# TraceDistanceCoherence

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 Other toolboxes required TraceDistanceCoherence Computes the trace distance of coherence of a quantum state CVX L1NormCoherenceRelEntCoherenceRobustnessCoherence Coherence and incoherence no

TraceDistanceCoherence is a function that computes the trace distance of coherence of a quantum state $\rho$, defined as follows:

$C_{\text{tr}}(\rho) := \min_{D \in \mathcal{I}}\big\{ \|\rho - D\|_{\text{tr}}\big\},$

where $\mathcal{I}$ is the set of incoherent quantum states (i.e., the set of density matrices that are diagonal in the standard basis).

## Syntax

• TDC = TraceDistanceCoherence(RHO)
• [TDC,D] = TraceDistanceCoherence(RHO)

## Argument descriptions

### Input arguments

• RHO: A state (either pure or mixed) to have its trace distance of coherence computed.

### Output arguments

• TDC: The trace distance of coherence of RHO.
• D: A vector such that diag(D) is the closest incoherent state to RHO.

## Examples

### Maximally coherent states

The largest possible value of the trace distance of coherence on $d$-dimensional states is $2 - 2/d$, and is attained exactly by the "maximally coherent states": pure states whose entries all have the same absolute value.

>> d = 5;
>> v = ones(d,1)/sqrt(d); % this is a maximally coherent state
>> TraceDistanceCoherence(v)

ans =

1.6000

>> 2 - 2/d

ans =

1.6000

## Source code

Click on "expand" to the right to view the MATLAB source code for this function.

1. %%  TraceDistanceCoherence    Computes the trace distance of coherence of a quantum state
2. %   This function has one required argument:
3. %     RHO: a pure state vector or a density matrix
4. %
5. %   [TDC,D] = TraceDistanceCoherence(RHO) is the trace distance of
6. %   coherence of the quantum state (density matrix) RHO (i.e., it is the
7. %   minimal distance in the trace norm between RHO and a diagonal density
8. %   matrix). The optional output argument D is a vector containing the
9. %   entries of the diagonal state that is closest to RHO
10. %   (in other words, TraceNorm(RHO - diag(D)) == TDC).
11. %
12. %   URL: http://www.qetlab.com/TraceDistanceCoherence
13.
14. %   requires: cvx (http://cvxr.com/cvx/)
15. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)
16. %   package: QETLAB
17. %   last updated: January 22, 2016
18.
19. function [TDC,D] = TraceDistanceCoherence(rho)
20.
21.     % Compute the size of RHO. If it's a pure state (vector), we can
22.     % compute it quickly. If not, we have to use a semidefinite program.
23.     [m,n] = size(rho);
24.     ispure = (min(m,n) == 1);
25.
26.     % If the state is single-qubit, we can compute it faster.
27.     if(max(m,n) == 1)
28.         TDC = 0;
29.         D = 1;
30.         return
31.     elseif(m == 2 && n == 2)
32.         TDC = 2*abs(rho(1,2));
33.         D = diag(diag(rho));
34.         return
35.     else
36.         if(m == n && rank(rho) == 1)
37.             [~,~,v] = svd(rho); % v(:,1) is the pure state form of rho
38.             rho = v(:,1);
39.             ispure = 1;
40.         end
41.
42.         % Fore pure states, we can compute it much faster than via SDP.
43.         % The algorithm is somewhat complicated though, so we contain it in
44.         % the auxiliary function TraceCoherencePure.
45.         if(ispure)
46.             [TDC,D] = TraceCoherencePure(rho);
47.             return
48.         end
49.     end
50.
51.     % In general, trace distance of coherence is computed by semidefinite
52.     % programming.
53.     cvx_begin sdp quiet
54.         cvx_precision best
55.         variable d(n)
56.
57.         minimize norm_nuc(rho - diag(d))
58.
59.         subject to
60.             d >= 0;
61.             sum(d) == 1;
62.     cvx_end
63.
64.     D = d/sum(d); % divide by sum(d) for numerical reasons
65.     TDC = real(cvx_optval);
66. end
67.
68. % The following function computes the trace distance coherence of a pure
69. % state using a method that is much faster than the naive SDP.
70. function [val,D] = TraceCoherencePure(x)
71.     x = abs(x(:));
72.     [xs,ind] = sort(x,'descend'); % sort the entries of x in decending order
73.
74.     n = length(xs);
75.
76.     % Take care of a degenerate case.
77.     if(n == 1)
78.         val = 0;
79.         D = 1;
80.         return
81.     end
82.
83.     % Now loop through to try to find k. Use binary search to make this as
84.     % fast as possible.
85.     klow = 1;
86.     khigh = n;
87.
89.     % might miss it.
90.     D = zeros(n,1); D(1) = 1;
91.     D = D(perm_inv(ind));
92.     q = ((2*xs(1)^2-2) + sqrt((2*xs(1)^2-2)^2 + 4*(1 - xs(1)^2)*xs(1)^2))/(2*xs(1));
93.     val = 2*q/(xs(1) - q);
94.
95.     while 1==1 % loop until we explicitly break out
96.         k = ceil((khigh+klow)/2);
97.         s = sum(xs(1:k));
98.         m = sum(xs((k+1):n).^2);
99.         r = (s^2-1-k*m);
100.         q = (r + sqrt(r^2 + 4*k*m*s^2))/(2*k*s);
101.
102.         if(q < xs(k))
103.             % We have found a potential k!
104.             % Now compute D.
105.             D = zeros(n,1);
106.             for j = 1:k
107.                 D(j) = (xs(j) - q)/(s - k*q);
108.             end
109.
110.             D = D(perm_inv(ind));
111.
112.             % We could compute val = sum(svd(x*x' - D)), but the following
113.             % method is much quicker.
114.             val = 2*q/(s - k*q);
115.
116.             % We want to increase klow (i.e., look at the top half of
117.             % our range).
118.             klow = k;
119.         end
120.
121.         if(k == khigh)
122.             % We're done!
123.             return
124.         end
125.
126.         if(q >= xs(k))
127.             % We want to decrease khigh (i.e., look at the bottom half of
128.             % our range).
129.             khigh = k;
130.         end
131.     end
132. end