unit_vec 4 0⟩ $$ (i,j)" by auto next show "dim_row (post_meas0 2 |β⇩_{1}⇩_{0}⟩ 1) = dim_row |unit_vec 4 0⟩" by(auto simp add: post_meas0_def ket_vec_def) show "dim_col (post_meas0 2 |β⇩_{1}⇩_{0}⟩ 1) = dim_col |unit_vec 4 0⟩" by(auto simp add: post_meas0_def ket_vec_def) qed lemma post_meas0_bell11_fst [simp]: "post_meas0 2 |β⇩_{1}⇩_{1}⟩ 0 = |unit_vec 4 1⟩" proof fix i j::nat assume "i < dim_row |unit_vec 4 1⟩" and "j < dim_col |unit_vec 4 1⟩" then have "i ∈ {0,1,2,3}" and "j = 0" by(auto simp add: ket_vec_def) moreover have "post_meas0 2 |β⇩_{1}⇩_{1}⟩ 0 $$ (0,0) = |unit_vec 4 1⟩ $$ (0,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell11_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{1}⇩_{1}⟩ 0 $$ (1,0) = |unit_vec 4 1⟩ $$ (1,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell11_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{1}⇩_{1}⟩ 0 $$ (2,0) = |unit_vec 4 1⟩ $$ (2,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell11_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{1}⇩_{1}⟩ 0 $$ (3,0) = |unit_vec 4 1⟩ $$ (3,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell11_def ket_vec_def real_sqrt_divide) ultimately show "post_meas0 2 |β⇩_{1}⇩_{1}⟩ 0 $$ (i,j) = |unit_vec 4 1⟩ $$ (i,j)" by auto next show "dim_row (post_meas0 2 |β⇩_{1}⇩_{1}⟩ 0) = dim_row |unit_vec 4 1⟩" by(auto simp add: post_meas0_def ket_vec_def) show "dim_col (post_meas0 2 |β⇩_{1}⇩_{1}⟩ 0) = dim_col |unit_vec 4 1⟩" by(auto simp add: post_meas0_def ket_vec_def) qed lemma post_meas0_bell11_snd [simp]: "post_meas0 2 |β⇩_{1}⇩_{1}⟩ 1 = - |unit_vec 4 2⟩" proof fix i j::nat assume "i < dim_row (- |unit_vec 4 2⟩)" and "j < dim_col (- |unit_vec 4 2⟩)" then have "i ∈ {0,1,2,3}" and "j = 0" by(auto simp add: ket_vec_def) moreover have "post_meas0 2 |β⇩_{1}⇩_{1}⟩ 1 $$ (0,0) = (- |unit_vec 4 2⟩) $$ (0,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell11_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{1}⇩_{1}⟩ 1 $$ (1,0) = (- |unit_vec 4 2⟩) $$ (1,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell11_def ket_vec_def real_sqrt_divide) moreover have "post_meas0 2 |β⇩_{1}⇩_{1}⟩ 1 $$ (2,0) = (- |unit_vec 4 2⟩) $$ (2,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell11_def ket_vec_def real_sqrt_divide del:One_nat_def) moreover have "post_meas0 2 |β⇩_{1}⇩_{1}⟩ 1 $$ (3,0) = (- |unit_vec 4 2⟩) $$ (3,0)" by(simp add: post_meas0_def unit_vec_def select_index_def bell11_def ket_vec_def real_sqrt_divide) ultimately show "post_meas0 2 |β⇩_{1}⇩_{1}⟩ 1 $$ (i,j) = (- |unit_vec 4 2⟩) $$ (i,j)" by auto next show "dim_row (post_meas0 2 |β⇩_{1}⇩_{1}⟩ 1) = dim_row (- |unit_vec 4 2⟩)" by(auto simp add: post_meas0_def ket_vec_def) show "dim_col (post_meas0 2 |β⇩_{1}⇩_{1}⟩ 1) = dim_col (- |unit_vec 4 2⟩)" by(auto simp add: post_meas0_def ket_vec_def) qed lemma post_meas1_bell00_fst [simp]: "post_meas1 2 |β⇩_{0}⇩_{0}⟩ 0 = |unit_vec 4 3⟩" proof fix i j::nat assume "i < dim_row |unit_vec 4 3⟩" and "j < dim_col |unit_vec 4 3⟩" then have "i ∈ {0,1,2,3}" and "j = 0" by(auto simp add: ket_vec_def) moreover have "post_meas1 2 |β⇩_{0}⇩_{0}⟩ 0 $$ (0,0) = |unit_vec 4 3⟩ $$ (0,0)" by(simp add: post_meas1_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide) moreover have "post_meas1 2 |β⇩_{0}⇩_{0}⟩ 0 $$ (1,0) = |unit_vec 4 3⟩ $$ (1,0)" by(simp add: post_meas1_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide) moreover have "post_meas1 2 |β⇩_{0}⇩_{0}⟩ 0 $$ (2,0) = |unit_vec 4 3⟩ $$ (2,0)" by(simp add: post_meas1_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide) moreover have "post_meas1 2 |β⇩_{0}⇩_{0}⟩ 0 $$ (3,0) = |unit_vec 4 3⟩ $$ (3,0)" by(simp add: post_meas1_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide) ultimately show "post_meas1 2 |β⇩_{0}⇩_{0}⟩ 0 $$ (i,j) = |unit_vec 4 3⟩ $$ (i,j)" by auto next show "dim_row (post_meas1 2 |β⇩_{0}⇩_{0}⟩ 0) = dim_row |unit_vec 4 3⟩" by(auto simp add: post_meas1_def ket_vec_def) show "dim_col (post_meas1 2 |β⇩_{0}⇩_{0}⟩ 0) = dim_col |unit_vec 4 3⟩" by(auto simp add: post_meas1_def ket_vec_def) qed lemma post_meas1_bell00_snd [simp]: "post_meas1 2 |β⇩_{0}⇩_{0}⟩ 1 = |unit_vec 4 3⟩" proof fix i j::nat assume "i < dim_row |unit_vec 4 3⟩" and "j < dim_col |unit_vec 4 3⟩" then have "i ∈ {0,1,2,3}" and "j = 0" by(auto simp add: ket_vec_def) moreover have "post_meas1 2 |β⇩_{0}⇩_{0}⟩ 1 $$ (0,0) = |unit_vec 4 3⟩ $$ (0,0)" by(simp add: post_meas1_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide) moreover have "post_meas1 2 |β⇩_{0}⇩_{0}⟩ 1 $$ (1,0) = |unit_vec 4 3⟩ $$ (1,0)" by(simp add: post_meas1_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide) moreover have "post_meas1 2 |β⇩_{0}⇩_{0}⟩ 1 $$ (2,0) = |unit_vec 4 3⟩ $$ (2,0)" by(simp add: post_meas1_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide) moreover have "post_meas1 2 |β⇩_{0}⇩_{0}⟩ 1 $$ (3,0) = |unit_vec 4 3⟩ $$ (3,0)" by(simp add: post_meas1_def unit_vec_def select_index_def bell00_def ket_vec_def real_sqrt_divide del: One_nat_def) ultimately show "post_meas1 2 |β⇩_{0}⇩_{0}⟩ 1 $$ (i,j) = |unit_vec 4 3⟩ $$ (i,j)" by auto next show "dim_row (post_meas1 2 |β⇩_{0}⇩_{0}⟩ 1) = dim_row |unit_vec 4 3⟩" by(auto simp add: post_meas1_def ket_vec_def) show "dim_col (post_meas1 2 |β⇩_{0}⇩_{0}⟩ 1) = dim_col |unit_vec 4 3⟩" by(auto simp add: post_meas1_def ket_vec_def) qed lemma post_meas1_bell01_fst [simp]: "post_meas1 2 |β⇩_{0}⇩_{1}⟩ 0 = |unit_vec 4 2⟩" proof fix i j::nat assume "i < dim_row |unit_vec 4 2⟩" and "j < dim_col |unit_vec 4 2⟩" then have "i ∈ {0,1,2,3}" and "j = 0" by(auto simp add: ket_vec_def) moreover have "post_meas1 2 |β⇩_{0}⇩_{1}⟩ 0 $$ (0,0) = |unit_vec 4 2⟩ $$ (0,0)" by(simp add: post_meas1_def unit_vec_def select_index_def bell01_def ket_vec_def real_sqrt_divide) moreover have "post_meas1 2 |β⇩_{0}⇩_{1}⟩ 0 $$ (1,0) = |unit_vec 4 2⟩ $$ (1,0)" by(simp add: post_meas1_def unit_vec_def select_index_def bell01_def ket_vec_def real_sqrt_divide) moreover have "post_meas1 2 |β⇩_{0}⇩_{1}⟩ 0 $$ (2,0) = |unit_vec 4 2⟩ $$ (2,0)" by(simp add: post_meas1_def unit_vec_def select_index_def bell01_def ket_vec_def real_sqrt_divide) moreover have "post_meas1 2 |β⇩_{0}⇩_{1}⟩ 0 $$ (3,0) = |unit_vec 4 2⟩ </